Lines of explanation

193 lines, walked from extends: and corrects: on THEORY documents. Each step explains itself; this page is the order they came in.

Grouped by tags, which every line holds in common — a line about two things is listed under both.

agency

From For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient

THEORY-026 — For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient (Active)
- THEORY-817 — Once an agent’s actions affect the percepts it receives, maximal prediction of those percepts is generally not what thermodynamic efficiency requires: in percept-action loops with feedback, maximally predictive agents generally extract less work than the most efficient agent (Proposed)

From A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape

THEORY-040 — A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape (Proposed)
- THEORY-1118 — McKenna and Coates argue, with Dryden and with Buss and Westlund, that a condition on free agency stated only in the agent’s present psychological structure is met by an agent whose structure a manipulator installed, so a mesh account must count that agent free or add a historical condition; Frankfurt, read first-hand, takes the first option (Active)

From A regulation is autonomous to the degree it is integrated with one’s other values and needs: autonomy is graded, and is not set by the order of the attitude that endorses it

THEORY-054 — A regulation is autonomous to the degree it is integrated with one’s other values and needs: autonomy is graded, and is not set by the order of the attitude that endorses it (Proposed)
- THEORY-047 — A regulation taken in under controlling conditions is introjected rather than integrated, so the social history of how a value was acquired shapes how autonomous acting on it is (Proposed)

From Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one

THEORY-067 — Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one (Proposed)
- THEORY-1117 — Ainslie proposes that self-control over time needs no faculty of will: hyperbolic discounting makes preferences that dominate at different delays, these grow into interests that bargain with no governor above them, and resolve is the recursive staking of expected future choices on the present one (Active)

From Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency

THEORY-071 — Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency (Proposed)
- THEORY-1119 — Barandiaran, Di Paolo and Rohde propose that minimal agency requires individuality, interactional asymmetry and normativity set by the system’s own organisation, each necessary and together, they say, sufficient; that individuality without the other two is not agency; and that a system optimizing an externally fixed function is not a model of agency (Active)

From Agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, and planning, autonomy, memory and coordination do not supply one

THEORY-219 — Agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, and planning, autonomy, memory and coordination do not supply one (Proposed)
- THEORY-1165 — Kahl proposes that agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, which is absent where revision is imposed from outside, escaped by rollback or chosen by a fixed higher-order rule; that allocating authority among a system’s own components coordinates execution without authoring its governing frame; and that planning, autonomy, memory and coordination do not supply that standpoint (Active)

From Self-production alone gives a system only an all-or-nothing norm, to go on or to die, and that distinction is available to an observer, not to the system; a graded norm the system itself has, by which viable states are better or worse for it, needs adaptivity, a further capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them, and sense-making and agency need adaptivity

THEORY-628 — Self-production alone gives a system only an all-or-nothing norm, to go on or to die, and that distinction is available to an observer, not to the system; a graded norm the system itself has, by which viable states are better or worse for it, needs adaptivity, a further capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them, and sense-making and agency need adaptivity (Proposed)
- THEORY-1166 — Di Paolo argues that autopoiesis as originally defined entails conservation and not active homeostasis, so it gives a system only an all-or-nothing norm, to go on or to die, which an observer and not the system can use; that a graded norm the system itself has needs adaptivity, a capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them; and that sense-making and agency need adaptivity (Active)

From Moral responsibility requires the normative competence to recognise and act on the True and the Good, which identification with one’s own motives does not supply

THEORY-122 — Moral responsibility requires the normative competence to recognise and act on the True and the Good, which identification with one’s own motives does not supply (Proposed)
- THEORY-118 — Praiseworthiness is compatible with being psychologically determined by good reasons, but blameworthiness requires that the agent could have done otherwise, that is, could have acted on the right reasons (Proposed)

From An agent can be morally responsible for what she did although she could not have done otherwise, when what made the action unavoidable played no part in bringing it about, so the principle of alternate possibilities is false

THEORY-119 — An agent can be morally responsible for what she did although she could not have done otherwise, when what made the action unavoidable played no part in bringing it about, so the principle of alternate possibilities is false (Proposed)
- THEORY-121 — Moral responsibility requires guidance control, moderate reasons-responsiveness of the agent’s own actual-sequence mechanism, and not regulative control over alternative possibilities, so responsibility is compatible with causal determinism (Active)

From Orseau, McGill and Legg cast whether an identified system is an agent or a device as Bayesian model comparison between descriptions of the function it optimises and descriptions of its input-output mapping, so that the verdict is relative to the observer’s prior; on seven toy gridworld trajectories goal-seeking comes out agent-like and wall-following device-like, and they argue that under a speed prior everything may look like an agent

THEORY-624 — Orseau, McGill and Legg cast whether an identified system is an agent or a device as Bayesian model comparison between descriptions of the function it optimises and descriptions of its input-output mapping, so that the verdict is relative to the observer’s prior; on seven toy gridworld trajectories goal-seeking comes out agent-like and wall-following device-like, and they argue that under a speed prior everything may look like an agent (Active)
- THEORY-1236 — Agency is relative to an observer: an identified system is an agent, for that observer, to the degree that a description of the function it optimises explains its behaviour better under the observer’s prior than any description of its input-output mapping (Proposed)

From If agents are the systems that would adapt their policy were their actions to affect the world differently, then in mechanised causal games meeting Kenton et al.’s five assumptions (one agent per connected component of the decision–utility graph, best response to every mechanism intervention, a fixed tie-breaking order, terminal edges only into decision mechanisms, and mechanism interventions able to set any deterministic function) the game graph, with its decisions, utilities and agents, is recovered exactly from interventions on mechanisms

THEORY-527 — If agents are the systems that would adapt their policy were their actions to affect the world differently, then in mechanised causal games meeting Kenton et al.’s five assumptions (one agent per connected component of the decision–utility graph, best response to every mechanism intervention, a fixed tie-breaking order, terminal edges only into decision mechanisms, and mechanism interventions able to set any deterministic function) the game graph, with its decisions, utilities and agents, is recovered exactly from interventions on mechanisms (Active)
- THEORY-801 — Agents are the systems that would adapt their policy were their actions to affect the world differently, so the agent is wherever that adaptation is: a thermostat is not an agent while it together with its design process is, an RL training process is while its learnt policy generally is not, and humans are (Proposed)

anthology-candidate

From In a linear-in-parameters model with Gaussian noise and a Gaussian prior, the evidence is exactly the best-fit likelihood times an Occam factor set by the posterior curvature, and when the design grows linearly with n that factor costs (k/2) log n

THEORY-718 — In a linear-in-parameters model with Gaussian noise and a Gaussian prior, the evidence is exactly the best-fit likelihood times an Occam factor set by the posterior curvature, and when the design grows linearly with n that factor costs (k/2) log n (Active)
- THEORY-087 — The Bayesian complexity penalty grows as (d/2) log n in a regular model, where the curvature at the optimum sets the Occam factor, and as λ log n in a singular one, where λ is the real log canonical threshold; with the prior positive on the optimal set, λ falls below d/2 when that set is infinite or has a point of degenerate curvature, and equals d/2 when it is finitely many points of non-degenerate curvature (Active)
  - THEORY-080 — In a linearised network with a Gaussian prior on its parameters, the Occam factor is a sum over the NTK’s eigenvalues, so on spherical data each harmonic pays ½ log(1 + βnµ_k/α) and the kernel’s eigenvalue decay schedules the complexity penalty across frequencies (Proposed)

From In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet

THEORY-083 — In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet (Active)
- THEORY-788 — Training a deep classifier past zero training error generally collapses its last-layer features onto their class means, drives the centred means toward a simplex equiangular tight frame aligned with the classifier, and makes the decision nearest class mean, as Papyan, Han and Donoho propose from their runs (Proposed)

THEORY-115 — Linear connectivity from a shared, already-stable start holds only when the objectives the runs optimise share structure, and breaks as that structure is removed (Proposed)
- THEORY-105 — For RoBERTa-base fine-tuned on English classification data, with loss measured by a linear probe refitted on the frozen encoder, points sampled from the convex hull of models fine-tuned on one dataset, one task family or twelve GLUE and SuperGLUE datasets lost no more than any model fine-tuned elsewhere and beat the group’s own members in 88% to 96.7% of comparisons, and the centroid of the general-level models was a better starting point for BitFit than the pretrained model (Active)
  - THEORY-828 — Fine-tuning from one pretrained model typically lands near the edge of a task-specific low-loss region in weight space that contains the convex hull of the fine-tuned models, and averaging their weights, as model soups and stochastic weight averaging do, works by moving inward (Proposed)

From Two ε-dropout-stable solutions of a multilayer ReLU net, under a loss convex in the output, are joined by a piecewise-linear path whose loss never exceeds the worse endpoint’s by more than ε, and joined pairwise paths compose with no higher barrier than their worst

THEORY-736 — Two ε-dropout-stable solutions of a multilayer ReLU net, under a loss convex in the output, are joined by a piecewise-linear path whose loss never exceeds the worse endpoint’s by more than ε, and joined pairwise paths compose with no higher barrier than their worst (Active)
- THEORY-109 — The solutions SGD finds in a deep network lie in one connected low-loss set, joined by simple curves and spanning volumes of measurable dimension, and this is a property of the solutions training finds, not of every minimum (Proposed)
  - THEORY-112 — Aligning independently trained solutions over the architecture’s symmetries removed the linear barrier between them for MNIST MLPs and 32×-wide ResNet-20s on CIFAR-10, two-thirds of it for an ImageNet ResNet-50, and not all of it for VGG-16 or ResNet-20 at 1× width, the barrier after weight matching falling with width; for 6- to 8-layer ViTs and GPT-2s, permutations alone removed 63–90% of it, and adding an orthogonal map on the residual stream, learned against the loss at the path’s midpoint, took it to zero or near zero except on BookCorpus (0.42) (Active)
    - THEORY-110 — A set of trained solutions has a centre that is linearly connected, after alignment, to the rest of them: the solution set is a star domain even where it is not convex (Proposed) — also carries further THEORY-109, THEORY-741
    - THEORY-1188 — Once the architecture’s symmetries are factored out, most of the linear barrier between independently trained solutions disappears; the group that matters is the full one, permutations for MLPs and CNNs but for transformers also an orthogonal map on the residual stream, without which a barrier remains; and the barrier left after alignment falls with width (Proposed)
  - THEORY-114 — In image classification, linear mode connectivity appeared during training and not at initialization: copies of ResNet-20 and VGG-16 on CIFAR-10 and of ResNet-50 and Inception-v3 on ImageNet, trained from a shared state, became stable to SGD noise in test error only after 1.5–20% of training (LeNet on MNIST was stable from the start), and between independently initialized MLPs on MNIST and CIFAR-10 the barrier after weight-matching alignment was high at initialization and fell gradually through training (Active)
    - THEORY-1172 — Linear mode connectivity emerges during training, not at initialization: copies trained from a shared state become stable to SGD noise after 1.5–20% of training, and independently trained networks become linearly connected after alignment only gradually (Proposed)
  - THEORY-557 — Low-loss paths between independently trained solutions are found outside supervised image classifiers: graph neural networks trained from different seeds on one graph are joined by quadratic Bezier curves where the straight line has a barrier, and one pair each of diffusion and contrastive models by chains of low-loss checkpoints (Proposed)
THEORY-741 — A two-layer ReLU student of width at least kM, fitting an orthonormal teacher of M neurons on the sphere, has for any k global minima a global minimum joined to each by a two-piece linear path of minima; and in a deep linear network of depth L and width above 1 + r(L − 1), any r global minima in general position share a global minimum linearly connected to each (Active)

From Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble’s accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model

THEORY-113 — Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble’s accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model (Active)
- THEORY-1185 — Merging models trained on different tasks from different initialisations fails where their features have no counterpart in the other model, and such features grow more common with depth (Proposed)

THEORY-117 — In Uselis, Dittadi and Oh’s two-concept grids, ResNet-50s trained from scratch, with checkpoints chosen on the unseen pairs, reach unseen combinations as the share of combinations seen grows and not as up to four times more data of the same combinations is added; and when embeddings are exactly linearly factored, seen pairs that connect every value of both concepts, as two per value arranged in one cycle do, determine the embedding of every unseen pair (Active)
- THEORY-794 — In vision models, compositional generalization to unseen combinations of concepts is driven by the diversity of the combinations seen in training rather than by data volume, and arrives with representations that are linearly factored by concept, as Uselis, Dittadi and Oh propose (Proposed)

From Cosine similarities to matched anchor points are unchanged by exactly the maps that preserve each sample’s cosines to the anchors, among them every orthogonal map times a positive scalar and every per-sample positive rescaling but not translations; in Moschella et al.’s experiments independently trained encoders agreed in them only approximately, enough for a decoder reading them to transfer untrained between seeds, architectures and languages with loss, while their nearest neighbours mostly differed

THEORY-175 — Cosine similarities to matched anchor points are unchanged by exactly the maps that preserve each sample’s cosines to the anchors, among them every orthogonal map times a positive scalar and every per-sample positive rescaling but not translations; in Moschella et al.’s experiments independently trained encoders agreed in them only approximately, enough for a decoder reading them to transfer untrained between seeds, architectures and languages with loss, while their nearest neighbours mostly differed (Active)
- THEORY-798 — Independently trained neural encoders of the same data learn latent spaces that differ mainly by an angle-preserving map, a rotation or reflection times a rescaling, so their cosine similarities to shared anchor samples agree across seeds, architectures and languages, as Moschella et al. propose (Proposed)

From In a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set fixes the generalising representation of addition when its same-answer pairs leave only translation and scale free; for non-modular addition with p = 10 and one-dimensional embeddings, generalisation sets in near the training fraction, about 0.4, at which that becomes likely

THEORY-181 — In a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set fixes the generalising representation of addition when its same-answer pairs leave only translation and scale free; for non-modular addition with p = 10 and one-dimensional embeddings, generalisation sets in near the training fraction, about 0.4, at which that becomes likely (Active)
- THEORY-782 — In networks that learn addition from summed embeddings, the critical training fraction of grokking is the fraction at which the training set’s same-answer pairs come to fix the generalising representation up to translation and scale, at every group size, as Liu, Kitouni, Nolte, Michaud, Tegmark and Williams propose (Proposed)

From In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²))

THEORY-779 — In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²)) (Active)
- THEORY-209 — A converged network trained with SGD is not at a point and does not sample its training loss: under a quadratic approximation it samples a Gibbs distribution on a modified loss, isotropic in the Hessian’s column space, with probability currents that keep it oscillating, so its speed and spread are set by the learning rate, momentum and batch size (Proposed)

From In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise

THEORY-211 — In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise (Active)
- THEORY-790 — In deep networks trained by SGD on real data, minibatch noise drives parameter groups related by a rescaling symmetry toward the point where their gradient noise balances, at a rate set by learning rate over batch size, so that at stationarity they sit neither where gradient flow would leave them nor spread out as under isotropic noise, as Ziyin, Li and Ueda propose (Proposed)

From Every mirror symmetry of a network’s per-sample loss makes its symmetric subspace invariant under gradient descent and SGD, and when the loss’s Hessian is bounded below by −c, weight decay γ‖θ‖² with γ > c/2 puts every minimum in that subspace; rescaling symmetries that hold at λ = −1, rotation symmetries and permutation symmetries contain such mirrors, whose subspaces are zeroed weights, reduced rank and identical neurons

THEORY-213 — Every mirror symmetry of a network’s per-sample loss makes its symmetric subspace invariant under gradient descent and SGD, and when the loss’s Hessian is bounded below by −c, weight decay γ‖θ‖² with γ > c/2 puts every minimum in that subspace; rescaling symmetries that hold at λ = −1, rotation symmetries and permutation symmetries contain such mirrors, whose subspaces are zeroed weights, reduced rank and identical neurons (Active)
- THEORY-781 — Strong gradient noise in SGD, like weight decay above a threshold, makes the subspace fixed by a mirror symmetry of a network’s loss attracting even where it is a saddle, so that networks trained with SGD at large learning rates or small batches reach sparse, low-rank and identical-neuron solutions, as Ziyin proposes (Proposed)

From In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary

THEORY-421 — In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary (Active)
- THEORY-825 — Hyperbolic graph embedding models trained on task losses that never mention the hierarchy do not place a tree’s root nearest the origin, and their distance from the origin orders hierarchical levels only partly (Proposed)

behavioral-integration

From Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one

THEORY-067 — Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one (Proposed)
- THEORY-1117 — Ainslie proposes that self-control over time needs no faculty of will: hyperbolic discounting makes preferences that dominate at different delays, these grow into interests that bargain with no governor above them, and resolve is the recursive staking of expected future choices on the present one (Active)

From Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning

THEORY-069 — Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning (Proposed)
- THEORY-1227 — Haken, Kelso and Bunz model the switch of rhythmic bimanual coordination from anti-phase to in-phase at a critical frequency as the loss of one stable state of an order parameter, relative phase, derived from two coupled oscillators; Kelso, Scholz and Schöner report the variability of relative phase rising before the switch in the anti-phase pattern only, in a few subjects and with no statistical test (Active)

From How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control

THEORY-070 — How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control (Proposed)
- THEORY-1110 — Shenhav, Botvinick and Cohen propose that control is allocated top-down by a dedicated specifying system: dACC computes the expected value of a control signal, its probability-weighted payoff less an intrinsic cost rising with intensity, and specifies the signal that lateral prefrontal and subcortical structures implement (Active)

From Between tests, uncorrected for publication bias, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); tests in which the plan was rehearsed had larger effects (.45 against .33, with no gain beyond one rehearsal), tests in which it was recorded had smaller ones (.31 against .44), and adding how or for how long to a time and place went with about half the effect (.24 against .46)

THEORY-908 — Between tests, uncorrected for publication bias, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); tests in which the plan was rehearsed had larger effects (.45 against .33, with no gain beyond one rehearsal), tests in which it was recorded had smaller ones (.31 against .44), and adding how or for how long to a time and place went with about half the effect (.24 against .46) (Active)
- THEORY-1175 — Between tests, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); rehearsing the plan once raised the effect, keeping a copy to revisit lowered it (.31 against .44), and adding how or for how long to a time-and-place cue roughly halved it (Proposed)

From Gollwitzer and Sheeran (2006) conclude, narratively from a handful of studies that measured or manipulated the goal intention, that implementation intentions affect performance only when the goal intention they serve is strong and activated, so the planned response is not elicited mechanically by its cue; in 2024, across eight tests, plan effects were larger with strong, activated or self-concordant goal intentions than with weak, unactivated or self-discordant ones (d = .79 for the difference), and in 31 physical-activity tests split at the median of intention, planning helped with strong intentions (d = .31) and had no effect with weak ones (d ≈ 0)

THEORY-904 — Gollwitzer and Sheeran (2006) conclude, narratively from a handful of studies that measured or manipulated the goal intention, that implementation intentions affect performance only when the goal intention they serve is strong and activated, so the planned response is not elicited mechanically by its cue; in 2024, across eight tests, plan effects were larger with strong, activated or self-concordant goal intentions than with weak, unactivated or self-discordant ones (d = .79 for the difference), and in 31 physical-activity tests split at the median of intention, planning helped with strong intentions (d = .31) and had no effect with weak ones (d ≈ 0) (Active)
- THEORY-1176 — Implementation intentions affect performance only when the goal intention they serve is strong and activated, so the cue-triggered response is gated by the goal and does not fire on the cue alone (Proposed)

THEORY-918 — Across 642 experimental tests to September 2019, forming implementation intentions improved outcomes, with uncorrected effects on behaviour (d = .27), cognitive performance (.47), affect regulation (.66) and social judgement (.52), but publication bias was substantial: the pooled d = .36 falls to a model-averaged d = .15 under robust Bayesian meta-analysis, which still found extreme evidence of an effect (BF10 = 319), with large heterogeneity (Active)
- THEORY-1192 — Across 642 experimental tests to September 2019, forming implementation intentions reliably improved behaviour, cognitive performance, affect regulation and social judgement, but publication bias was substantial: the pooled d = .36 falls to a model-averaged d = .15 under robust Bayesian meta-analysis, with large heterogeneity (Proposed)

causality

From On Pythia language models from 410M to 12B parameters, in three single prompt pairs and two small prompt distributions, verifying attention or neuron nodes by activation patching in the order given by attribution patching, a first-order gradient estimate of each node’s noising effect, found the nodes with the largest effects at about 2 to 50 times the cost of an oracle ordering, below unordered patching, subsampling and block search in every setting; its misses were concentrated in queries and keys behind saturated attention and in nodes whose direct and indirect effects nearly cancel

THEORY-996 — On Pythia language models from 410M to 12B parameters, in three single prompt pairs and two small prompt distributions, verifying attention or neuron nodes by activation patching in the order given by attribution patching, a first-order gradient estimate of each node’s noising effect, found the nodes with the largest effects at about 2 to 50 times the cost of an oracle ordering, below unordered patching, subsampling and block search in every setting; its misses were concentrated in queries and keys behind saturated attention and in nodes whose direct and indirect effects nearly cancel (Active)
- THEORY-1000 — For fine-grained nodes of transformer language models generally, attribution patching corrected for attention saturation is a reliable prefilter for activation patching: nodes it ranks low rarely have large single-node patching effects, so verifying nodes in its order finds the important ones at a small multiple of the cost of an oracle, as Kramár et al. expect (Proposed)

From On four subject–verb agreement tasks in Pythia-70M and Gemma-2-2B, circuits of sparse-autoencoder features and autoencoder error nodes, selected by linear estimates of indirect effect, carried half or more of the behaviour on held-out pairs with about 100 and 500 nodes, where circuits of neurons needed about 1,500 and more than 30,000; with the error nodes removed, the same feature circuits carried little of it, under about a quarter in Pythia and almost none in Gemma

THEORY-1007 — On four subject–verb agreement tasks in Pythia-70M and Gemma-2-2B, circuits of sparse-autoencoder features and autoencoder error nodes, selected by linear estimates of indirect effect, carried half or more of the behaviour on held-out pairs with about 100 and 500 nodes, where circuits of neurons needed about 1,500 and more than 30,000; with the error nodes removed, the same feature circuits carried little of it, under about a quarter in Pythia and almost none in Gemma (Active)
- THEORY-1014 — In a language model, a behaviour that looks like one mechanism, whether picked out by one contrast or by one cluster of next-token predictions, is often carried by several distinct pathways of sparse features that converge on the output without fully intersecting, as in Pythia-70M’s agreement across a relative clause, its prediction of “to” before an infinitival object, and its incrementing of sequences (Proposed)

From Attribution patching estimates the effect of patching an activation from a clean run into a corrupted run as the difference between the two activations dotted with the gradient of the metric on the corrupted run, so the estimates for every activation come from two forward passes and one backward pass and add over sub-components; the estimate is exact when the metric is affine in the activation, gives no effect for a nonlinearity in saturation whatever the true change, and through LayerNorm ignores the component of a patch along the vector being normalized

THEORY-1023 — Attribution patching estimates the effect of patching an activation from a clean run into a corrupted run as the difference between the two activations dotted with the gradient of the metric on the corrupted run, so the estimates for every activation come from two forward passes and one backward pass and add over sub-components; the estimate is exact when the metric is affine in the activation, gives no effect for a nonlinearity in saturation whatever the true change, and through LayerNorm ignores the component of a patch along the vector being normalized (Active)
- THEORY-1021 — In transformer language models, attribution patching agrees well with activation patching for small patches of single heads and their queries, keys, values and attention patterns, for nodes late in a circuit relative to the metric, and for circuits that route information through attention, and agrees badly for large patches such as whole residual streams, for the first MLP layer and the early layers of deep circuits, where nonlinearities move out of saturation, and where an effect needs several features at once, as Nanda proposes (Proposed)

From In GPT-J, for 652 CounterFact facts the model knows, the per-fact causal-tracing effect at an MLP layer adds at most 3.2 points of explained variance in the success of editing that fact at that layer, beyond what the choice of layer alone explains, across ROME, MEMIT and constrained fine-tuning and five editing objectives; ROME rewrites the fact at over 96% at every layer but the last, including layers where tracing does not place it, and the only reliable link appears when the edit is made on the noised input that tracing uses

THEORY-1024 — In GPT-J, for 652 CounterFact facts the model knows, the per-fact causal-tracing effect at an MLP layer adds at most 3.2 points of explained variance in the success of editing that fact at that layer, beyond what the choice of layer alone explains, across ROME, MEMIT and constrained fine-tuning and five editing objectives; ROME rewrites the fact at over 96% at every layer but the last, including layers where tracing does not place it, and the only reliable link appears when the edit is made on the noised input that tracing uses (Active)
- THEORY-1026 — In transformer language models, locating where a forward pass carries information about a fact and finding where a weight edit best changes the expressed fact are different questions, because an edit at one layer can override information accumulated across other layers, so that many layers could hold a fact and some happen to, as Hase, Bansal, Kim and Ghandeharioun propose (Proposed)

From With uniform interventions imposed separately at each level, a coarse-grained macro description of a known discrete mechanism can have more effective information than the micro description it supervenes on, so the level at which effective information is highest can be a macro level

THEORY-222 — With uniform interventions imposed separately at each level, a coarse-grained macro description of a known discrete mechanism can have more effective information than the micro description it supervenes on, so the level at which effective information is highest can be a macro level (Active)
- THEORY-787 — A system’s causal structure can be most effective at a macro scale, so that causation emerges at a coarse-grained level and the macro description is the one at which the system’s causes are found, as Hoel, Albantakis and Tononi propose from effective information (Rejected)
  - THEORY-1167 — When one intervention distribution over micro states is used at both levels, the effective information of a coarse-graining of a discrete mechanism, with each macro intervention realized by that distribution within its block, cannot exceed the micro effective information; so a positive causal emergence of effective information, scored with a uniform distribution at each level, equals the micro gain from changing the distribution minus a non-negative loss, and is a gain from the change of distribution (Active) — also carries further THEORY-222
    - THEORY-1169 — For a discrete mechanism scored with one full-support intervention distribution over micro causes at both levels, a coarse-graining of cause and effect keeps all of the micro effective information if and only if it merges only causes with identical rows P(Y | do(x)) and only effects with proportional columns, whatever the distribution; so the coarsest coarse-graining that ties the micro is the causal partition of causal feature learning, with effects grouped by normalized likelihood, and any merge of causes with different effects loses information under every such distribution (Active)

THEORY-889 — In Acharya, Blackwell and Sen’s analysis, white Southerners surveyed in 2006 to 2011 who live in counties where a larger share of the 1860 population was enslaved are, with state fixed effects and 1860 county controls, somewhat less Democratic, less supportive of affirmative action and more racially resentful, by about 0.09 to 0.13 standard deviations per standard deviation (about 20 points) of slave share, and in 1984 to 1998 survey data colder toward black Americans, by about 0.32; a cotton-suitability instrument gives larger estimates; the direct effect estimated with present black share held fixed is about as large; and in county presidential vote, which counts black voters until disenfranchisement, the difference appeared after Reconstruction rather than before the war (Active)
- THEORY-1225 — Among white Southerners today, those living in counties where a larger share of the 1860 population was enslaved are somewhat less Democratic, less supportive of affirmative action and more racially resentful, by about a tenth of a standard deviation per standard deviation of slave share, and colder toward black Americans, by about a third; the difference is not accounted for by the present black share of the population, and in voting it appeared after Reconstruction rather than before the war, which supports its origin in the incentives that emancipation gave planters to repress black labor and votes, and its local transmission thereafter (Proposed)

THEORY-243 — In the three-unit identity network x ↦ x(1, 0, 1) ↦ (0, 2, 1)·h, aligning the input with the third unit and aligning it with the direction (h₁ + h₂)/√2, two alignments related by an orthogonal change of basis, pass every interchange intervention on the input equally, so interchange interventions on that variable alone do not fix which internal direction mediates it (Active)
- THEORY-802 — In trained networks as in constructed ones, interchange interventions on a fixed set of variables can be satisfied equally by alignments related by a change of basis, so which internal direction mediates a high-level variable is fixed only relative to a chosen decomposition into units, and complex models will generally have several distinct, fully correct abstractions (Proposed)

From If agents are the systems that would adapt their policy were their actions to affect the world differently, then in mechanised causal games meeting Kenton et al.’s five assumptions (one agent per connected component of the decision–utility graph, best response to every mechanism intervention, a fixed tie-breaking order, terminal edges only into decision mechanisms, and mechanism interventions able to set any deterministic function) the game graph, with its decisions, utilities and agents, is recovered exactly from interventions on mechanisms

THEORY-527 — If agents are the systems that would adapt their policy were their actions to affect the world differently, then in mechanised causal games meeting Kenton et al.’s five assumptions (one agent per connected component of the decision–utility graph, best response to every mechanism intervention, a fixed tie-breaking order, terminal edges only into decision mechanisms, and mechanism interventions able to set any deterministic function) the game graph, with its decisions, utilities and agents, is recovered exactly from interventions on mechanisms (Active)
- THEORY-801 — Agents are the systems that would adapt their policy were their actions to affect the world differently, so the agent is wherever that adaptation is: a thermostat is not an agent while it together with its design process is, an RL training process is while its learnt policy generally is not, and humans are (Proposed)

THEORY-674 — In Garg and Fetzer’s GPT-4o-mini extraction from NBER and CEPR working papers, edges kept in at least 4 of 9 passes and never checked against human labels, the mean per-paper share of edges labelled as backed by difference-in-differences, instrumental variables, an RCT, regression discontinuity or synthetic control rose from 7.7% in 1990 to 31.7% in 2020, at every overlap threshold; the paper’s per-decade slope fell from +0.0187 to +0.0011 once method mix and graph volume were controlled (Active)
- THEORY-824 — Economics working papers increasingly back their claims with design-based identification: in NBER and CEPR working papers the share of claims supported by difference-in-differences, instrumental variables, an RCT, regression discontinuity or synthetic control rose about fourfold from 1990 to 2020 (Proposed)

cognition

From How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control

THEORY-070 — How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control (Proposed)
- THEORY-1110 — Shenhav, Botvinick and Cohen propose that control is allocated top-down by a dedicated specifying system: dACC computes the expected value of a control signal, its probability-weighted payoff less an intrinsic cost rising with intensity, and specifies the signal that lateral prefrontal and subcortical structures implement (Active)

From In Claude 3.5 Haiku, writing the second line of a rhyming couplet whose first line ends “grab it”, features for the candidate rhymes “rabbit” and “habit” are active on the newline token before the line, steering them changes the line’s final word only when applied at that token, and suppressing “rabbit” turns the line towards “habit”; injecting a “rabbit” or “green” word feature at that point in 25 other poems ended the line with the injected word in 70% of cases, and such planned-word features were found in about half of the poems the authors examined

THEORY-1017 — In Claude 3.5 Haiku, writing the second line of a rhyming couplet whose first line ends “grab it”, features for the candidate rhymes “rabbit” and “habit” are active on the newline token before the line, steering them changes the line’s final word only when applied at that token, and suppressing “rabbit” turns the line towards “habit”; injecting a “rabbit” or “green” word feature at that point in 25 other poems ended the line with the injected word in 70% of cases, and such planned-word features were found in about half of the poems the authors examined (Active)
- THEORY-982 — Claude 3.5 Haiku routinely computes intermediate steps within a single forward pass and plans its outputs ahead, representing a target before the text that leads to it and writing towards it, as Lindsey et al. propose from their case studies (Proposed) — also carries further THEORY-984
THEORY-984 — In Claude 3.5 Haiku, on the prompt “Fact: the capital of the state containing Dallas is”, which it completes with “Austin”, an attribution graph shows Dallas features activating Texas features that, with “say a capital” features, promote Austin, beside a direct edge from Dallas to “say Austin”; inhibiting the Dallas features lowers the Texas features and leaves “say a capital” largely intact, and replacing the Texas features with those of California, Georgia, British Columbia, China or the Byzantine Empire makes the model name that place’s capital (Active)

From In Vishne, Gerber, Knight and Deouell’s intracranial recordings from ten epilepsy patients viewing clearly visible images for up to 1.5 s, the pattern of high-frequency activity across ventral temporal cortex carried image category, and exemplar identity among 60 images, in a time-invariant code for as long as the image stayed on while amplitude fell about 80%; at the sampled prefrontal and parietal sites, category content was a burst about 150 to 600 ms after onset that did not track duration

THEORY-102 — In Vishne, Gerber, Knight and Deouell’s intracranial recordings from ten epilepsy patients viewing clearly visible images for up to 1.5 s, the pattern of high-frequency activity across ventral temporal cortex carried image category, and exemplar identity among 60 images, in a time-invariant code for as long as the image stayed on while amplitude fell about 80%; at the sampled prefrontal and parietal sites, category content was a burst about 150 to 600 ms after onset that did not track duration (Active)
- THEORY-830 — In human visual cortex the content of a sustained seen image is carried by a stable population code that lasts as long as the image, not by a sustained level of activity, and frontoparietal cortex marks only its onset; so a sustained experience rests on sensory representation and its discrete updates on frontoparietal representation (Proposed)

From Simon proposes that ill structured problems can be handled by an alternation between a solver working on a problem that is well structured at each moment and a recognition and retrieval system that keeps modifying the problem space from memory or the environment, that a serial solver gains nothing by assembling the problem space at the outset, and that nothing but the size of the knowledge base may separate ill from well structured problems, so that no qualitatively new mechanisms are needed

THEORY-1084 — Simon proposes that ill structured problems can be handled by an alternation between a solver working on a problem that is well structured at each moment and a recognition and retrieval system that keeps modifying the problem space from memory or the environment, that a serial solver gains nothing by assembling the problem space at the outset, and that nothing but the size of the knowledge base may separate ill from well structured problems, so that no qualitatively new mechanisms are needed (Active)
- THEORY-1216 — Ill structured problems can be handled by an alternation between a solver working on a problem that is well structured at each moment and a recognition and retrieval system that keeps modifying the problem space from memory or the environment, a serial solver gains nothing by assembling the problem space at the outset, and nothing but the size of the knowledge base may separate ill from well structured problems, so that no qualitatively new mechanisms are needed (Proposed)

From Between tests, uncorrected for publication bias, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); tests in which the plan was rehearsed had larger effects (.45 against .33, with no gain beyond one rehearsal), tests in which it was recorded had smaller ones (.31 against .44), and adding how or for how long to a time and place went with about half the effect (.24 against .46)

THEORY-908 — Between tests, uncorrected for publication bias, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); tests in which the plan was rehearsed had larger effects (.45 against .33, with no gain beyond one rehearsal), tests in which it was recorded had smaller ones (.31 against .44), and adding how or for how long to a time and place went with about half the effect (.24 against .46) (Active)
- THEORY-1175 — Between tests, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); rehearsing the plan once raised the effect, keeping a copy to revisit lowered it (.31 against .44), and adding how or for how long to a time-and-place cue roughly halved it (Proposed)

From If belief is individuated by functional role, an external store that is constantly consulted, directly available and automatically endorsed can carry a standing belief, so the material vehicles of some mental states extend beyond the skin; the step from coupling to parthood rests on that premise and a parity principle the authors state rather than argue

THEORY-579 — If belief is individuated by functional role, an external store that is constantly consulted, directly available and automatically endorsed can carry a standing belief, so the material vehicles of some mental states extend beyond the skin; the step from coupling to parthood rests on that premise and a parity principle the authors state rather than argue (Proposed)
- THEORY-1194 — Clark and Chalmers argue that if belief is individuated by functional role, an external store that is constantly consulted, directly available and automatically endorsed can carry a standing belief, so that the material vehicles of some mental states extend beyond the skin; the step from coupling to parthood rests on that premise and on a parity principle they state rather than argue (Active)
- THEORY-512 — The extended-mind thesis worth disputing is that external entities partly constitute a subject’s mental states in virtue of the subject’s sensorimotor interaction with them; so stated it can reach standing belief but not consciousness, if consciousness needs information directly available for global control, since perception makes availability indirect (Proposed)

From In Posani et al.’s analysis of the IBL Brainwide Map, recorded while trained mice performed one visual decision task, the eight-variable selectivity profiles of neurons within one cortical area clustered beyond a covariance-matched Gaussian null only in VISp, AUDp and SSp-ul of about 20 areas tested, while profiles pooled across areas clustered along anatomical lines; and over the task conditions each area distinguishes, at least 95% of random balanced dichotomies were linearly decodable above a label-shuffled null in 15 of 16 areas, a lenient criterion

THEORY-124 — In Posani et al.’s analysis of the IBL Brainwide Map, recorded while trained mice performed one visual decision task, the eight-variable selectivity profiles of neurons within one cortical area clustered beyond a covariance-matched Gaussian null only in VISp, AUDp and SSp-ul of about 20 areas tested, while profiles pooled across areas clustered along anatomical lines; and over the task conditions each area distinguishes, at least 95% of random balanced dichotomies were linearly decodable above a label-shuffled null in 15 of 16 areas, a lenient criterion (Active)
- THEORY-1196 — During a visual decision task, neurons within a single area of mouse cortex rarely form discrete functional types, yet the task conditions the area distinguishes are linearly separable in nearly every way (Proposed)

From In Claude Sonnet 4.5, Opus 4.5 and Opus 4.6, Gurnee, Sofroniew et al. found that projecting out the ten most active J-space directions in layers 38 to 54 of 100 lowered an experiential-language score, graded by Opus 4.6-based model graders on five or six prompts per condition, both in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions left it near baseline and the text stayed fluent; on Haiku 4.5 coherence failed first

THEORY-869 — In Claude Sonnet 4.5, Opus 4.5 and Opus 4.6, Gurnee, Sofroniew et al. found that projecting out the ten most active J-space directions in layers 38 to 54 of 100 lowered an experiential-language score, graded by Opus 4.6-based model graders on five or six prompts per condition, both in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions left it near baseline and the text stayed fluent; on Haiku 4.5 coherence failed first (Active)
- THEORY-1224 — In Claude language models, projecting out the most active J-space directions in early workspace layers lowers the rate of felt, first-person, sensory language in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions do not and the text stays coherent; so in these models the experiential register of report depends on the access-like workspace and is not specific to self-report (Proposed)

THEORY-865 — In Gurnee, Sofroniew et al.’s experiments on Claude Sonnet 4.5, Haiku 4.5, Opus 4.5 and Opus 4.6, the residual-stream directions a corpus-averaged Jacobian lens marks as verbalizable (the J-space) carried most of the causal effect on verbal report and on unspoken two-hop intermediates while holding a small share of their variance (swaps along them succeeding on 59% and 61% of trials, against 5% and 28% for the remainder, which fell to 0% and 6% with the J-space clamped), held instructed concepts, and passed one swapped argument to several functions (76 of 192 swaps); in two small task sets (n = 8 and 11), report and flexible inference followed a swap while continuation, anomaly detection and line wrapping did not, a split the authors partly define by this result; and by the lens’s own statistics the J-space works in a middle band of layers, holds about 25 active directions and under 10% of activation variance, and is amplified and relayed by the weights more than other directions (Active)
- THEORY-1226 — In Claude language models, a sparse set of residual-stream directions that a corpus-averaged Jacobian lens identifies as verbalizable (the J-space) is causally privileged for verbal report, for holding instructed concepts, for unspoken intermediate steps of reasoning and for passing one argument to many functions, while continuation, anomaly detection and line wrapping use the same information without routing through it; it carries workspace-like content only in a middle band of layers, holds on the order of tens of active directions and under a tenth of activation variance, and is amplified and relayed by the weights more than other directions (Proposed)

From If commonsense knowledge includes defaults that later evidence displaces, a monotonic logic, in which an added axiom can never remove a theorem, cannot represent that displacement as it stands; Minsky’s further claim, that no logistic system, one that separates propositions from general laws of inference, can carry commonsense knowledge even once repaired, is asserted, not shown

THEORY-462 — If commonsense knowledge includes defaults that later evidence displaces, a monotonic logic, in which an added axiom can never remove a theorem, cannot represent that displacement as it stands; Minsky’s further claim, that no logistic system, one that separates propositions from general laws of inference, can carry commonsense knowledge even once repaired, is asserted, not shown (Active)
- THEORY-1239 — No logistic system, one that separates propositions from general laws of inference, can carry commonsense knowledge, whose defaults later evidence displaces: attempts to repair its monotonicity will continue to fail (Proposed)

From A language model idealized as continuing any text with a plausible next word from the collection of all plausible texts follows narrative coherence and not truth, so its false statements are confabulations within the story the context sets up; whether trained and aligned chat models keep that nature is an open empirical question the idealization does not settle

THEORY-547 — A language model idealized as continuing any text with a plausible next word from the collection of all plausible texts follows narrative coherence and not truth, so its false statements are confabulations within the story the context sets up; whether trained and aligned chat models keep that nature is an open empirical question the idealization does not settle (Active)
- THEORY-1240 — Trained, instruction-tuned and preference-tuned chat models keep the nature of the idealized fiction machine: what they assert follows the narrative coherence of the frame, role or story the conversation sets up, not a representation of truth that stays fixed across frames (Proposed)

From An ideal observer is optimal relative to the information it is given, so a constrained ideal is the optimal decoder of a degraded input, not a restricted decoder of the full one; comparing real observers with ideals taken at successive stages of the visual system places losses by stage, and in early-vision tasks human performance is reported often to parallel the retinal-stage ideal across conditions while falling well below it

THEORY-336 — An ideal observer is optimal relative to the information it is given, so a constrained ideal is the optimal decoder of a degraded input, not a restricted decoder of the full one; comparing real observers with ideals taken at successive stages of the visual system places losses by stage, and in early-vision tasks human performance is reported often to parallel the retinal-stage ideal across conditions while falling well below it (Active)
- THEORY-1242 — In early-vision tasks, human performance varies across conditions as the information available at the retinal output does, while falling well below the retinal-stage ideal, so substantial losses lie in central factors (Proposed)

From In Zaslavsky, Kemp, Regier and Tishby’s analysis, the colour-naming systems of the World Color Survey languages and English lie near the information-bottleneck bound for compressing perceptual meanings into words, at a trade-off parameter just above 1 and nearer than hue-rotated variants of themselves, with the need distribution over meanings estimated from the naming data

THEORY-163 — In Zaslavsky, Kemp, Regier and Tishby’s analysis, the colour-naming systems of the World Color Survey languages and English lie near the information-bottleneck bound for compressing perceptual meanings into words, at a trade-off parameter just above 1 and nearer than hue-rotated variants of themselves, with the need distribution over meanings estimated from the naming data (Active)
- THEORY-791 — The colour-naming systems of the world’s languages lie near the information-bottleneck bound for compressing perceptual meanings into words, and differences in one trade-off parameter place each language along it, as Zaslavsky, Kemp, Regier and Tishby propose (Proposed)

From In the 66 Pew surveys of 2001–2011 that varied the order of two back-to-back binary questions, question order changed the answers but not the probability of giving the same answer to both, the equality a Lüders projection model predicts for any projectors; a classical model with order events can meet the same equality, so the regularity does not by itself favour quantum over classical probability

THEORY-166 — In the 66 Pew surveys of 2001–2011 that varied the order of two back-to-back binary questions, question order changed the answers but not the probability of giving the same answer to both, the equality a Lüders projection model predicts for any projectors; a classical model with order events can meet the same equality, so the regularity does not by itself favour quantum over classical probability (Active)
- THEORY-796 — Question-order effects in surveys generally leave the probability of giving the same answer to both questions unchanged, the QQ equality that a projection model of judgment predicts, as Wang, Solloway, Shiffrin and Busemeyer propose (Proposed)

From A speaker who chooses words to minimise the information lost about a single intended referent, with words’ literal meanings uniform over their extensions, picks each true word with probability inversely proportional to a power of its extension, so the size principle follows from informative communication; in one betting experiment, learners who knew the referent bet on a novel word’s meaning in line with the inversion of that speaker (r = .93 on condition means), and in one free-description study speakers’ word use followed it only weakly (r = .19)

THEORY-472 — A speaker who chooses words to minimise the information lost about a single intended referent, with words’ literal meanings uniform over their extensions, picks each true word with probability inversely proportional to a power of its extension, so the size principle follows from informative communication; in one betting experiment, learners who knew the referent bet on a novel word’s meaning in line with the inversion of that speaker (r = .93 on condition means), and in one free-description study speakers’ word use followed it only weakly (r = .19) (Active)
- THEORY-172 — In Frank and Goodman’s one-shot reference games, listeners’ mean bets matched a parameter-free listener inverting a speaker who chooses among true utterances by informativeness, with a separately measured salience prior, and not salience alone; and in Bergen, Goodman and Levy’s signalling games with novel symbols, listeners read a signal as excluding what the available alternative would have conveyed (Active)
  - THEORY-785 — Listeners interpret an utterance by inverting a model of a speaker who chooses among alternatives by informativeness, so what a fixed form conveys depends on what else the speaker could have said, as the rational speech act account of Frank, Goodman and coauthors proposes (Proposed)
- THEORY-816 — Learners inferring what a novel word means from its use reason about a speaker who chose the word to be informative about the referent, so the size principle in word learning is a consequence of that inference and not a separate rule of generalisation (Proposed)

From In Aerts’s subspace model of Hampton’s membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice

THEORY-255 — In Aerts’s subspace model of Hampton’s membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice (Active)
- THEORY-807 — The guppy-type overextension in Hampton’s membership data arises because a combined concept is in part a new, emergent concept and not only a logical combination of its parts, so human concept combination runs in two superposed layers, one classical and logical and one quantum and conceptual (Proposed)

From In Hopfield’s 1982 simulations at N = 30 and 100, a network of threshold units with symmetric Hebbian couplings acts as a content-addressable memory whose stored random patterns are attractors: recall degrades severely near 0.15N patterns, a pattern placed 10 bits from another at N = 100 often fuses with it, and at 25-fold overload no stored pattern stays stable

THEORY-272 — In Hopfield’s 1982 simulations at N = 30 and 100, a network of threshold units with symmetric Hebbian couplings acts as a content-addressable memory whose stored random patterns are attractors: recall degrades severely near 0.15N patterns, a pattern placed 10 bits from another at N = 100 often fuses with it, and at 25-fold overload no stored pattern stays stable (Active)
- THEORY-800 — A network of N threshold units with symmetric Hebbian couplings stores random patterns as attractors up to a load proportional to N, about 0.15N for all large N, past which recall fails for every pattern at once, and a provision for forgetting old memories, such as couplings that saturate, keeps the most recent ones retrievable however many are added (Proposed)

From In Kirsh and Maglio’s Tetris keystroke data, rotation began in earnest by 400 to 600 ms after a piece appeared, sometimes before it was fully in view, and at each of three skill levels drops preceded by a move to the wall and back were longer than ordinary drops, about 19 to 20 squares against 13 to 16

THEORY-360 — In Kirsh and Maglio’s Tetris keystroke data, rotation began in earnest by 400 to 600 ms after a piece appeared, sometimes before it was fully in view, and at each of three skill levels drops preceded by a move to the wall and back were longer than ordinary drops, about 19 to 20 squares against 13 to 16 (Active)
- THEORY-823 — Tetris players rotate pieces more often than planning a placement and then executing it would require, rotate them early and check the column more the higher the drop, because these moves are epistemic actions: physical action that stands in for mental computation at the moment of use (Proposed)

complex-systems

From Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning

THEORY-069 — Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning (Proposed)
- THEORY-1227 — Haken, Kelso and Bunz model the switch of rhythmic bimanual coordination from anti-phase to in-phase at a critical frequency as the loss of one stable state of an order parameter, relative phase, derived from two coupled oscillators; Kelso, Scholz and Schöner report the variability of relative phase rising before the switch in the anti-phase pattern only, in a few subjects and with no statistical test (Active)

From In Friedkin’s simulations of reflected appraisal with Friedkin–Johnsen anchoring on initial positions, 99.0–100% of 100,000 random strongly connected networks of 5 to 25 members, complete or with random zero weights, evolve to a single leader and the rest to two

THEORY-1161 — In Friedkin’s simulations of reflected appraisal with Friedkin–Johnsen anchoring on initial positions, 99.0–100% of 100,000 random strongly connected networks of 5 to 25 members, complete or with random zero weights, evolve to a single leader and the rest to two (Active)
- THEORY-1232 — Under reflected appraisal across issues, whether power concentrates in one member depends on the single-issue step: with Friedkin–Johnsen anchoring on initial positions almost every random strongly connected group evolves to a single leader, and with DeGroot consensus the same random complete networks keep power spread (Proposed)

From In an illustrative active-inference simulation of post-traumatic context inference, run once per phenotype, a prior that one is in a harmful context, made overly precise by a run of harmful trials under the “vulnerable” learning and forgetting weights, makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision makes the attenuation policy less vigorous without correcting the prior, and the policy returns when precision is restored

THEORY-285 — In an illustrative active-inference simulation of post-traumatic context inference, run once per phenotype, a prior that one is in a harmful context, made overly precise by a run of harmful trials under the “vulnerable” learning and forgetting weights, makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision makes the attenuation policy less vigorous without correcting the prior, and the policy returns when precision is restored (Active)
- THEORY-1179 — In an active-inference model of post-traumatic context inference, an overly precise prior that one is in a harmful context makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision removes the symptoms without correcting the prior (Proposed)

From In one case study of the 2017 r/place Mona Lisa, what was added to or removed from the image was, once its forum opened, decided off the canvas, in a forum of 161 accounts negotiating through linked design images, where a few central members admitted or refused additions, and the forum’s changes then appeared in a region where about 77,000 accounts placed tiles, most of whom never posted; how many placers followed the designs is not measured, and the authors argue that the placement record alone cannot show who directed the image or who cooperated

THEORY-218 — In one case study of the 2017 r/place Mona Lisa, what was added to or removed from the image was, once its forum opened, decided off the canvas, in a forum of 161 accounts negotiating through linked design images, where a few central members admitted or refused additions, and the forum’s changes then appeared in a region where about 77,000 accounts placed tiles, most of whom never posted; how many placers followed the designs is not measured, and the authors argue that the placement record alone cannot show who directed the image or who cooperated (Active)
- THEORY-1181 — The content of a crowd-built r/place artwork is decided off the canvas, in negotiation through shared design images, where a few central members admit or refuse additions, and carried out by placers who mostly never join the discussion, so the placement record shows the outcome of coordination but not who directed it or who cooperated (Proposed)

From On the 2022 r/place canvas, the abrupt overthrow of a crowd-built image from a stable period is predicted up to hours ahead, on held-out images and on the 2023 edition, by a combination of many image-level series far better than by any generic slowing-down indicator; the series the predictor relies on include low past restoring of attacked pixels as well as attack outrunning restoration, and about a quarter of overthrows give no warning at the 20-minute range

THEORY-214 — On the 2022 r/place canvas, the abrupt overthrow of a crowd-built image from a stable period is predicted up to hours ahead, on held-out images and on the 2023 edition, by a combination of many image-level series far better than by any generic slowing-down indicator; the series the predictor relies on include low past restoring of attacked pixels as well as attack outrunning restoration, and about a quarter of overthrows give no warning at the 20-minute range (Active)
- THEORY-1183 — Crowd-maintained r/place images persist as an equilibrium of attack and repair, and their abrupt overthrow is preceded by several distinct precursors, including the defenders’ own conduct, rather than by one generic warning signal, while many overthrows give no warning (Proposed)

From In Bhattacharyya et al.’s single-cell swarm assays of E. coli MG1655, the daughters of one mother swarmed alike at the fourth generation sampled and varied again at the seventh; swarm size correlated with a plasmid fepA-promoter reporter of Fur derepression by low iron (Spearman r = 0.71), and an iron chelator or added iron shifted swarming and held the shared state longer; a three-class switching model fitted to two time points matched the lineages only with a delay of three to four generations before switching, which the authors read as ruling out a bistable switch

THEORY-179 — In Bhattacharyya et al.’s single-cell swarm assays of E. coli MG1655, the daughters of one mother swarmed alike at the fourth generation sampled and varied again at the seventh; swarm size correlated with a plasmid fepA-promoter reporter of Fur derepression by low iron (Spearman r = 0.71), and an iron chelator or added iron shifted swarming and held the shared state longer; a three-class switching model fitted to two time points matched the lineages only with a delay of three to four generations before switching, which the authors read as ruling out a bistable switch (Active)
- THEORY-1209 — In genetically identical E. coli, a cell’s swarming potential is a transient inherited state: its descendants share it for about four generations and lose it by about seven, and it tracks the cell’s iron status rather than a bistable switch (Proposed)

From In a sparsely-gated mixture-of-experts layer trained without balancing losses, the gate came to give large weights to the same few experts, which the authors attribute to self-reinforcement, favoured experts being trained more and so selected more; in one 256-expert language model, the squared coefficient of variation of expert importance or load as an auxiliary loss reduced the peak-to-mean load from 17.8 to at most 1.47 and test perplexity from 39.8 to 35.6–35.7

THEORY-927 — In a sparsely-gated mixture-of-experts layer trained without balancing losses, the gate came to give large weights to the same few experts, which the authors attribute to self-reinforcement, favoured experts being trained more and so selected more; in one 256-expert language model, the squared coefficient of variation of expert importance or load as an auxiliary loss reduced the peak-to-mean load from 17.8 to at most 1.47 and test perplexity from 39.8 to 35.6–35.7 (Active)
- THEORY-1218 — In a sparsely-gated mixture-of-experts layer trained without balancing losses, the gate’s concentration on a few experts arises by self-reinforcement: favoured experts are trained more rapidly and so are selected still more (Proposed)

From When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; Yang et al. state that with homogeneous agents mean field Q-learning converges to the Nash Q-value under three assumptions, one of which they call a strong constraint, by a proof whose contraction step needs a temperature above K/2 while its greedy-in-the-limit assumption needs the temperature to fall to zero; in their Gaussian Squeeze runs with 500 and 1000 agents, mean field Q-learning and actor-critic learned the optimal allocation where independent learners, FMQ, Rec-FMQ and a MADDPG variant failed

THEORY-945 — When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; Yang et al. state that with homogeneous agents mean field Q-learning converges to the Nash Q-value under three assumptions, one of which they call a strong constraint, by a proof whose contraction step needs a temperature above K/2 while its greedy-in-the-limit assumption needs the temperature to fall to zero; in their Gaussian Squeeze runs with 500 and 1000 agents, mean field Q-learning and actor-critic learned the optimal allocation where independent learners, FMQ, Rec-FMQ and a MADDPG variant failed (Active)
- THEORY-1231 — When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; with homogeneous agents, mean field Q-learning converges to the Nash Q-value under assumptions the authors call strong, and learned tasks of 500 and 1000 agents where independent and centralized-critic baselines failed (Proposed)

From The α-Rank chain over pure strategy profiles, built from Fermi-imitation fixation probabilities in finite populations with rare mutation, has a unique stationary distribution at every finite ranking intensity; as the intensity grows without bound its transition matrix converges to that of a chain that moves only along weakly better unilateral deviations, whose closed classes are the sink strongly connected components of the game’s weakly-better-response graph, on each of which it is a Markov-Conley chain, and each of which lies inside a single chain component of the replicator flow

THEORY-942 — The α-Rank chain over pure strategy profiles, built from Fermi-imitation fixation probabilities in finite populations with rare mutation, has a unique stationary distribution at every finite ranking intensity; as the intensity grows without bound its transition matrix converges to that of a chain that moves only along weakly better unilateral deviations, whose closed classes are the sink strongly connected components of the game’s weakly-better-response graph, on each of which it is a Markov-Conley chain, and each of which lies inside a single chain component of the replicator flow (Active)
- THEORY-1233 — For the replicator flow of a finite game, every asymptotically stable sink chain component contains at least one sink strongly connected component of the weakly-better-response graph, a Markov-Conley chain (Proposed)

From In an ensemble of sign-consistent Boolean rules on the 87-gene core of E. coli’s regulatory network, a transient knockout or overexpression of one gene can leave the network in a different attractor after release, and every gene that does so reaches a strongly connected component containing a positive circuit

THEORY-178 — In an ensemble of sign-consistent Boolean rules on the 87-gene core of E. coli’s regulatory network, a transient knockout or overexpression of one gene can leave the network in a different attractor after release, and every gene that does so reaches a strongly connected component containing a positive circuit (Active)
- THEORY-780 — Transcriptional regulation alone can give a bacterium’s gene expression a memory: in E. coli, a transient knockdown or overexpression of one regulator, with the environment held fixed, can leave other genes in a different self-maintaining expression state after it is released, as Zhao, Wytock, Reynolds and Motter predict from a Boolean-network model of its regulatory core (Proposed)

From Under geometric coarse-graining of the S¹ hidden-metric network model, which merges blocks of r angular neighbours into supernodes linked if any of their members were, the supernodes’ link probability approaches the S¹ form with β kept, κ′ = (Σκ^β)^{1/β}, μ′ = μ/r and R′ = R/r, its ratio to that form tending to 1 as their separation grows large against the block width and the coupling μκ′κ″; between nearby supernodes it exceeds that form

THEORY-190 — Under geometric coarse-graining of the S¹ hidden-metric network model, which merges blocks of r angular neighbours into supernodes linked if any of their members were, the supernodes’ link probability approaches the S¹ form with β kept, κ′ = (Σκ^β)^{1/β}, μ′ = μ/r and R′ = R/r, its ratio to that form tending to 1 as their separation grows large against the block width and the coupling μκ′κ″; between nearby supernodes it exceeds that form (Active)
- THEORY-799 — Under geometric coarse-graining of the S¹ hidden-metric network model the average degree flows as r^ν, with ν = 2/(γ − 1) − 1 when the degree exponent γ sets it and 2/β − 1 when β does, so the coarse-grained networks run toward a complete graph for γ < 3 or β < 2 and toward a ring otherwise (Proposed)
- THEORY-803 — Real complex networks embedded in the hyperbolic plane are self-similar under geometric renormalization, their coarse-grained layers keeping the degree distribution, clustering and degree–degree correlations of the original, and this self-similarity is evidence that a hidden metric space underlies them (Proposed)

From On the 2017 r/place log, with coalitions recovered from co-placement, which coalitions survive is barely predictable early from their size, area, start time and colour entropy, the size distribution of recovered coalitions shifts toward large ones as the event proceeds, and some lasting artworks were held by coalitions that kept working under sustained overwriting

THEORY-216 — On the 2017 r/place log, with coalitions recovered from co-placement, which coalitions survive is barely predictable early from their size, area, start time and colour entropy, the size distribution of recovered coalitions shifts toward large ones as the event proceeds, and some lasting artworks were held by coalitions that kept working under sustained overwriting (Active)
- THEORY-795 — On a contested shared canvas, the artworks that last are those whose coalitions keep repairing them under continual overwriting, not those left unattacked, as Wu and Silva propose from the 2017 r/place lifecycles (Proposed)

From For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose’s invariant subspaces, which exist on open sets but not generically at every scale

THEORY-733 — For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose’s invariant subspaces, which exist on open sets but not generically at every scale (Active)
- THEORY-257 — Dynamical independence, zero information flow from the microscopic history into a coarse-graining given its own history, fixes emergent macroscopic variables by the dynamics rather than by a chosen query, but for linear VAR models of order above 1 generically none is exact, and the optima are many (Proposed)

From In Hopfield’s 1982 simulations at N = 30 and 100, a network of threshold units with symmetric Hebbian couplings acts as a content-addressable memory whose stored random patterns are attractors: recall degrades severely near 0.15N patterns, a pattern placed 10 bits from another at N = 100 often fuses with it, and at 25-fold overload no stored pattern stays stable

THEORY-272 — In Hopfield’s 1982 simulations at N = 30 and 100, a network of threshold units with symmetric Hebbian couplings acts as a content-addressable memory whose stored random patterns are attractors: recall degrades severely near 0.15N patterns, a pattern placed 10 bits from another at N = 100 often fuses with it, and at 25-fold overload no stored pattern stays stable (Active)
- THEORY-800 — A network of N threshold units with symmetric Hebbian couplings stores random patterns as attractors up to a load proportional to N, about 0.15N for all large N, past which recall fails for every pattern at once, and a provision for forgetting old memories, such as couplings that saturate, keeps the most recent ones retrievable however many are added (Proposed)

From For a fixed linear operator in an inner-product norm, growth over time t is the top singular value of the propagator, which the eigenvalues bound only from below; it starts at the numerical abscissa, which can be positive when every mode decays; and the forced response is bounded by the condition number of the eigenvectors over the distance to the spectrum

THEORY-727 — For a fixed linear operator in an inner-product norm, growth over time t is the top singular value of the propagator, which the eigenvalues bound only from below; it starts at the numerical abscissa, which can be positive when every mode decays; and the forced response is bounded by the condition number of the eigenvectors over the distance to the spectrum (Active)
- THEORY-323 — Under spatially uncorrelated white noise, a stable non-normal linear system holds more variance than the sum of its pole contributions, and equality holds only when the operator is normal, so its spectrum understates the standing variance (Proposed)

From For a linear system du/dt = A(t)u, non-normality is necessary for time dependence to destabilize an operator that is uniformly stable at every instant, and a non-normal operator neutral at every instant can be asymptotically unstable under switching

THEORY-724 — For a linear system du/dt = A(t)u, non-normality is necessary for time dependence to destabilize an operator that is uniformly stable at every instant, and a non-normal operator neutral at every instant can be asymptotically unstable under switching (Active)
- THEORY-325 — Random variation of a linear operator destabilizes it generically, needing less variation the more non-normal and less damped the mean operator (Proposed)

From For a system driven by a protocol fixed in advance, with microscopic reversibility and with forward and reverse processes started uniformly on their macrostates, the log-odds of two outcomes II and III reached from I equal the difference in their log state counts, plus the log-ratio of their reversal probabilities, plus the difference in minus the log mean of e^(−βQ) over the paths reaching each, where Q is the heat released; so when the first two terms are equal the likelier outcome is the one whose paths release more heat on average with less variation

THEORY-831 — For a system driven by a protocol fixed in advance, with microscopic reversibility and with forward and reverse processes started uniformly on their macrostates, the log-odds of two outcomes II and III reached from I equal the difference in their log state counts, plus the log-ratio of their reversal probabilities, plus the difference in minus the log mean of e^(−βQ) over the paths reaching each, where Q is the heat released; so when the first two terms are equal the likelier outcome is the one whose paths release more heat on average with less variation (Active)
- THEORY-412 — Driven matter tends toward states that look adapted to the drive, with no replication needed: of two outcomes of a system driven by a fixed external field, equal in internal entropy and in the probability of reversal, the likelier is the one whose formation absorbed work from the drive and dissipated it more reliably (Proposed)

From Ince’s I_ccs, the expected local co-information over only those events where every source, the joint source and the overlap change the target’s surprisal in the same direction, under the maximum-entropy joint that preserves the source–target and joint-source marginals, is symmetric and self-redundant and satisfies the independent identity property; it fails Harder et al.’s identity and monotonicity, so its atoms can be negative, and, though its paper says otherwise, it fails subset equality

THEORY-714 — Ince’s I_ccs, the expected local co-information over only those events where every source, the joint source and the overlap change the target’s surprisal in the same direction, under the maximum-entropy joint that preserves the source–target and joint-source marginals, is symmetric and self-redundant and satisfies the independent identity property; it fails Harder et al.’s identity and monotonicity, so its atoms can be negative, and, though its paper says otherwise, it fails subset equality (Active)
- THEORY-804 — Ince’s common change in surprisal is a correct measure of the redundancy among any number of sources, and the negative partial-information atoms it gives are local misinformation and mechanistic redundancy, not defects of the measure (Proposed)

compositionality

THEORY-167 — The Russian wondertale is built from a small fixed set of character functions in a fixed order, so that all wondertales share one composition and differ in who performs the functions and how (Proposed)
- THEORY-1164 — Propp proposes, from his analysis of Afanasyev’s tales 50–151, that the Russian wondertale is built from 31 character functions, each defined by its consequence for the action, occurring in one order and falling to seven roles, so that wondertales share one composition and differ in who performs the functions and how (Active)

THEORY-117 — In Uselis, Dittadi and Oh’s two-concept grids, ResNet-50s trained from scratch, with checkpoints chosen on the unseen pairs, reach unseen combinations as the share of combinations seen grows and not as up to four times more data of the same combinations is added; and when embeddings are exactly linearly factored, seen pairs that connect every value of both concepts, as two per value arranged in one cycle do, determine the embedding of every unseen pair (Active)
- THEORY-794 — In vision models, compositional generalization to unseen combinations of concepts is driven by the diversity of the combinations seen in training rather than by data volume, and arrives with representations that are linearly factored by concept, as Uselis, Dittadi and Oh propose (Proposed)

From As Raviv, Meyer and Lev-Ari report it, in a laboratory game where 12 fully connected groups of eight and 12 of four built a new communication system with total interaction held fixed, the larger groups’ languages grew more systematic faster, ended more systematic and differed less from one another, while communicative success and convergence did not differ by size; input variability, higher in the larger groups, predicted the next round’s gain in structure, an association measured and not manipulated

THEORY-164 — As Raviv, Meyer and Lev-Ari report it, in a laboratory game where 12 fully connected groups of eight and 12 of four built a new communication system with total interaction held fixed, the larger groups’ languages grew more systematic faster, ended more systematic and differed less from one another, while communicative success and convergence did not differ by size; input variability, higher in the larger groups, predicted the next round’s gain in structure, an association measured and not manipulated (Active)
- THEORY-1171 — In groups that build a new communication system, larger groups make it more systematic, and the greater variety of input they face drives the effect (Proposed)

From On the Random Hierarchy Model, in the experiments reported (s = 2, L ≤ 4, small v), deep CNNs matched to the tree and trained in a feature-learning regime reach a tenth of chance error from about n_c m^L examples, polynomial in the input dimension, at the training-set size at which their hidden layers become invariant to swapping synonymous tuples; two-layer networks and the same CNNs as fixed kernels need a number proportional to the whole input space, exponential in the input dimension, and deep CNNs stay near chance when every tuple is made equally likely in every class

THEORY-195 — On the Random Hierarchy Model, in the experiments reported (s = 2, L ≤ 4, small v), deep CNNs matched to the tree and trained in a feature-learning regime reach a tenth of chance error from about n_c m^L examples, polynomial in the input dimension, at the training-set size at which their hidden layers become invariant to swapping synonymous tuples; two-layer networks and the same CNNs as fixed kernels need a number proportional to the whole input space, exponential in the input dimension, and deep CNNs stay near chance when every tuple is made equally likely in every class (Active)
- THEORY-786 — Deep networks learn hierarchically generated data from a number of examples set by when the correlations between input patches and the class rise above sampling noise, so detecting those correlations is how they learn the grammar, as Cagnetta and coauthors propose (Proposed)

From In the Random Hierarchy Model with v symbols per level, m rules per symbol and s-tuples, the covariance between two tokens whose lowest common ancestor is ℓ levels up has zero mean over grammars and root-mean-square size about ((1 − m/v^(s−1))/(v³ m^(2ℓ−1)))^(1/2), falling by a factor m per level, while its estimate from P sequences carries noise of about (v²P)^(−1/2), so the two meet at P_ℓ ≈ v m^(2ℓ−1)/(1 − m/v^(s−1)); all to leading order in large v and m

THEORY-778 — In the Random Hierarchy Model with v symbols per level, m rules per symbol and s-tuples, the covariance between two tokens whose lowest common ancestor is ℓ levels up has zero mean over grammars and root-mean-square size about ((1 − m/v^(s−1))/(v³ m^(2ℓ−1)))^(1/2), falling by a factor m per level, while its estimate from P sequences carries noise of about (v²P)^(−1/2), so the two meet at P_ℓ ≈ v m^(2ℓ−1)/(1 − m/v^(s−1)); all to leading order in large v and m (Active)
- THEORY-201 — In next-token prediction on hierarchically generated sequences, the correlation between two tokens falls by about a factor m per level of their common ancestor, so a training set of P sequences resolves correlations only out to a distance t*(P) at which they meet sampling noise; learners come to represent the hidden symbols up to that depth, giving loss steps at about v m^(2ℓ−1) examples, and on character-level text the context at which the loss saturates grows as P^(1/(2β)) with β the measured correlation decay (Proposed)

consciousness

From Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled

THEORY-045 — Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled (Proposed)
- THEORY-090 — A basic, affective form of consciousness is realised subcortically, in the upper brainstem and thalamus, before and without cortex, and its first function is alarm: survival behaviour, care for the body and generalised one-shot learning, with no choice between options (Proposed)
- THEORY-093 — A content of mind is conscious when a continuous flow of homeostatic feelings identifies it as the organism’s own; the feelings are conscious in themselves and hybrid, partly made of the body state they map, because interoceptive pathways let the body act directly on the neurons that map it (Proposed)

From In Vishne, Gerber, Knight and Deouell’s intracranial recordings from ten epilepsy patients viewing clearly visible images for up to 1.5 s, the pattern of high-frequency activity across ventral temporal cortex carried image category, and exemplar identity among 60 images, in a time-invariant code for as long as the image stayed on while amplitude fell about 80%; at the sampled prefrontal and parietal sites, category content was a burst about 150 to 600 ms after onset that did not track duration

THEORY-102 — In Vishne, Gerber, Knight and Deouell’s intracranial recordings from ten epilepsy patients viewing clearly visible images for up to 1.5 s, the pattern of high-frequency activity across ventral temporal cortex carried image category, and exemplar identity among 60 images, in a time-invariant code for as long as the image stayed on while amplitude fell about 80%; at the sampled prefrontal and parietal sites, category content was a burst about 150 to 600 ms after onset that did not track duration (Active)
- THEORY-830 — In human visual cortex the content of a sustained seen image is carried by a stable population code that lasts as long as the image, not by a sustained level of activity, and frontoparietal cortex marks only its onset; so a sustained experience rests on sensory representation and its discrete updates on frontoparietal representation (Proposed)

THEORY-953 — On Isaac’s structuralist account, the strongest evidence that a quality-space model targets phenomenal experience, and not only behaviour or neural activity, is convergence between models of the same space derived through different behavioural procedures, since experience is what those procedures share; he concedes that the convergence observed is coarse, with discrepancies that in many cases exceed each method’s own uncertainty, and that because the models average over subjects what they converge on is a standard observer (Active)
- THEORY-1204 — The strongest evidence that a quality-space model targets phenomenal experience, and not only behaviour or neural activity, is convergence between models of the same space derived through different behavioural procedures, since experience is what those procedures share (Proposed)

From In Claude Sonnet 4.5, Opus 4.5 and Opus 4.6, Gurnee, Sofroniew et al. found that projecting out the ten most active J-space directions in layers 38 to 54 of 100 lowered an experiential-language score, graded by Opus 4.6-based model graders on five or six prompts per condition, both in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions left it near baseline and the text stayed fluent; on Haiku 4.5 coherence failed first

THEORY-869 — In Claude Sonnet 4.5, Opus 4.5 and Opus 4.6, Gurnee, Sofroniew et al. found that projecting out the ten most active J-space directions in layers 38 to 54 of 100 lowered an experiential-language score, graded by Opus 4.6-based model graders on five or six prompts per condition, both in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions left it near baseline and the text stayed fluent; on Haiku 4.5 coherence failed first (Active)
- THEORY-1224 — In Claude language models, projecting out the most active J-space directions in early workspace layers lowers the rate of felt, first-person, sensory language in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions do not and the text stays coherent; so in these models the experiential register of report depends on the access-like workspace and is not specific to self-report (Proposed)

THEORY-865 — In Gurnee, Sofroniew et al.’s experiments on Claude Sonnet 4.5, Haiku 4.5, Opus 4.5 and Opus 4.6, the residual-stream directions a corpus-averaged Jacobian lens marks as verbalizable (the J-space) carried most of the causal effect on verbal report and on unspoken two-hop intermediates while holding a small share of their variance (swaps along them succeeding on 59% and 61% of trials, against 5% and 28% for the remainder, which fell to 0% and 6% with the J-space clamped), held instructed concepts, and passed one swapped argument to several functions (76 of 192 swaps); in two small task sets (n = 8 and 11), report and flexible inference followed a swap while continuation, anomaly detection and line wrapping did not, a split the authors partly define by this result; and by the lens’s own statistics the J-space works in a middle band of layers, holds about 25 active directions and under 10% of activation variance, and is amplified and relayed by the weights more than other directions (Active)
- THEORY-1226 — In Claude language models, a sparse set of residual-stream directions that a corpus-averaged Jacobian lens identifies as verbalizable (the J-space) is causally privileged for verbal report, for holding instructed concepts, for unspoken intermediate steps of reasoning and for passing one argument to many functions, while continuation, anomaly detection and line wrapping use the same information without routing through it; it carries workspace-like content only in a middle band of layers, holds on the order of tens of active directions and under a tenth of activation variance, and is amplified and relayed by the weights more than other directions (Proposed)

From In one study of 33 experienced Transcendental Meditation practitioners and 33 matched controls who counted instead, the practitioners rated their session higher on a retrospective pure-awareness scale whose top was defined for them as free from content (d = 0.77), and a classifier on EEG separated meditation from counting at 65% balanced accuracy on held-out subjects and, within practitioners, meditation from rest with 88% sensitivity; the authors found no robust association between the ratings and the EEG features

THEORY-870 — In one study of 33 experienced Transcendental Meditation practitioners and 33 matched controls who counted instead, the practitioners rated their session higher on a retrospective pure-awareness scale whose top was defined for them as free from content (d = 0.77), and a classifier on EEG separated meditation from counting at 65% balanced accuracy on held-out subjects and, within practitioners, meditation from rest with 88% sensitivity; the authors found no robust association between the ratings and the EEG features (Active)
- THEORY-1241 — Experienced Transcendental Meditation practitioners report more pure awareness than matched counting controls, and multivariate EEG separates the meditation period from counting and from rest, but the reports are not shown to track any EEG feature (Proposed)

From Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness

THEORY-866 — Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness (Active)
- THEORY-1246 — The so-called contentless goal-states of Shamatha, Transcendental and Stillness Meditation are neither truly contentless nor identical: they lack thoughts, perceptions, images and memories but keep wakefulness, calm, stillness, a non-conceptual knowing and the like, and they differ across the practices in attentional stability (Proposed)

contextuality

From An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality

THEORY-012 — An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality (Active)
- THEORY-011 — Generalized contextuality is strictly broader than Kochen–Specker contextuality: a qubit already has no preparation-noncontextual model, though no Kochen–Specker argument with sharp measurements exists in dimension two (Active)
- THEORY-014 — In both formalisms of contextuality, classicality is the existence of a nonnegative version of a linear representation that always exists over the reals, so negativity is universal and only its unavoidability is nonclassical (Active) — also carries further THEORY-016
- THEORY-015 — Kochen–Specker noncontextuality is measurement noncontextuality plus outcome determinism for sharp measurements, and preparation noncontextuality implies that determinism, so every Kochen–Specker proof refutes universal generalized noncontextuality while measurement noncontextuality alone is consistent with quantum theory (Active) — also carries further THEORY-016
- THEORY-526 — The Čech cohomology obstruction of a support section is a sound but incomplete witness of contextuality: it vanishes exactly when the section lies in a compatible family of integer combinations of support sections, so it is non-zero on every section of every All-vs-Nothing model, over any coefficient ring, on a connected cover, yet it misses the Hardy model entirely, part of a strongly contextual Kochen–Specker cover, and every section of a strongly contextual model on the fully symmetric (2,2,4) Bell cover (Active)
  - THEORY-524 — On cyclic measurement scenarios, where the contexts’ overlaps form one chordless N-cycle, the ℤ/2 Čech obstruction computed on the (N−1)-th iterated joint model is exact for logical and strong contextuality, the bound N − 1 being tight; beyond cycles no such result is established, the published extension is false on its own example, and the completeness conjecture fails on a Kochen–Specker cover that is never detected at any level (Active)
THEORY-016 — An operational theory admits a generalized-noncontextual model exactly when its GPT admits a positive quasiprobability representation, and for a tomographically local theory any diagram-preserving such model is an exact frame with exactly as many ontic states as the GPT’s dimension (Active)

From Classical Boolean logic is restored when a presheaf of contextual data acquires a global section or is sheafified

THEORY-024 — Classical Boolean logic is restored when a presheaf of contextual data acquires a global section or is sheafified (Rejected)
- THEORY-516 — For any presheaf of propositions over a small category of contexts, a sieve-valued valuation satisfies the generalized functional-composition condition exactly when it is a subobject of that presheaf, and the construction runs unchanged in classical physics, where every microstate gives a global section yet truth values are still non-Boolean sieves; so the logic of the truth values is fixed by the base category, not by whether a presheaf has global sections (Active)

From Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned

THEORY-720 — Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned (Active)
- THEORY-174 — Classical simulations between no-signalling empirical models on different scenarios never create contextuality, and a convex map between two scenarios’ model sets is induced by a classical non-adaptive procedure exactly when some non-contextual model of the hom scenario that satisfies its predicate induces it (Active)
  - THEORY-1170 — A convex map between two scenarios’ model sets is induced by a classical adaptive procedure exactly when some non-contextual model of the corresponding adaptive hom scenario induces it (Proposed)

From In Contextuality-by-Default 2.0, whether a system of non-binary variables is contextual depends on which variables represent the measurements: coarse-graining can turn a noncontextual system into a contextual one, while for connections of two variables, and for consistently connected systems, CbD 1.0 and 2.0 agree

THEORY-722 — In Contextuality-by-Default 2.0, whether a system of non-binary variables is contextual depends on which variables represent the measurements: coarse-graining can turn a noncontextual system into a contextual one, while for connections of two variables, and for consistently connected systems, CbD 1.0 and 2.0 agree (Active)
- THEORY-168 — In Contextuality-by-Default, whether a fixed set of measurements is contextual depends on how the system is represented: which couplings are imposed, and which dichotomizations of the variables are included (Active)

databases

From An access method’s read, update and memory overheads, each a ratio to the base data alone with a floor of 1.0, are conjectured to trade so that bounding any two forces a lower bound on the third; the conjecture is unproved, its formal wording says no more than the floor already does, and the paper’s extreme designs show conflicts between pairs, read against memory and update against read, not a symmetric three-way trade

THEORY-344 — An access method’s read, update and memory overheads, each a ratio to the base data alone with a floor of 1.0, are conjectured to trade so that bounding any two forces a lower bound on the third; the conjecture is unproved, its formal wording says no more than the floor already does, and the paper’s extreme designs show conflicts between pairs, read against memory and update against read, not a symmetric three-way trade (Active)
- THEORY-1244 — No access method is optimal on read, update and memory overhead at once: bounding any two of the three forces the third above a hard lower bound that cannot be further reduced (Proposed)

From When stored views are never updated, greedily storing the view of largest marginal saving in query cost gets at least 1 − 1/e of the optimal saving on a data-cube lattice or any OR or AND view graph, against the optimum for the same number of views or for the space the greedy ends up using, which may exceed the budget by one view

THEORY-356 — When stored views are never updated, greedily storing the view of largest marginal saving in query cost gets at least 1 − 1/e of the optimal saving on a data-cube lattice or any OR or AND view graph, against the optimum for the same number of views or for the space the greedy ends up using, which may exceed the budget by one view (Active)
- THEORY-350 — Under a budget on maintenance time, a stored view’s upkeep depends on which other views are stored, so upkeep is not additive and adding a view can lower it, and greedily adding the single view of best saving per unit of upkeep can be arbitrarily bad; on OR view graphs a greedy over inverted tree sets gets 1 − 1/e of the saving of an optimum given as much upkeep, using up to twice the budget, if no tree-piece of the optimum alone exceeds the budget (Active)

From In one study of a stack of proximity graphs on nested random subsets of the data, each element’s top layer drawn geometrically, greedy search on uniform random data in 8 dimensions needed a number of distance computations that grew no worse than logarithmically with the number of items and stayed below a single-layer navigable graph’s at every size tested, and in 4 dimensions the beam width needed for a fixed recall saturated; the logarithmic bound is derived only for exact Delaunay graphs of bounded degree, which the index does not build, and on 200 million SIFT vectors the measured growth departs from it

THEORY-365 — In one study of a stack of proximity graphs on nested random subsets of the data, each element’s top layer drawn geometrically, greedy search on uniform random data in 8 dimensions needed a number of distance computations that grew no worse than logarithmically with the number of items and stayed below a single-layer navigable graph’s at every size tested, and in 4 dimensions the beam width needed for a fixed recall saturated; the logarithmic bound is derived only for exact Delaunay graphs of bounded degree, which the index does not build, and on 200 million SIFT vectors the measured growth departs from it (Active)
- THEORY-815 — Layering a navigable small-world graph on nested random subsets of the data, with links separated by length scale and each layer’s degree capped, makes greedy nearest-neighbour search cost grow logarithmically with the number of items on data of low intrinsic dimension, where the single-layer graph’s cost grows polylogarithmically (Proposed)

From Locality-sensitive hashing answers approximate near-neighbour queries over n points with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, where ρ = ln(1/p1)/ln(1/p2) is set by how much more often near pairs collide than far ones and is at most 1/(1+ε) for approximation 1+ε in Hamming space, with constant success probability per query; the paper gives upper bounds only and proves no limit on what organizing can buy

THEORY-388 — Locality-sensitive hashing answers approximate near-neighbour queries over n points with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, where ρ = ln(1/p1)/ln(1/p2) is set by how much more often near pairs collide than far ones and is at most 1/(1+ε) for approximation 1+ε in Hamming space, with constant success probability per query; the paper gives upper bounds only and proves no limit on what organizing can buy (Active)
- THEORY-372 — For c-approximate near-neighbour search over n points in Euclidean space, space n^{1+ρu+o(1)} and query time n^{ρq+o(1)} are achievable on any data set whenever c²√ρq + (c²−1)√ρu ≥ √(2c²−1) (c√ρq + (c−1)√ρu ≥ √(2c−1) in Hamming space), and no list-of-points structure, the model of known data-independent hashing, does better on random instances; without that model the curve is matched only at query time n^{o(1)} by one or two cell probes (Active)

From For positive relational algebra on relations whose tuples are annotated from a commutative semiring, every query’s annotated answer equals its answer over provenance polynomials N[X] in the input tuple ids, evaluated at the input annotations, so set, bag, c-table and event-table semantics all factor through the polynomials; revaluing inputs can remove output tuples but never add them, and nothing is shown for negation, difference or aggregation

THEORY-373 — For positive relational algebra on relations whose tuples are annotated from a commutative semiring, every query’s annotated answer equals its answer over provenance polynomials N[X] in the input tuple ids, evaluated at the input annotations, so set, bag, c-table and event-table semantics all factor through the polynomials; revaluing inputs can remove output tuples but never add them, and nothing is shown for negation, difference or aggregation (Active)
- THEORY-380 — For datalog over a finite input instance annotated from a commutative ω-continuous semiring, with an answer’s tag defined as the sum over its derivation trees of the product of its leaves’ tags, the tags are the least fixed point of a finite polynomial system and factor through formal power series N∞⟦X⟧ in the input tuple ids; an answer tuple’s series has an infinite coefficient exactly when one of its derivation trees contains a tuple lying on a cycle of unit rules (Active)

From When a cached interval of width W is kept valid by its source, which pushes a refresh at cost C_vr whenever the value leaves it, and queries fetch the exact value at cost C_qr whenever it is too wide, and the chances of the two per step go as 1/W² and as W, the expected refresh cost is least at W* = (ρK₁/K₂)^(1/3) with ρ = 2C_vr/C_qr, where ρ times the chance of a pushed refresh equals the chance of a fetch

THEORY-386 — When a cached interval of width W is kept valid by its source, which pushes a refresh at cost C_vr whenever the value leaves it, and queries fetch the exact value at cost C_qr whenever it is too wide, and the chances of the two per step go as 1/W² and as W, the expected refresh cost is least at W* = (ρK₁/K₂)^(1/3) with ρ = 2C_vr/C_qr, where ρ times the chance of a pushed refresh equals the chance of a fetch (Active)
- THEORY-377 — In Olston, Loo and Widom’s simulations, a rule that multiplies a cached interval’s width by 1 + α on a pushed refresh with probability min{ρ, 1}, and divides it by 1 + α on a query’s exact fetch with probability min{1/ρ, 1}, settled within 1% of the best fixed width on one random walk and within 5% over eight walk settings; on a 50-host network-traffic trace, restricted to exact copies, it almost matched a tuned adaptive cache-or-not rule, and with a 50-value cache, where queries tolerated imprecision, it cost less; that it converges to the best width is argued, not proved (Active)

From Fagin’s Algorithm finds the top k of N objects held in m attribute-sorted lists under any monotone combining function, and when the lists’ orderings are independent with at least m − 1 of them uniformly random it does so with O(N^((m−1)/m) k^(1/m)) sorted and random accesses with arbitrarily high probability; with dependent orderings the bound can fail

THEORY-379 — Fagin’s Algorithm finds the top k of N objects held in m attribute-sorted lists under any monotone combining function, and when the lists’ orderings are independent with at least m − 1 of them uniformly random it does so with O(N^((m−1)/m) k^(1/m)) sorted and random accesses with arbitrarily high probability; with dependent orderings the bound can fail (Active)
- THEORY-832 — For strict combining functions and m attribute-sorted lists of N objects whose orderings are probabilistically independent, no correct top-k algorithm does essentially better than Fagin’s Algorithm in the worst case with high probability: every one needs on the order of N^((m−1)/m) k^(1/m) accesses (Proposed)

ecology

From On the Black Queen Hypothesis of Morris, Lenski and Zinser, losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required

THEORY-1048 — On the Black Queen Hypothesis of Morris, Lenski and Zinser, losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required (Active)
- THEORY-1235 — Losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required (Proposed)

embodied-cognition

From Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning

THEORY-069 — Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning (Proposed)
- THEORY-1227 — Haken, Kelso and Bunz model the switch of rhythmic bimanual coordination from anti-phase to in-phase at a critical frequency as the loss of one stable state of an order parameter, relative phase, derived from two coupled oscillators; Kelso, Scholz and Schöner report the variability of relative phase rising before the switch in the anti-phase pattern only, in a few subjects and with no statistical test (Active)

From Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency

THEORY-071 — Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency (Proposed)
- THEORY-1119 — Barandiaran, Di Paolo and Rohde propose that minimal agency requires individuality, interactional asymmetry and normativity set by the system’s own organisation, each necessary and together, they say, sufficient; that individuality without the other two is not agency; and that a system optimizing an externally fixed function is not a model of agency (Active)

From Self-production alone gives a system only an all-or-nothing norm, to go on or to die, and that distinction is available to an observer, not to the system; a graded norm the system itself has, by which viable states are better or worse for it, needs adaptivity, a further capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them, and sense-making and agency need adaptivity

THEORY-628 — Self-production alone gives a system only an all-or-nothing norm, to go on or to die, and that distinction is available to an observer, not to the system; a graded norm the system itself has, by which viable states are better or worse for it, needs adaptivity, a further capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them, and sense-making and agency need adaptivity (Proposed)
- THEORY-1166 — Di Paolo argues that autopoiesis as originally defined entails conservation and not active homeostasis, so it gives a system only an all-or-nothing norm, to go on or to die, which an observer and not the system can use; that a graded norm the system itself has needs adaptivity, a capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them; and that sense-making and agency need adaptivity (Active)

From If belief is individuated by functional role, an external store that is constantly consulted, directly available and automatically endorsed can carry a standing belief, so the material vehicles of some mental states extend beyond the skin; the step from coupling to parthood rests on that premise and a parity principle the authors state rather than argue

THEORY-579 — If belief is individuated by functional role, an external store that is constantly consulted, directly available and automatically endorsed can carry a standing belief, so the material vehicles of some mental states extend beyond the skin; the step from coupling to parthood rests on that premise and a parity principle the authors state rather than argue (Proposed)
- THEORY-1194 — Clark and Chalmers argue that if belief is individuated by functional role, an external store that is constantly consulted, directly available and automatically endorsed can carry a standing belief, so that the material vehicles of some mental states extend beyond the skin; the step from coupling to parthood rests on that premise and on a parity principle they state rather than argue (Active)
- THEORY-512 — The extended-mind thesis worth disputing is that external entities partly constitute a subject’s mental states in virtue of the subject’s sensorimotor interaction with them; so stated it can reach standing belief but not consciousness, if consciousness needs information directly available for global control, since perception makes availability indirect (Proposed)

From In Kirsh and Maglio’s Tetris keystroke data, rotation began in earnest by 400 to 600 ms after a piece appeared, sometimes before it was fully in view, and at each of three skill levels drops preceded by a move to the wall and back were longer than ordinary drops, about 19 to 20 squares against 13 to 16

THEORY-360 — In Kirsh and Maglio’s Tetris keystroke data, rotation began in earnest by 400 to 600 ms after a piece appeared, sometimes before it was fully in view, and at each of three skill levels drops preceded by a move to the wall and back were longer than ordinary drops, about 19 to 20 squares against 13 to 16 (Active)
- THEORY-823 — Tetris players rotate pieces more often than planning a placement and then executing it would require, rotate them early and check the column more the higher the drop, because these moves are epistemic actions: physical action that stands in for mental computation at the moment of use (Proposed)

From Brushing a seen rubber hand in synchrony with the hidden real hand makes touch be felt at the rubber hand, makes it be reported as one’s own, and shifts the felt position of the real hand toward it; a small asynchrony removes these effects

THEORY-765 — Brushing a seen rubber hand in synchrony with the hidden real hand makes touch be felt at the rubber hand, makes it be reported as one’s own, and shifts the felt position of the real hand toward it; a small asynchrony removes these effects (Rejected)
- THEORY-766 — Brushing a seen rubber hand in synchrony with the hidden real hand, against asynchronous brushing, raises reports that touch is felt at the rubber hand and, more weakly and less reliably, that the rubber hand is one’s own; the proprioceptive drift toward the rubber hand that accompanies these reports is not a measure of them; the standard comparison conditions do not equalise participants’ expectations, so whether the stimulation or the expectation produces the report difference is open (Active)
  - THEORY-827 — The rubber hand illusion is a perceptual illusion driven by multisensory integration: the difference in reports between synchronous and asynchronous brushing is produced by the stimulation, not by participants’ expectations (Proposed)

emotion-and-affect

From Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled

THEORY-045 — Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled (Proposed)
- THEORY-090 — A basic, affective form of consciousness is realised subcortically, in the upper brainstem and thalamus, before and without cortex, and its first function is alarm: survival behaviour, care for the body and generalised one-shot learning, with no choice between options (Proposed)
- THEORY-093 — A content of mind is conscious when a continuous flow of homeostatic feelings identifies it as the organism’s own; the feelings are conscious in themselves and hybrid, partly made of the body state they map, because interoceptive pathways let the body act directly on the neurons that map it (Proposed)

From As Chou, Wilson and Smith review them, in about six foraging studies, mostly sub-clinical, higher stress or anxiety went with less exploration, and in one task with a predator to check for (Trier et al. 2023), people higher in trait somatic anxiety checked less and did worse, a reduction the study’s further analyses associated with greater aversion to detecting the predator

THEORY-907 — As Chou, Wilson and Smith review them, in about six foraging studies, mostly sub-clinical, higher stress or anxiety went with less exploration, and in one task with a predator to check for (Trier et al. 2023), people higher in trait somatic anxiety checked less and did worse, a reduction the study’s further analyses associated with greater aversion to detecting the predator (Active)
- THEORY-1180 — In foraging tasks, higher stress or anxiety goes consistently with less exploration, and in one task with a predator to check for, people higher in somatic anxiety checked less and did worse, a reduction associated with greater aversion to detecting the predator (Proposed)

From Chou, Wilson and Smith read about eight computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples, with mixed results and none replicated, as suggesting that directed exploration rises with trait or cognitive anxiety and falls with state or somatic anxiety; none they review links random exploration to anxiety

THEORY-917 — Chou, Wilson and Smith read about eight computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples, with mixed results and none replicated, as suggesting that directed exploration rises with trait or cognitive anxiety and falls with state or somatic anxiety; none they review links random exploration to anxiety (Active)
- THEORY-1186 — Across computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples and not yet replicated, directed exploration tends to rise with trait or cognitive anxiety and to fall with state or somatic anxiety (Proposed)

From In Gangneung, South Korea, in 2022, people the authors observed by chance in beach-front cafés outwardly ignored routine retaliatory missile launches and military sites on the beach, while a missile malfunction at night under a news embargo was followed by tweets voicing fear of war with North Korea and blaming the government and military

THEORY-898 — In Gangneung, South Korea, in 2022, people the authors observed by chance in beach-front cafés outwardly ignored routine retaliatory missile launches and military sites on the beach, while a missile malfunction at night under a news embargo was followed by tweets voicing fear of war with North Korea and blaming the government and military (Active)
- THEORY-1202 — A securitised public keeps its routines by not attending to routine security practices, which keeps an anxiety about the threat latent rather than absent; when the security apparatus itself disrupts those routines the anxiety surfaces, and with it blame of the apparatus (Proposed)

THEORY-329 — In 32,479 Weibo posts of 2011–2016 scored with an emotion lexicon, posts tagged fake by the platform’s fact-checkers carry more anger and less joy than true posts from verified accounts, and a post’s share of anger words goes with more retweets after controls; whether the gap is due to falsity, and whether anger causes the spread, is not shown (Active)
- THEORY-1221 — Anger makes fake news viral online: on Weibo fake news carries more anger and less joy than true news, and the anger in a post causes it to be shared more (Proposed)

THEORY-123 — In ad hoc groups of students negotiating a bonus, a confederate’s randomly assigned pleasant or unpleasant mood spread to the other members, by their self-reports and by blind coders, and the more pleasant mood a group caught, the more cooperative and the less conflicted coders rated it, a correlation with measured contagion rather than a manipulated effect; members’ moods also converged without a confederate, and members did not credit their mood for how effective they judged themselves; the numbers are from the 2001 manuscript (Active)
- THEORY-126 — A collective emotion is the synchronous convergence of individuals’ affective responses to one event or object, produced bottom-up by shared appraisal, contagion and group membership, and it needs no group subject; it takes a We-mode form when the appraisals rest on collectively intentional states (Proposed)
- THEORY-826 — In work groups, a member’s displayed mood spreads to the others, and pleasant emotional contagion improves the group’s cooperation and reduces its conflict (Proposed)

From In two midlife community samples (n = 182 and 176), instructed reappraisal of aversive pictures, mostly by reinterpretation, lowered reported negative affect, while the BOLD responses of the amygdala, periaqueductal gray, hypothalamus and the other subcortical regions that responded to the pictures showed evidence of no change (Bayes factors below 1/10); the emotion-responsive regions it reduced were all cortical, mainly higher visual and parietal

THEORY-871 — In two midlife community samples (n = 182 and 176), instructed reappraisal of aversive pictures, mostly by reinterpretation, lowered reported negative affect, while the BOLD responses of the amygdala, periaqueductal gray, hypothalamus and the other subcortical regions that responded to the pictures showed evidence of no change (Bayes factors below 1/10); the emotion-responsive regions it reduced were all cortical, mainly higher visual and parietal (Active)
- THEORY-1237 — Instructed reappraisal lowers reported negative affect to aversive pictures while the responses of the amygdala, periaqueductal gray, hypothalamus and other subcortical regions to the same pictures show evidence of no change; the regions it reduces are cortical, mainly higher visual and parietal (Proposed)

endosymbiosis

From On the Black Queen Hypothesis of Morris, Lenski and Zinser, losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required

THEORY-1048 — On the Black Queen Hypothesis of Morris, Lenski and Zinser, losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required (Active)
- THEORY-1235 — Losing a costly function whose product leaks to neighbours is favoured by ordinary individual selection in large free-living microbial populations, and the loss proceeds until the remaining producers just supply the community, so that most cells come to depend on a rare few with no relatedness, cooperation or collective required (Proposed)

From On McCutcheon and colleagues’ survey, bacterial endosymbionts lose their own cell-envelope and envelope-transport genes together below about 400 kb, and begin to lose translation genes (ribosomal proteins, tRNAs, aminoacyl-tRNA synthetases) below about 200 kb, so that the smallest-genome symbionts cannot be self-sufficient and their membranes, transport and protein synthesis are presumably supplied by the host, as for organelles

THEORY-1049 — On McCutcheon and colleagues’ survey, bacterial endosymbionts lose their own cell-envelope and envelope-transport genes together below about 400 kb, and begin to lose translation genes (ribosomal proteins, tRNAs, aminoacyl-tRNA synthetases) below about 200 kb, so that the smallest-genome symbionts cannot be self-sufficient and their membranes, transport and protein synthesis are presumably supplied by the host, as for organelles (Active)
- THEORY-1208 — Bacterial endosymbionts lose their cell-envelope and envelope-transport genes together below about 400 kb and begin to lose translation genes below about 200 kb, and the host supplies the membranes, transport and protein synthesis that the smallest-genome symbionts no longer encode, as for organelles (Proposed)

From In Athreya, Czuppon and Gokhale’s adaptive-dynamics model of an incipient endosymbiosis, mutual dependence (the product of the partners’ obligacies) evolves ahead of reproductive cohesion (the product of their adhesions) early on, but only under the mapping in which each partner’s free-living growth depends on its own obligacy alone while dissociation depends on both partners’ adhesions; the authors’ own appendix shows the bias vanishing under matched mappings and reversing under the opposite one, and their survey of real symbioses cannot test it

THEORY-1050 — In Athreya, Czuppon and Gokhale’s adaptive-dynamics model of an incipient endosymbiosis, mutual dependence (the product of the partners’ obligacies) evolves ahead of reproductive cohesion (the product of their adhesions) early on, but only under the mapping in which each partner’s free-living growth depends on its own obligacy alone while dissociation depends on both partners’ adhesions; the authors’ own appendix shows the bias vanishing under matched mappings and reversing under the opposite one, and their survey of real symbioses cannot test it (Active)
- THEORY-1243 — In incipient endosymbioses in nature, mutual dependence evolves ahead of reproductive cohesion, so that endosymbioses are more likely to be mutually dependent than cohesive (Proposed)

From On Karnkowska and colleagues’ genome analysis, the oxymonad Monocercomonoides exilis lost its mitochondrion only after the organelle’s essential functions had been removed one by one, Fe–S cluster assembly to a bacterial-type SUF system shared with mitochondriate relatives and then the glycine cleavage system by loss, and the loss left the rest of the cell canonical, with no compensating changes found

THEORY-1098 — On Karnkowska and colleagues’ genome analysis, the oxymonad Monocercomonoides exilis lost its mitochondrion only after the organelle’s essential functions had been removed one by one, Fe–S cluster assembly to a bacterial-type SUF system shared with mitochondriate relatives and then the glycine cleavage system by loss, and the loss left the rest of the cell canonical, with no compensating changes found (Active)
- THEORY-1223 — Monocercomonoides exilis lost its mitochondrion only after the organelle’s essential functions had been removed in steps, Fe–S cluster assembly to a cytosolic SUF system and then the glycine cleavage system by loss, and the loss required no compensating change elsewhere in the cell (Proposed)

From On Kay and colleagues’ relaxed-clock dating of pre-LECA gene duplications, most duplications in eukaryotic gene families of archaeal origin, including actin, tubulin, vesicle-trafficking and nuclear-targeted families, predate the onset of duplications in genes of alphaproteobacterial origin at about 2.2 billion years ago, which the authors take as the start of mitochondrial endosymbiosis, so the host was elaborating its cytoskeleton, endomembranes and nuclear machinery before the mitochondrion; each date is a maximum age for its paralogues’ functions, many intervals reach below 2.2 billion years, and the order of phagocytosis and the mitochondrion is left open

THEORY-1099 — On Kay and colleagues’ relaxed-clock dating of pre-LECA gene duplications, most duplications in eukaryotic gene families of archaeal origin, including actin, tubulin, vesicle-trafficking and nuclear-targeted families, predate the onset of duplications in genes of alphaproteobacterial origin at about 2.2 billion years ago, which the authors take as the start of mitochondrial endosymbiosis, so the host was elaborating its cytoskeleton, endomembranes and nuclear machinery before the mitochondrion; each date is a maximum age for its paralogues’ functions, many intervals reach below 2.2 billion years, and the order of phagocytosis and the mitochondrion is left open (Active)
- THEORY-1203 — The host lineage elaborated its cytoskeleton, endomembrane system and nuclear machinery before mitochondrial endosymbiosis began, about 2.2 billion years ago: most pre-LECA duplications in eukaryotic gene families of archaeal origin predate it (Proposed)

From On Sørensen and colleagues’ review, a bacterial endosymbiont becomes genetically integrated, the authors’ mark of an organelle, when nuclear genes whose products are imported into it functionally replace essential genes it has lost, and this does not require gene transfer from the endosymbiont: in the Paulinella chromatophore and the nitroplast, the two systems with hundreds of imported proteins, many of which fill gaps left by lost genes, the imported proteins come mostly from the host’s own genes or from other bacteria, and in the nitroplast no gene transfer from the endosymbiont has been found

THEORY-1100 — On Sørensen and colleagues’ review, a bacterial endosymbiont becomes genetically integrated, the authors’ mark of an organelle, when nuclear genes whose products are imported into it functionally replace essential genes it has lost, and this does not require gene transfer from the endosymbiont: in the Paulinella chromatophore and the nitroplast, the two systems with hundreds of imported proteins, many of which fill gaps left by lost genes, the imported proteins come mostly from the host’s own genes or from other bacteria, and in the nitroplast no gene transfer from the endosymbiont has been found (Active)
- THEORY-1228 — A bacterial endosymbiont becomes genetically integrated, an organelle, when host proteins imported into it replace genes it has lost, and this does not require gene transfer from the endosymbiont: in the Paulinella chromatophore and the nitroplast, the two systems with hundreds of imported proteins, the replacing genes are mostly the host’s own or come from other bacteria (Proposed)

From Tobiasson, Luo, Wolf and Koonin find, on 5,850 eukaryotic gene stems, that normalized stem length does not behave as a clock of acquisition time: alphaproteobacterial stems are longer than Asgard ones for the 70% of stems below 0.3 and shorter only above 0.35, oxidative-phosphorylation genes presumed acquired together vary by more than two orders of magnitude, and information-processing genes have short stems; they conclude that even normalized stem lengths are set primarily by factors other than acquisition time, at least over the long timescales analysed, and suggest that adaptation after acquisition lengthens stems

THEORY-1101 — Tobiasson, Luo, Wolf and Koonin find, on 5,850 eukaryotic gene stems, that normalized stem length does not behave as a clock of acquisition time: alphaproteobacterial stems are longer than Asgard ones for the 70% of stems below 0.3 and shorter only above 0.35, oxidative-phosphorylation genes presumed acquired together vary by more than two orders of magnitude, and information-processing genes have short stems; they conclude that even normalized stem lengths are set primarily by factors other than acquisition time, at least over the long timescales analysed, and suggest that adaptation after acquisition lengthens stems (Active)
- THEORY-1201 — Normalized stem length is not a clock of gene acquisition time: it is set mainly by factors other than when a gene was acquired, such as adaptation to the new cell after acquisition, so relative timings of eukaryogenesis drawn from stem lengths alone are not supported (Proposed)

From On Pittis and Gabaldón’s stem-length analysis of the LECA proteome, families of alphaproteobacterial origin and mitochondrial location sit closer to their prokaryotic relatives, after correction for each family’s rate, than families of other bacterial origin (P = 0.044), which sit closer than archaeal families, and the authors read this as late acquisition of the mitochondrion by a host whose proteome was already chimaeric, under their stated assumption that rates before and after LECA are correlated

THEORY-1102 — On Pittis and Gabaldón’s stem-length analysis of the LECA proteome, families of alphaproteobacterial origin and mitochondrial location sit closer to their prokaryotic relatives, after correction for each family’s rate, than families of other bacterial origin (P = 0.044), which sit closer than archaeal families, and the authors read this as late acquisition of the mitochondrion by a host whose proteome was already chimaeric, under their stated assumption that rates before and after LECA are correlated (Active)
- THEORY-1206 — The mitochondrion was acquired late in eukaryogenesis, by a host whose proteome was already chimaeric, with genes from several bacterial sources acquired before it (Proposed)

From On Shigenobu and colleagues’ genome of Buchnera sp. APS, the aphid symbiont keeps the genes for the amino acids its host cannot make and has lost almost all of those its host can, uses host-made glutamate and aspartate as precursors, and lacks phospholipid synthesis and nearly all its own transcriptional regulators, so that its metabolism interlocks with the host’s and it presumably lives only inside the bacteriocyte, a state reached by gene loss from an E. coli-like ancestor

THEORY-1103 — On Shigenobu and colleagues’ genome of Buchnera sp. APS, the aphid symbiont keeps the genes for the amino acids its host cannot make and has lost almost all of those its host can, uses host-made glutamate and aspartate as precursors, and lacks phospholipid synthesis and nearly all its own transcriptional regulators, so that its metabolism interlocks with the host’s and it presumably lives only inside the bacteriocyte, a state reached by gene loss from an E. coli-like ancestor (Active)
- THEORY-1219 — Buchnera’s metabolism interlocks with its aphid host’s: it makes the essential amino acids its host cannot from host-supplied glutamate and aspartate, depends on the host for the non-essential amino acids and its membrane phospholipids, and cannot live outside the bacteriocyte (Proposed)

From In Tobiasson, Luo, Wolf and Koonin’s constrained-tree survey of widespread eukaryotic protein families, Asgard archaea are the largest single prokaryotic source, with about half the likelihood weight in the best-resolved families and the lead in nearly every functional system, while Alphaproteobacteria contribute a small, coherent set centred on the mitochondrion and other bacteria about a third of families, scattered across functions with no pathway led by one donor except a narrow Myxococcota signal in nucleotide-sugar and nicotinate metabolism; the authors read this as one Asgard host lineage, one endosymbiont and piecemeal gene transfer rather than further symbioses

THEORY-1104 — In Tobiasson, Luo, Wolf and Koonin’s constrained-tree survey of widespread eukaryotic protein families, Asgard archaea are the largest single prokaryotic source, with about half the likelihood weight in the best-resolved families and the lead in nearly every functional system, while Alphaproteobacteria contribute a small, coherent set centred on the mitochondrion and other bacteria about a third of families, scattered across functions with no pathway led by one donor except a narrow Myxococcota signal in nucleotide-sugar and nicotinate metabolism; the authors read this as one Asgard host lineage, one endosymbiont and piecemeal gene transfer rather than further symbioses (Active)
- THEORY-1197 — Eukaryogenesis involved one archaeal host lineage, of Asgard descent, and one bacterial endosymbiont, the alphaproteobacterial ancestor of mitochondria; the eukaryotic genes from other bacteria came by piecemeal gene transfer, not from further symbiotic partners (Proposed)

THEORY-1105 — In Bernabeu, Manzano-Morales, Marcet-Houben and Gabaldón’s reconstruction of the last eukaryotic common ancestor’s proteome, two bacterial phyla besides the mitochondrial ancestor, Myxococcota and Planctomycetota, each account for a share of the proteome comparable to that of Alphaproteobacteria and Asgard archaea under thresholds calibrated on those accepted partners, and stem-length distributions order the acquisitions as waves, Asgard first, then Planctomycetota, then the mitochondrial ancestor and Myxococcota together; the authors read this as serial associations with several partners, while conceding that their data cannot say whether any donor was a symbiont (Active)
- THEORY-1215 — Besides Asgard archaea and the alphaproteobacterial ancestor of mitochondria, Myxococcota and Planctomycetota were major donors of genes to the eukaryotic lineage before its last common ancestor, in waves: Asgard first, then Planctomycetota before the mitochondrion, then the mitochondrial ancestor and Myxococcota together (Proposed)

From Campbell and colleagues report, from four cicada species with five or six eggs each, that Hodgkinia cells per egg range from about 4,900 in a species with one Hodgkinia lineage to about 30,000, some six times as many, in Magicicada septendecim, whose Hodgkinia has split into many interdependent lineages and the only species that differs significantly, while Sulcia numbers are not raised, the transmission route is the same, and every egg sampled carries every lineage tracked by marker genes; the authors take the raised titre as a host adaptation that compensates for the symbiont’s fragmentation, and a cost of it

THEORY-1106 — Campbell and colleagues report, from four cicada species with five or six eggs each, that Hodgkinia cells per egg range from about 4,900 in a species with one Hodgkinia lineage to about 30,000, some six times as many, in Magicicada septendecim, whose Hodgkinia has split into many interdependent lineages and the only species that differs significantly, while Sulcia numbers are not raised, the transmission route is the same, and every egg sampled carries every lineage tracked by marker genes; the authors take the raised titre as a host adaptation that compensates for the symbiont’s fragmentation, and a cost of it (Active)
- THEORY-1214 — Cicada hosts whose endosymbiont Hodgkinia has split into many interdependent lineages compensate by sending more Hodgkinia cells to each egg, with Sulcia numbers and the transmission route unchanged, so that every egg still receives every lineage; the raised titre is a host adaptation driven by the symbiont’s fragmentation, and a cost of it (Proposed)

From On Sørensen and colleagues’ scenario, an endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations, and host effector control without vertical transmission, as in rhizobia, does not lead to it

THEORY-1107 — On Sørensen and colleagues’ scenario, an endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations, and host effector control without vertical transmission, as in rhizobia, does not lead to it (Active)
- THEORY-1211 — An endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations (Proposed)

From On Imachi and colleagues’ cultivation of Candidatus Prometheoarchaeum syntrophicum MK-D1, the first Asgard archaeon grown in the laboratory is a small, internally simple coccus with long branching protrusions that degrades amino acids only in syntrophy, handing hydrogen and formate to partners that can be exchanged, and from its genome and other Asgard genomes the authors infer that the Asgard ancestor was likewise dependent on metabolic partners; the isolate is a modern relative separated from the host lineage by about two billion years, grown only in co-culture, and the entangle–engulf–endogenize model built on it is a hypothesis the authors call not definitive

THEORY-1108 — On Imachi and colleagues’ cultivation of Candidatus Prometheoarchaeum syntrophicum MK-D1, the first Asgard archaeon grown in the laboratory is a small, internally simple coccus with long branching protrusions that degrades amino acids only in syntrophy, handing hydrogen and formate to partners that can be exchanged, and from its genome and other Asgard genomes the authors infer that the Asgard ancestor was likewise dependent on metabolic partners; the isolate is a modern relative separated from the host lineage by about two billion years, grown only in co-culture, and the entangle–engulf–endogenize model built on it is a hypothesis the authors call not definitive (Active)
- THEORY-1200 — The Asgard archaeal ancestor of eukaryotes depended on metabolic partners: like Candidatus Prometheoarchaeum syntrophicum, it degraded amino acids in syntrophy, handing hydrogen and formate to partners that removed them (Proposed)

From Santana-Molina, Williams, Snel and Spang find that the last eukaryotic common ancestor’s central carbon metabolism was chimeric: a few Asgard-archaeal enzymes in cytosolic glycolysis and the pentose phosphate pathway, four placed with Asgard archaea at bootstrap 39–97% and two ambiguously, most alphaproteobacterial ones in the mitochondrial TCA cycle and pyruvate conversion, and others from various bacteria with no third dominant donor, since remodelled by loss, horizontal replacement and retargeting so that compartment no longer follows origin; the authors take the cytosol–mitochondrion split as support for syntrophy

THEORY-1109 — Santana-Molina, Williams, Snel and Spang find that the last eukaryotic common ancestor’s central carbon metabolism was chimeric: a few Asgard-archaeal enzymes in cytosolic glycolysis and the pentose phosphate pathway, four placed with Asgard archaea at bootstrap 39–97% and two ambiguously, most alphaproteobacterial ones in the mitochondrial TCA cycle and pyruvate conversion, and others from various bacteria with no third dominant donor, since remodelled by loss, horizontal replacement and retargeting so that compartment no longer follows origin; the authors take the cytosol–mitochondrion split as support for syntrophy (Active)
- THEORY-1205 — Eukaryotes began in a syntrophic partnership: an archaeal host fermented sugars and passed reducing equivalents to an alphaproteobacterial endosymbiont that contributed a TCA cycle and an electron transport chain, which is why the last eukaryotic common ancestor’s central carbon metabolism is archaeal in cytosolic glycolysis and the pentose phosphate pathway and alphaproteobacterial in the mitochondrial TCA cycle (Proposed)

epistemology

From Simon proposes that ill structured problems can be handled by an alternation between a solver working on a problem that is well structured at each moment and a recognition and retrieval system that keeps modifying the problem space from memory or the environment, that a serial solver gains nothing by assembling the problem space at the outset, and that nothing but the size of the knowledge base may separate ill from well structured problems, so that no qualitatively new mechanisms are needed

THEORY-1084 — Simon proposes that ill structured problems can be handled by an alternation between a solver working on a problem that is well structured at each moment and a recognition and retrieval system that keeps modifying the problem space from memory or the environment, that a serial solver gains nothing by assembling the problem space at the outset, and that nothing but the size of the knowledge base may separate ill from well structured problems, so that no qualitatively new mechanisms are needed (Active)
- THEORY-1216 — Ill structured problems can be handled by an alternation between a solver working on a problem that is well structured at each moment and a recognition and retrieval system that keeps modifying the problem space from memory or the environment, a serial solver gains nothing by assembling the problem space at the outset, and nothing but the size of the knowledge base may separate ill from well structured problems, so that no qualitatively new mechanisms are needed (Proposed)

THEORY-327 — Fricker, in her 2003 paper and her 2017 retrospective, holds that testimonial injustice is an ingenuous misjudgement of a speaker’s credibility caused by prejudice, and that deliberate manipulation of credibility is a related but distinct wrong; and she accepts Anderson’s structural form, which is pre-emptive: a speaker’s word is never sought because of a structural prejudice, such as an un-refreshed list, that is no one’s fault (Active)
- THEORY-1173 — Testimonial injustice is an ingenuous misjudgement of a speaker’s credibility caused by prejudice, not a deliberate manipulation of it, and it can be structural and pre-emptive, when a speaker’s word is never sought because of a structural prejudice that is no one’s fault (Proposed)

THEORY-146 — In Holman and Bruner’s bandit model of honest Bayesian researchers whose beliefs funding cannot touch, who differ in method and whose influence and training of newcomers follow productivity, industry funding of those whose methods already favour its drug lowers, in the runs their figures report, the share of the community on the better treatment, and has no effect without methodological diversity; at the one setting reported, an independent funder that counts industry-funded trials toward its merit threshold lowers that share further, and one that excludes them raises it (Active)
- THEORY-1174 — Industry can bias a scientific community toward a worse treatment without corrupting any individual, by selectively funding researchers whose methods already favour its product, provided researchers differ in method and both influence and the training of newcomers follow productivity; an independent funder that discounts industry-funded work counteracts this, and one that funds on productivity alone makes it worse (Proposed)

From When a group learns only about the options its members pursue and good evidence is scarce, sparser communication makes it more likely to settle on the better option, at a large cost in speed; when evidence is plentiful, sparsity only slows learning

THEORY-143 — When a group learns only about the options its members pursue and good evidence is scarce, sparser communication makes it more likely to settle on the better option, at a large cost in speed; when evidence is plentiful, sparsity only slows learning (Proposed)
- THEORY-148 — A group learning by trying needs diversity in what its members pursue that lasts long enough to survive misleading early evidence and then gives way to consensus; limited communication and stubborn priors each supply it, and together they make it permanent, so the group never settles (Proposed)
- THEORY-422 — In the standard formal models of group learning, no communication network is best for every epistemic problem: costly transmission favours minimally connected or star networks, pooling of estimates by averaging favours regular networks, and learning by trying in bandit problems can favour sparse ones, so a group’s network cannot be judged until its learning problem is known (Active)

From In Rasooly and Rozzi’s pre-registered randomised experiment on 817 Manifold play-money markets, about 59% of a ±5-point price shock that no trader’s information supported was still present after 60 days in unresolved markets, with about 25% undone in the first week and reversion slowing after it; their model of finitely many risk-averse traders gives such partial, slowing reversion even with no learning, and the data cannot tell that from learning

THEORY-267 — In Rasooly and Rozzi’s pre-registered randomised experiment on 817 Manifold play-money markets, about 59% of a ±5-point price shock that no trader’s information supported was still present after 60 days in unresolved markets, with about 25% undone in the first week and reversion slowing after it; their model of finitely many risk-averse traders gives such partial, slowing reversion even with no learning, and the data cannot tell that from learning (Active)
- THEORY-810 — Prediction-market prices keep much of an arbitrary price shock that no trader’s information supports for weeks, rather than correcting it within a short transition phase (Proposed)

From In a community whose consensus model is replaced whenever a single sound pairwise model comparison favours the challenger, which models get proposed can hold the consensus on a false model while results reproduce almost perfectly, so reproducibility need not track convergence to truth

THEORY-295 — In a community whose consensus model is replaced whenever a single sound pairwise model comparison favours the challenger, which models get proposed can hold the consensus on a false model while results reproduce almost perfectly, so reproducibility need not track convergence to truth (Active)
- THEORY-814 — In scientific communities that settle on a consensus model by sound model comparisons, high reproducibility of results is not evidence that the consensus is true, because which alternatives get proposed can hold a false consensus whose results reproduce almost perfectly (Proposed)

From For a logically closed theory, the contractions satisfying the six basic AGM postulates are exactly the partial meet contractions, and those satisfying the two supplementary postulates as well are exactly the ones whose selection of remainders is generated by a transitive relation; the theory alone does not fix the contraction, and the results concern one change of one fixed theory

THEORY-353 — For a logically closed theory, the contractions satisfying the six basic AGM postulates are exactly the partial meet contractions, and those satisfying the two supplementary postulates as well are exactly the ones whose selection of remainders is generated by a transitive relation; the theory alone does not fix the contraction, and the results concern one change of one fixed theory (Active)
- THEORY-347 — Belief bases with the same logical closure can revise to opposite conclusions on the same input, so a closed belief set does not fix its own revision; partial meet contraction of a base need not satisfy Recovery, and bases keep distinct inconsistent states apart, while a belief-set model reproduces the base difference only by attaching a different selection mechanism to the same set (Active)

From In Hrbacek’s relative analysis, if belief at a context level is credence ultraclose to 1 at that level, the conjunction of any standard number of believed propositions is believed, while for an ultralarge number it can fail: in a fair lottery with an ultralarge number of tickets each ‘this ticket loses’ is believed and their conjunction has probability 0

THEORY-829 — In Hrbacek’s relative analysis, if belief at a context level is credence ultraclose to 1 at that level, the conjunction of any standard number of believed propositions is believed, while for an ultralarge number it can fail: in a fair lottery with an ultralarge number of tickets each ‘this ticket loses’ is believed and their conjunction has probability 0 (Active)
- THEORY-647 — If outright belief is credence ultraclose to 1 relative to a context level, conjunction holds for any standard (‘few’) number of believed propositions and fails for an ultralarge number, which dissolves the Lottery Paradox as an accumulation of negligible rounding errors (Proposed)

ethics

THEORY-327 — Fricker, in her 2003 paper and her 2017 retrospective, holds that testimonial injustice is an ingenuous misjudgement of a speaker’s credibility caused by prejudice, and that deliberate manipulation of credibility is a related but distinct wrong; and she accepts Anderson’s structural form, which is pre-emptive: a speaker’s word is never sought because of a structural prejudice, such as an un-refreshed list, that is no one’s fault (Active)
- THEORY-1173 — Testimonial injustice is an ingenuous misjudgement of a speaker’s credibility caused by prejudice, not a deliberate manipulation of it, and it can be structural and pre-emptive, when a speaker’s word is never sought because of a structural prejudice that is no one’s fault (Proposed)

From Moral responsibility requires the normative competence to recognise and act on the True and the Good, which identification with one’s own motives does not supply

THEORY-122 — Moral responsibility requires the normative competence to recognise and act on the True and the Good, which identification with one’s own motives does not supply (Proposed)
- THEORY-118 — Praiseworthiness is compatible with being psychologically determined by good reasons, but blameworthiness requires that the agent could have done otherwise, that is, could have acted on the right reasons (Proposed)

From An agent can be morally responsible for what she did although she could not have done otherwise, when what made the action unavoidable played no part in bringing it about, so the principle of alternate possibilities is false

THEORY-119 — An agent can be morally responsible for what she did although she could not have done otherwise, when what made the action unavoidable played no part in bringing it about, so the principle of alternate possibilities is false (Proposed)
- THEORY-121 — Moral responsibility requires guidance control, moderate reasons-responsiveness of the agent’s own actual-sequence mechanism, and not regulative control over alternative possibilities, so responsibility is compatible with causal determinism (Active)

free-will

From A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape

THEORY-040 — A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape (Proposed)
- THEORY-1118 — McKenna and Coates argue, with Dryden and with Buss and Westlund, that a condition on free agency stated only in the agent’s present psychological structure is met by an agent whose structure a manipulator installed, so a mesh account must count that agent free or add a historical condition; Frankfurt, read first-hand, takes the first option (Active)

From Moral responsibility requires the normative competence to recognise and act on the True and the Good, which identification with one’s own motives does not supply

THEORY-122 — Moral responsibility requires the normative competence to recognise and act on the True and the Good, which identification with one’s own motives does not supply (Proposed)
- THEORY-118 — Praiseworthiness is compatible with being psychologically determined by good reasons, but blameworthiness requires that the agent could have done otherwise, that is, could have acted on the right reasons (Proposed)

From An agent can be morally responsible for what she did although she could not have done otherwise, when what made the action unavoidable played no part in bringing it about, so the principle of alternate possibilities is false

THEORY-119 — An agent can be morally responsible for what she did although she could not have done otherwise, when what made the action unavoidable played no part in bringing it about, so the principle of alternate possibilities is false (Proposed)
- THEORY-121 — Moral responsibility requires guidance control, moderate reasons-responsiveness of the agent’s own actual-sequence mechanism, and not regulative control over alternative possibilities, so responsibility is compatible with causal determinism (Active)

game-theory

From Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one

THEORY-067 — Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one (Proposed)
- THEORY-1117 — Ainslie proposes that self-control over time needs no faculty of will: hyperbolic discounting makes preferences that dominate at different delays, these grow into interests that bargain with no governor above them, and resolve is the recursive staking of expected future choices on the present one (Active)

From In empirical games built from about a thousand sampled search agents in eight OpenSpiel board games, most on 3×3 or 4×4 boards, and in the AlphaStar league, Nash clusters taken from maximum-entropy equilibria are, read against a fitted skew-normal curve with no test, largest at intermediate transitive strength and narrower toward the strongest and the weakest strategies, while Blotto, the disc game, Kuhn poker and noisy Elo games do not show that profile

THEORY-930 — In empirical games built from about a thousand sampled search agents in eight OpenSpiel board games, most on 3×3 or 4×4 boards, and in the AlphaStar league, Nash clusters taken from maximum-entropy equilibria are, read against a fitted skew-normal curve with no test, largest at intermediate transitive strength and narrower toward the strongest and the weakest strategies, while Blotto, the disc game, Kuhn poker and noisy Elo games do not show that profile (Active)
- THEORY-1199 — Real-world games of skill have a spinning-top geometry: their non-transitive part, measured as Nash cluster size, is largest at intermediate strength and narrows toward the strongest and the weakest strategies (Proposed)

From When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; Yang et al. state that with homogeneous agents mean field Q-learning converges to the Nash Q-value under three assumptions, one of which they call a strong constraint, by a proof whose contraction step needs a temperature above K/2 while its greedy-in-the-limit assumption needs the temperature to fall to zero; in their Gaussian Squeeze runs with 500 and 1000 agents, mean field Q-learning and actor-critic learned the optimal allocation where independent learners, FMQ, Rec-FMQ and a MADDPG variant failed

THEORY-945 — When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; Yang et al. state that with homogeneous agents mean field Q-learning converges to the Nash Q-value under three assumptions, one of which they call a strong constraint, by a proof whose contraction step needs a temperature above K/2 while its greedy-in-the-limit assumption needs the temperature to fall to zero; in their Gaussian Squeeze runs with 500 and 1000 agents, mean field Q-learning and actor-critic learned the optimal allocation where independent learners, FMQ, Rec-FMQ and a MADDPG variant failed (Active)
- THEORY-1231 — When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; with homogeneous agents, mean field Q-learning converges to the Nash Q-value under assumptions the authors call strong, and learned tasks of 500 and 1000 agents where independent and centralized-critic baselines failed (Proposed)

From The α-Rank chain over pure strategy profiles, built from Fermi-imitation fixation probabilities in finite populations with rare mutation, has a unique stationary distribution at every finite ranking intensity; as the intensity grows without bound its transition matrix converges to that of a chain that moves only along weakly better unilateral deviations, whose closed classes are the sink strongly connected components of the game’s weakly-better-response graph, on each of which it is a Markov-Conley chain, and each of which lies inside a single chain component of the replicator flow

THEORY-942 — The α-Rank chain over pure strategy profiles, built from Fermi-imitation fixation probabilities in finite populations with rare mutation, has a unique stationary distribution at every finite ranking intensity; as the intensity grows without bound its transition matrix converges to that of a chain that moves only along weakly better unilateral deviations, whose closed classes are the sink strongly connected components of the game’s weakly-better-response graph, on each of which it is a Markov-Conley chain, and each of which lies inside a single chain component of the replicator flow (Active)
- THEORY-1233 — For the replicator flow of a finite game, every asymptotically stable sink chain component contains at least one sink strongly connected component of the weakly-better-response graph, a Markov-Conley chain (Proposed)

From The replicator-plus-entropy model of Boltzmann Q-learning in two-player stateless games is, in the small-step limit, the expected dynamics of a learner that updates every action’s value each step, and so of frequency-adjusted Q-learning wherever every action’s probability is at least its adjustment parameter; it is not that of Q-learning, whose expected update carries an extra factor of each action’s selection probability, so that its policy dynamics depend on the Q-values; in Kaisers and Tuyls’s three 2×2 games at one temperature, compared by eye, Q-learning’s trajectories departed from the model depending on initialization and the frequency-adjusted learner’s followed it

THEORY-932 — The replicator-plus-entropy model of Boltzmann Q-learning in two-player stateless games is, in the small-step limit, the expected dynamics of a learner that updates every action’s value each step, and so of frequency-adjusted Q-learning wherever every action’s probability is at least its adjustment parameter; it is not that of Q-learning, whose expected update carries an extra factor of each action’s selection probability, so that its policy dynamics depend on the Q-values; in Kaisers and Tuyls’s three 2×2 games at one temperature, compared by eye, Q-learning’s trajectories departed from the model depending on initialization and the frequency-adjusted learner’s followed it (Active)
- THEORY-1234 — The replicator-plus-entropy model of Boltzmann Q-learning in two-player stateless games is exact for frequency-adjusted Q-learning, not for Q-learning, whose expected update carries an extra factor of each action’s selection probability and whose trajectories in 2×2 games depart from the model depending on initialization (Proposed)

From In a finite partial-monitoring game, the signals of a set D of actions determine the expected loss difference of actions i and j under every outcome distribution iff ℓ_i − ℓ_j lies in the sum of the row spaces of their signal matrices; when it does not, two outcome distributions differing along a vector in the common kernel give every action in D the same signal law and reverse the preference between i and j, so no amount of play restricted to D can learn the comparison

THEORY-962 — In a finite partial-monitoring game, the signals of a set D of actions determine the expected loss difference of actions i and j under every outcome distribution iff ℓ_i − ℓ_j lies in the sum of the row spaces of their signal matrices; when it does not, two outcome distributions differing along a vector in the common kernel give every action in D the same signal law and reverse the preference between i and j, so no amount of play restricted to D can learn the comparison (Active)
- THEORY-961 — In a finite partial-monitoring game, an action optimal wherever two neighbouring actions tie has a loss vector on the segment between theirs, so a degenerate or duplicate action never creates a neighbouring pair or decides the trivial class, yet its signals count toward the pair’s local observability; and the T^2/3 lower bound for a pair that is not locally observable needs the separating direction in the kernel of the signal matrices of all such actions, not only of the two neighbours (Active)

individuation

From A Markov-blanket partition is defined relative to a chosen internal set, so a graph has one around almost any set of nodes, and the formalism alone does not say which set is the system

THEORY-066 — A Markov-blanket partition is defined relative to a chosen internal set, so a graph has one around almost any set of nodes, and the formalism alone does not say which set is the system (Active)
- THEORY-073 — In Friston’s 2013 formulation and the applications the record holds, a Markov blanket does not individuate a system: where it falls is fixed by modelling choices made before it is traced (the coupling graph, the internal set and the time at which steady state is assumed), and a blanket read from a fixed graph is the same whether the system produced the boundary or it was imposed (Active)
  - THEORY-1097 — A Markov blanket, in any formulation including sparse coupling at nonequilibrium steady state and time-varying blanket states, does not individuate a system: where it falls is fixed by modelling choices made before it is found, so it presupposes the boundary it is used to find, and it cannot represent a boundary the system produces (Proposed)

From Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency

THEORY-071 — Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency (Proposed)
- THEORY-1119 — Barandiaran, Di Paolo and Rohde propose that minimal agency requires individuality, interactional asymmetry and normativity set by the system’s own organisation, each necessary and together, they say, sufficient; that individuality without the other two is not agency; and that a system optimizing an externally fixed function is not a model of agency (Active)

From In Athreya, Czuppon and Gokhale’s adaptive-dynamics model of an incipient endosymbiosis, mutual dependence (the product of the partners’ obligacies) evolves ahead of reproductive cohesion (the product of their adhesions) early on, but only under the mapping in which each partner’s free-living growth depends on its own obligacy alone while dissociation depends on both partners’ adhesions; the authors’ own appendix shows the bias vanishing under matched mappings and reversing under the opposite one, and their survey of real symbioses cannot test it

THEORY-1050 — In Athreya, Czuppon and Gokhale’s adaptive-dynamics model of an incipient endosymbiosis, mutual dependence (the product of the partners’ obligacies) evolves ahead of reproductive cohesion (the product of their adhesions) early on, but only under the mapping in which each partner’s free-living growth depends on its own obligacy alone while dissociation depends on both partners’ adhesions; the authors’ own appendix shows the bias vanishing under matched mappings and reversing under the opposite one, and their survey of real symbioses cannot test it (Active)
- THEORY-1243 — In incipient endosymbioses in nature, mutual dependence evolves ahead of reproductive cohesion, so that endosymbioses are more likely to be mutually dependent than cohesive (Proposed)

From On Sørensen and colleagues’ scenario, an endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations, and host effector control without vertical transmission, as in rhizobia, does not lead to it

THEORY-1107 — On Sørensen and colleagues’ scenario, an endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations, and host effector control without vertical transmission, as in rhizobia, does not lead to it (Active)
- THEORY-1211 — An endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations (Proposed)

From Griesemer and Shavit propose, as a first and rough sketch, that a collective becomes a new level of individuality only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it, some core interactions have become generatively entrenched, and it has organizational closure, and that such a community can be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone

THEORY-1247 — Griesemer and Shavit propose, as a first and rough sketch, that a collective becomes a new level of individuality only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it, some core interactions have become generatively entrenched, and it has organizational closure, and that such a community can be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone (Active)
- THEORY-418 — A collective becomes a new level of individuality, on this sketch, only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it and some core interactions have become generatively entrenched and, as the paper asserts without deriving it from entrenchment, it has organizational closure; such a community can then be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone (Proposed)

From A structured group is not identical to its members: two groups of one kind can have the same members at every time and still be two, which theories that identify a group with a plurality, fusion, aggregate or set of its members cannot allow, and a view of groups as realizations of structures with functionally defined nodes can

THEORY-549 — A structured group is not identical to its members: two groups of one kind can have the same members at every time and still be two, which theories that identify a group with a plurality, fusion, aggregate or set of its members cannot allow, and a view of groups as realizations of structures with functionally defined nodes can (Proposed)
- THEORY-575 — ‘Social group’ names two kinds of thing: organized social groups, such as teams, courts and families, are structured wholes whose people occupy the nodes of an internal social structure, while feature social groups, such as women, racial groups and classes, are social kinds, nodes in external social structures, and are not concrete wholes (Proposed)

information-geometry

From For softmax cross-entropy with balanced classes, the Gauss–Newton matrix splits exactly into four positive semidefinite second-moment terms, the class-centre term has rank at most C with eigenvectors in the span of the class centres and the spectrum of a C × C Gram matrix, and so the k-th eigenvalue of the whole is never below the k-th eigenvalue of the class-centre term

THEORY-731 — For softmax cross-entropy with balanced classes, the Gauss–Newton matrix splits exactly into four positive semidefinite second-moment terms, the class-centre term has rank at most C with eigenvectors in the span of the class centres and the spectrum of a C × C Gram matrix, and so the k-th eigenvalue of the whole is never below the k-th eigenvalue of the class-centre term (Active)
- THEORY-078 — The top C eigenvalues of a trained C-class classifier’s Fisher are carried by the second moment of its class-mean logit derivatives, so a cut at the bulk edge keeps the span of the class means, and they stand clear of the bulk to the extent between-class separation exceeds within-class variation (Proposed)
  - THEORY-085 — A classifier’s class-driven Fisher outliers pair each class’s feature mean with that class’s own error mean, so K-FAC’s single Kronecker product of class-averaged factors adds every cross-class pairing and misplaces them (Proposed)

information-retrieval

From In one study of a stack of proximity graphs on nested random subsets of the data, each element’s top layer drawn geometrically, greedy search on uniform random data in 8 dimensions needed a number of distance computations that grew no worse than logarithmically with the number of items and stayed below a single-layer navigable graph’s at every size tested, and in 4 dimensions the beam width needed for a fixed recall saturated; the logarithmic bound is derived only for exact Delaunay graphs of bounded degree, which the index does not build, and on 200 million SIFT vectors the measured growth departs from it

THEORY-365 — In one study of a stack of proximity graphs on nested random subsets of the data, each element’s top layer drawn geometrically, greedy search on uniform random data in 8 dimensions needed a number of distance computations that grew no worse than logarithmically with the number of items and stayed below a single-layer navigable graph’s at every size tested, and in 4 dimensions the beam width needed for a fixed recall saturated; the logarithmic bound is derived only for exact Delaunay graphs of bounded degree, which the index does not build, and on 200 million SIFT vectors the measured growth departs from it (Active)
- THEORY-815 — Layering a navigable small-world graph on nested random subsets of the data, with links separated by length scale and each layer’s degree capped, makes greedy nearest-neighbour search cost grow logarithmically with the number of items on data of low intrinsic dimension, where the single-layer graph’s cost grows polylogarithmically (Proposed)

From Locality-sensitive hashing answers approximate near-neighbour queries over n points with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, where ρ = ln(1/p1)/ln(1/p2) is set by how much more often near pairs collide than far ones and is at most 1/(1+ε) for approximation 1+ε in Hamming space, with constant success probability per query; the paper gives upper bounds only and proves no limit on what organizing can buy

THEORY-388 — Locality-sensitive hashing answers approximate near-neighbour queries over n points with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, where ρ = ln(1/p1)/ln(1/p2) is set by how much more often near pairs collide than far ones and is at most 1/(1+ε) for approximation 1+ε in Hamming space, with constant success probability per query; the paper gives upper bounds only and proves no limit on what organizing can buy (Active)
- THEORY-372 — For c-approximate near-neighbour search over n points in Euclidean space, space n^{1+ρu+o(1)} and query time n^{ρq+o(1)} are achievable on any data set whenever c²√ρq + (c²−1)√ρu ≥ √(2c²−1) (c√ρq + (c−1)√ρu ≥ √(2c−1) in Hamming space), and no list-of-points structure, the model of known data-independent hashing, does better on random instances; without that model the curve is matched only at query time n^{o(1)} by one or two cell probes (Active)

From Fagin’s Algorithm finds the top k of N objects held in m attribute-sorted lists under any monotone combining function, and when the lists’ orderings are independent with at least m − 1 of them uniformly random it does so with O(N^((m−1)/m) k^(1/m)) sorted and random accesses with arbitrarily high probability; with dependent orderings the bound can fail

THEORY-379 — Fagin’s Algorithm finds the top k of N objects held in m attribute-sorted lists under any monotone combining function, and when the lists’ orderings are independent with at least m − 1 of them uniformly random it does so with O(N^((m−1)/m) k^(1/m)) sorted and random accesses with arbitrarily high probability; with dependent orderings the bound can fail (Active)
- THEORY-832 — For strict combining functions and m attribute-sorted lists of N objects whose orderings are probabilistically independent, no correct top-k algorithm does essentially better than Fagin’s Algorithm in the worst case with high probability: every one needs on the order of N^((m−1)/m) k^(1/m) accesses (Proposed)

information-theory

From For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient

THEORY-026 — For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient (Active)
- THEORY-817 — Once an agent’s actions affect the percepts it receives, maximal prediction of those percepts is generally not what thermodynamic efficiency requires: in percept-action loops with feedback, maximally predictive agents generally extract less work than the most efficient agent (Proposed)

From In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet

THEORY-083 — In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet (Active)
- THEORY-788 — Training a deep classifier past zero training error generally collapses its last-layer features onto their class means, drives the centred means toward a simplex equiangular tight frame aligned with the classifier, and makes the decision nearest class mean, as Papyan, Han and Donoho propose from their runs (Proposed)

From For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix

THEORY-717 — For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix (Active)
- THEORY-156 — For experiments on a finite parameter set whose outcome spaces are standard Borel, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation (Active)
  - THEORY-1096 — Blackwell’s comparison of experiments, its easy direction: an experiment is a family of distributions indexed by a common parameter, and if one experiment is a garbling of another by a Markov kernel, every risk attainable with the garbling in any decision problem with bounded losses is attainable with the original, on any outcome spaces (Active)
  - THEORY-1177 — For experiments on a finite parameter set, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation (Proposed)

From With uniform interventions imposed separately at each level, a coarse-grained macro description of a known discrete mechanism can have more effective information than the micro description it supervenes on, so the level at which effective information is highest can be a macro level

THEORY-222 — With uniform interventions imposed separately at each level, a coarse-grained macro description of a known discrete mechanism can have more effective information than the micro description it supervenes on, so the level at which effective information is highest can be a macro level (Active)
- THEORY-787 — A system’s causal structure can be most effective at a macro scale, so that causation emerges at a coarse-grained level and the macro description is the one at which the system’s causes are found, as Hoel, Albantakis and Tononi propose from effective information (Rejected)
  - THEORY-1167 — When one intervention distribution over micro states is used at both levels, the effective information of a coarse-graining of a discrete mechanism, with each macro intervention realized by that distribution within its block, cannot exceed the micro effective information; so a positive causal emergence of effective information, scored with a uniform distribution at each level, equals the micro gain from changing the distribution minus a non-negative loss, and is a gain from the change of distribution (Active) — also carries further THEORY-222
    - THEORY-1169 — For a discrete mechanism scored with one full-support intervention distribution over micro causes at both levels, a coarse-graining of cause and effect keeps all of the micro effective information if and only if it merges only causes with identical rows P(Y | do(x)) and only effects with proportional columns, whatever the distribution; so the coarsest coarse-graining that ties the micro is the causal partition of causal feature learning, with effects grouped by normalized likelihood, and any merge of causes with different effects loses information under every such distribution (Active)

From Evaluating expected free energy recursively over the beliefs an agent would hold after each plausible outcome makes active inference a pruned tree search over belief states, formally analogous to Bellman recursion; in single deterministic runs on a T-maze and an 8 × 8 maze it planned through short-term costs that agents with a shorter horizon did not; that the pruning keeps deep search tractable beyond toy problems is argued, not shown

THEORY-426 — Evaluating expected free energy recursively over the beliefs an agent would hold after each plausible outcome makes active inference a pruned tree search over belief states, formally analogous to Bellman recursion; in single deterministic runs on a T-maze and an 8 × 8 maze it planned through short-term costs that agents with a shorter horizon did not; that the pruning keeps deep search tractable beyond toy problems is argued, not shown (Active)
- THEORY-1245 — Evaluating expected free energy recursively over the beliefs an agent would hold after each plausible outcome makes active inference a pruned tree search over belief states, formally analogous to Bellman recursion, which lets the agent plan through short-term costs that one-step agents cannot; that the pruning keeps deep search tractable beyond toy problems is argued, not shown (Proposed)

From In Zaslavsky, Kemp, Regier and Tishby’s analysis, the colour-naming systems of the World Color Survey languages and English lie near the information-bottleneck bound for compressing perceptual meanings into words, at a trade-off parameter just above 1 and nearer than hue-rotated variants of themselves, with the need distribution over meanings estimated from the naming data

THEORY-163 — In Zaslavsky, Kemp, Regier and Tishby’s analysis, the colour-naming systems of the World Color Survey languages and English lie near the information-bottleneck bound for compressing perceptual meanings into words, at a trade-off parameter just above 1 and nearer than hue-rotated variants of themselves, with the need distribution over meanings estimated from the naming data (Active)
- THEORY-791 — The colour-naming systems of the world’s languages lie near the information-bottleneck bound for compressing perceptual meanings into words, and differences in one trade-off parameter place each language along it, as Zaslavsky, Kemp, Regier and Tishby propose (Proposed)

From On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers’ low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead

THEORY-194 — On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers’ low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead (Active)
- THEORY-789 — The block coarse-graining that maximizes mutual information with the system beyond a buffer picks out the renormalization-group-relevant variables in general, so those variables can be found from samples by information alone, without being known beforehand, as Koch-Janusz and Ringel propose (Proposed)

From On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses

THEORY-204 — On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses (Active)
- THEORY-784 — Language models trained on limited data show a test loss falling as P^(−γ/(2β)) because P training tokens resolve token correlations only out to a horizon growing as P^(1/(2β)), so that more data mainly lengthens the context a model can use, when the model learns within that horizon faster than it grows, as Cagnetta, Raventós, Ganguli and Wyart propose (Proposed)

From For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose’s invariant subspaces, which exist on open sets but not generically at every scale

THEORY-733 — For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose’s invariant subspaces, which exist on open sets but not generically at every scale (Active)
- THEORY-257 — Dynamical independence, zero information flow from the microscopic history into a coarse-graining given its own history, fixes emergent macroscopic variables by the dynamics rather than by a chosen query, but for linear VAR models of order above 1 generically none is exact, and the optima are many (Proposed)

From In Goldwasser, Gruber, Kalai and Paradise’s common-nonsense model, where the source language and the plausibility prior are uniform over sets from which shared random nonsense is removed and translators are one-to-one, maximum likelihood with the prior alone translates without parallel data with error O(log|Θ| / (α·min(m, |T|))) with high probability, falling in the shared nonsense α, in the size of the source language and in the number of samples

THEORY-270 — In Goldwasser, Gruber, Kalai and Paradise’s common-nonsense model, where the source language and the plausibility prior are uniform over sets from which shared random nonsense is removed and translators are one-to-one, maximum likelihood with the prior alone translates without parallel data with error O(log|Θ| / (α·min(m, |T|))) with high probability, falling in the shared nonsense α, in the size of the source language and in the number of samples (Active)
- THEORY-806 — Translation without parallel data, as of an unknown animal communication system into a human language, is learnable from a plausibility prior over the target language alone, with error falling as the common ground between the languages grows and as the source language grows richer (Proposed)

From For an optimal capacity-bounded learner on an orthogonal mixture of equally frequent random facts and a second domain, the facts are learned not at all below a capacity threshold set by the mixing ratio and in full once capacity exceeds it by the facts’ total entropy, and Pythia models trained on synthetic biographies mixed into web text show a threshold in model size and in mixing ratio

THEORY-293 — For an optimal capacity-bounded learner on an orthogonal mixture of equally frequent random facts and a second domain, the facts are learned not at all below a capacity threshold set by the mixing ratio and in full once capacity exceeds it by the facts’ total entropy, and Pythia models trained on synthetic biographies mixed into web text show a threshold in model size and in mixing ratio (Active)
- THEORY-809 — Language models trained by gradient descent on a mixture of random facts and other data reach the capacity-optimal allocation, so the model size and the mixing ratio at which they begin to learn the facts are those of an optimal capacity-bounded learner (Proposed)

From With a kernel normalised to k(x,x) = 1, the covariance operator of a distribution is a density operator, and its quantum relative entropy is a lower bound on the Kullback–Leibler divergence that is tight for every pair of distributions only for an orthonormal, hence discrete, embedding, lies above the KL of the kernel-smoothed distributions, and on the torus, with Fourier coefficients proportional to e^(−σ‖ω‖₁) and strictly positive Lipschitz densities, is within O(σ²) of KL as the bandwidth σ shrinks

THEORY-313 — With a kernel normalised to k(x,x) = 1, the covariance operator of a distribution is a density operator, and its quantum relative entropy is a lower bound on the Kullback–Leibler divergence that is tight for every pair of distributions only for an orthonormal, hence discrete, embedding, lies above the KL of the kernel-smoothed distributions, and on the torus, with Fourier coefficients proportional to e^(−σ‖ω‖₁) and strictly positive Lipschitz densities, is within O(σ²) of KL as the bandwidth σ shrinks (Active)
- THEORY-811 — With a kernel normalised to unit diagonal, the kernel quantum relative entropy of two distributions is within O(σ²) of their Kullback–Leibler divergence as the bandwidth σ shrinks, for translation-invariant kernels on the torus generally and for kernels on other compact groups (Proposed)

From For jointly Gaussian stationary processes, Granger causality in Geweke’s generalized-variance form is exactly twice the transfer entropy, conditional and multivariate, at the regressions’ lags, because least-squares residual covariance is the partial covariance and a Gaussian’s conditional entropy is half its log-determinant plus a constant

THEORY-700 — For jointly Gaussian stationary processes, Granger causality in Geweke’s generalized-variance form is exactly twice the transfer entropy, conditional and multivariate, at the regressions’ lags, because least-squares residual covariance is the partial covariance and a Gaussian’s conditional entropy is half its log-determinant plus a constant (Active)
- THEORY-641 — Under a well-specified, identifiable parametric Markov model of the target given the joint past, with ergodic lagged states, transfer entropy is the limit of the normalised log-likelihood ratio for the null of no dependence on the source’s past, so it is Granger-sense predictive improvement; for finite-state chains the plug-in estimator is exactly that statistic, and under the null and the usual regularity it is asymptotically χ² with the difference in parameter counts as degrees of freedom (Active)

From Ince’s I_ccs, the expected local co-information over only those events where every source, the joint source and the overlap change the target’s surprisal in the same direction, under the maximum-entropy joint that preserves the source–target and joint-source marginals, is symmetric and self-redundant and satisfies the independent identity property; it fails Harder et al.’s identity and monotonicity, so its atoms can be negative, and, though its paper says otherwise, it fails subset equality

THEORY-714 — Ince’s I_ccs, the expected local co-information over only those events where every source, the joint source and the overlap change the target’s surprisal in the same direction, under the maximum-entropy joint that preserves the source–target and joint-source marginals, is symmetric and self-redundant and satisfies the independent identity property; it fails Harder et al.’s identity and monotonicity, so its atoms can be negative, and, though its paper says otherwise, it fails subset equality (Active)
- THEORY-804 — Ince’s common change in surprisal is a correct measure of the redundancy among any number of sources, and the negative partial-information atoms it gives are local misinformation and mechanistic redundancy, not defects of the measure (Proposed)

learning-and-conditioning

From As reviewed by Gerstner et al. (2018) from five papers, mostly in slices and one laboratory per region, pairing pre- and postsynaptic activity changed the synapse when a neuromodulator or plateau potential followed within about 1 s in striatum, within 5 s (noradrenaline, LTP after pre-before-post pairing) or 2.5 s (serotonin, LTD after post-before-pre pairing) in visual and prefrontal cortex, and within about 2 s in CA1, and not at longer delays; the reviewers read this as a decaying synaptic eligibility trace and as experimental support for three-factor learning rules on behavioural time scales

THEORY-909 — As reviewed by Gerstner et al. (2018) from five papers, mostly in slices and one laboratory per region, pairing pre- and postsynaptic activity changed the synapse when a neuromodulator or plateau potential followed within about 1 s in striatum, within 5 s (noradrenaline, LTP after pre-before-post pairing) or 2.5 s (serotonin, LTD after post-before-pre pairing) in visual and prefrontal cortex, and within about 2 s in CA1, and not at longer delays; the reviewers read this as a decaying synaptic eligibility trace and as experimental support for three-factor learning rules on behavioural time scales (Active)
- THEORY-1178 — In slices of striatum, visual and prefrontal cortex and hippocampus, Hebbian co-activation leaves a synaptic eligibility trace, lasting about 1 s in striatum, 3 to 10 s in cortex and about 2 s in CA1, that a later neuromodulator or plateau potential converts into a weight change and whose sign it can set: experimental support for three-factor learning rules on behavioural time scales (Proposed)

From As Chou, Wilson and Smith review them, in about six foraging studies, mostly sub-clinical, higher stress or anxiety went with less exploration, and in one task with a predator to check for (Trier et al. 2023), people higher in trait somatic anxiety checked less and did worse, a reduction the study’s further analyses associated with greater aversion to detecting the predator

THEORY-907 — As Chou, Wilson and Smith review them, in about six foraging studies, mostly sub-clinical, higher stress or anxiety went with less exploration, and in one task with a predator to check for (Trier et al. 2023), people higher in trait somatic anxiety checked less and did worse, a reduction the study’s further analyses associated with greater aversion to detecting the predator (Active)
- THEORY-1180 — In foraging tasks, higher stress or anxiety goes consistently with less exploration, and in one task with a predator to check for, people higher in somatic anxiety checked less and did worse, a reduction associated with greater aversion to detecting the predator (Proposed)

From In Brown, Price and Dombrovski’s reading of seven computational learning studies they selected without a systematic search, with mixed directions, anxiety goes with altered adjustment of learning rate to uncertainty, mostly for aversive outcomes, while overall learning rate, choice consistency and outcome valuation show little relation to it

THEORY-897 — In Brown, Price and Dombrovski’s reading of seven computational learning studies they selected without a systematic search, with mixed directions, anxiety goes with altered adjustment of learning rate to uncertainty, mostly for aversive outcomes, while overall learning rate, choice consistency and outcome valuation show little relation to it (Active)
- THEORY-1182 — Across seven computational learning studies selected without a systematic search, anxiety goes with altered adjustment of learning rate to uncertainty, mostly for aversive outcomes, while overall learning rate, choice consistency and outcome valuation show little relation to it (Proposed)

From As Mancinelli and Bach’s systematic review reports it, rating it B, in three laboratory studies of healthy humans a signalled, costless avoidance response offered for a stimulus never paired with the aversive outcome was used, and withdrawing it raised expectancy of the outcome for that stimulus, though Xia et al. (2019) found no change in skin conductance or startle; every study signalled the response’s availability, which may itself have read as a danger cue

THEORY-920 — As Mancinelli and Bach’s systematic review reports it, rating it B, in three laboratory studies of healthy humans a signalled, costless avoidance response offered for a stimulus never paired with the aversive outcome was used, and withdrawing it raised expectancy of the outcome for that stimulus, though Xia et al. (2019) found no change in skin conductance or startle; every study signalled the response’s availability, which may itself have read as a danger cue (Active)
- THEORY-1184 — In three laboratory studies of healthy humans, offering a costless avoidance response for a stimulus never paired with the aversive outcome led people to use it, and withdrawing the response then raised their expectancy of the outcome for that stimulus; signalling of the response is a confound (Proposed)

From Chou, Wilson and Smith read about eight computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples, with mixed results and none replicated, as suggesting that directed exploration rises with trait or cognitive anxiety and falls with state or somatic anxiety; none they review links random exploration to anxiety

THEORY-917 — Chou, Wilson and Smith read about eight computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples, with mixed results and none replicated, as suggesting that directed exploration rises with trait or cognitive anxiety and falls with state or somatic anxiety; none they review links random exploration to anxiety (Active)
- THEORY-1186 — Across computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples and not yet replicated, directed exploration tends to rise with trait or cognitive anxiety and to fall with state or somatic anxiety (Proposed)

From As Mancinelli and Bach’s systematic review of avoidance learning in healthy humans reports it, rating both A, an avoidance response left available during extinction persists, in some participants for hundreds of unpunished trials, and keeps expectancy of the aversive outcome higher than in people denied it (nine studies); and less effective or costlier responses extinguish faster (six studies, one exception)

THEORY-901 — As Mancinelli and Bach’s systematic review of avoidance learning in healthy humans reports it, rating both A, an avoidance response left available during extinction persists, in some participants for hundreds of unpunished trials, and keeps expectancy of the aversive outcome higher than in people denied it (nine studies); and less effective or costlier responses extinguish faster (six studies, one exception) (Active)
- THEORY-1189 — In laboratory avoidance learning in healthy humans, an avoidance response that remains available during extinction persists, in some participants for hundreds of unpunished trials, and keeps expectancy of the aversive outcome higher than in people denied it; the more effective and the cheaper the response, the slower it extinguishes (Proposed)

From In Zorowitz, Momennejad and Daw’s simulations of a Markov decision process, an agent that evaluates actions as if it will fail to avoid danger at later steps, while in fact choosing optimally at each step, spreads threat value to states far from the danger and avoids early and widely, which reproduces qualitatively exaggerated threat appraisal, fear generalization and several laboratory findings associated with anxiety

THEORY-915 — In Zorowitz, Momennejad and Daw’s simulations of a Markov decision process, an agent that evaluates actions as if it will fail to avoid danger at later steps, while in fact choosing optimally at each step, spreads threat value to states far from the danger and avoids early and widely, which reproduces qualitatively exaggerated threat appraisal, fear generalization and several laboratory findings associated with anxiety (Active)
- THEORY-1190 — In a Markov decision process, an agent that evaluates actions as if it will fail to avoid danger at later steps, while in fact choosing optimally at each step, spreads threat value to states far from the danger and avoids early and widely, which reproduces qualitatively exaggerated threat appraisal, fear generalization and several laboratory findings associated with anxiety (Proposed)

From In one simulated T-maze task with hand-set parameters, run once per condition, an active inference agent’s habit, learned by counting its own state transitions and policy choices, emerged from goal-directed behaviour, and once acquired it persisted, and reinforced itself, after the reward was devalued, while an agent devalued before acquisition switched at once

THEORY-923 — In one simulated T-maze task with hand-set parameters, run once per condition, an active inference agent’s habit, learned by counting its own state transitions and policy choices, emerged from goal-directed behaviour, and once acquired it persisted, and reinforced itself, after the reward was devalued, while an agent devalued before acquisition switched at once (Active)
- THEORY-1191 — In a simulated active inference agent, a habit learned by counting the agent’s own state transitions and policy choices emerges from goal-directed behaviour, and once acquired it persists, and reinforces itself, after the reward is devalued, while an agent devalued before acquisition switches at once (Proposed)

From As Craske et al.’s 2014 review reports it, citing six studies and reporting no data of its own, how far fear falls during an extinction or exposure session, or across sessions, does not predict how much fear is expressed at follow-up, in animals, human laboratory samples and fearful, mostly analog, human samples

THEORY-921 — As Craske et al.’s 2014 review reports it, citing six studies and reporting no data of its own, how far fear falls during an extinction or exposure session, or across sessions, does not predict how much fear is expressed at follow-up, in animals, human laboratory samples and fearful, mostly analog, human samples (Active)
- THEORY-1193 — How far fear falls during an extinction or exposure session, or across sessions, does not predict how much fear is expressed at follow-up, in animals, human laboratory samples and fearful human samples (Proposed)

learning-theory

From Under a local quadratic model of the loss with the greedy optimal step, per-step progress at batch size B is ΔL_max/(1 + B_noise/B), and at fixed B the steps and examples needed satisfy (S/S_min − 1)(E/E_min − 1) = 1, with E_min/S_min the progress-weighted mean of B_noise

THEORY-735 — Under a local quadratic model of the loss with the greedy optimal step, per-step progress at batch size B is ΔL_max/(1 + B_noise/B), and at fixed B the steps and examples needed satisfy (S/S_min − 1)(E/E_min − 1) = 1, with E_min/S_min the progress-weighted mean of B_noise (Active)
- THEORY-033 — The measured noise scale B_simple tracks the critical batch size within about 10× on six of eight tasks and about 20× on two generative models, so the gradient’s noise-to-signal ratio sets the critical batch only as far as the local quadratic model with an optimal step holds (Proposed)

From In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet

THEORY-083 — In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet (Active)
- THEORY-788 — Training a deep classifier past zero training error generally collapses its last-layer features onto their class means, drives the centred means toward a simplex equiangular tight frame aligned with the classifier, and makes the decision nearest class mean, as Papyan, Han and Donoho propose from their runs (Proposed)

THEORY-117 — In Uselis, Dittadi and Oh’s two-concept grids, ResNet-50s trained from scratch, with checkpoints chosen on the unseen pairs, reach unseen combinations as the share of combinations seen grows and not as up to four times more data of the same combinations is added; and when embeddings are exactly linearly factored, seen pairs that connect every value of both concepts, as two per value arranged in one cycle do, determine the embedding of every unseen pair (Active)
- THEORY-794 — In vision models, compositional generalization to unseen combinations of concepts is driven by the diversity of the combinations seen in training rather than by data volume, and arrives with representations that are linearly factored by concept, as Uselis, Dittadi and Oh propose (Proposed)

From In empirical games built from about a thousand sampled search agents in eight OpenSpiel board games, most on 3×3 or 4×4 boards, and in the AlphaStar league, Nash clusters taken from maximum-entropy equilibria are, read against a fitted skew-normal curve with no test, largest at intermediate transitive strength and narrower toward the strongest and the weakest strategies, while Blotto, the disc game, Kuhn poker and noisy Elo games do not show that profile

THEORY-930 — In empirical games built from about a thousand sampled search agents in eight OpenSpiel board games, most on 3×3 or 4×4 boards, and in the AlphaStar league, Nash clusters taken from maximum-entropy equilibria are, read against a fitted skew-normal curve with no test, largest at intermediate transitive strength and narrower toward the strongest and the weakest strategies, while Blotto, the disc game, Kuhn poker and noisy Elo games do not show that profile (Active)
- THEORY-1199 — Real-world games of skill have a spinning-top geometry: their non-transitive part, measured as Nash cluster size, is largest at intermediate strength and narrows toward the strongest and the weakest strategies (Proposed)

From In a sparsely-gated mixture-of-experts layer trained without balancing losses, the gate came to give large weights to the same few experts, which the authors attribute to self-reinforcement, favoured experts being trained more and so selected more; in one 256-expert language model, the squared coefficient of variation of expert importance or load as an auxiliary loss reduced the peak-to-mean load from 17.8 to at most 1.47 and test perplexity from 39.8 to 35.6–35.7

THEORY-927 — In a sparsely-gated mixture-of-experts layer trained without balancing losses, the gate came to give large weights to the same few experts, which the authors attribute to self-reinforcement, favoured experts being trained more and so selected more; in one 256-expert language model, the squared coefficient of variation of expert importance or load as an auxiliary loss reduced the peak-to-mean load from 17.8 to at most 1.47 and test perplexity from 39.8 to 35.6–35.7 (Active)
- THEORY-1218 — In a sparsely-gated mixture-of-experts layer trained without balancing losses, the gate’s concentration on a few experts arises by self-reinforcement: favoured experts are trained more rapidly and so are selected still more (Proposed)

From When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; Yang et al. state that with homogeneous agents mean field Q-learning converges to the Nash Q-value under three assumptions, one of which they call a strong constraint, by a proof whose contraction step needs a temperature above K/2 while its greedy-in-the-limit assumption needs the temperature to fall to zero; in their Gaussian Squeeze runs with 500 and 1000 agents, mean field Q-learning and actor-critic learned the optimal allocation where independent learners, FMQ, Rec-FMQ and a MADDPG variant failed

THEORY-945 — When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; Yang et al. state that with homogeneous agents mean field Q-learning converges to the Nash Q-value under three assumptions, one of which they call a strong constraint, by a proof whose contraction step needs a temperature above K/2 while its greedy-in-the-limit assumption needs the temperature to fall to zero; in their Gaussian Squeeze runs with 500 and 1000 agents, mean field Q-learning and actor-critic learned the optimal allocation where independent learners, FMQ, Rec-FMQ and a MADDPG variant failed (Active)
- THEORY-1231 — When each agent’s Q-function is approximated as an average of pairwise terms and expanded about the mean action of its neighbours, the first-order term vanishes and the approximation error is a second-order remainder bounded by the curvature of the pairwise Q-function, leaving a two-body problem between an agent and its neighbours’ average; with homogeneous agents, mean field Q-learning converges to the Nash Q-value under assumptions the authors call strong, and learned tasks of 500 and 1000 agents where independent and centralized-critic baselines failed (Proposed)

From The replicator-plus-entropy model of Boltzmann Q-learning in two-player stateless games is, in the small-step limit, the expected dynamics of a learner that updates every action’s value each step, and so of frequency-adjusted Q-learning wherever every action’s probability is at least its adjustment parameter; it is not that of Q-learning, whose expected update carries an extra factor of each action’s selection probability, so that its policy dynamics depend on the Q-values; in Kaisers and Tuyls’s three 2×2 games at one temperature, compared by eye, Q-learning’s trajectories departed from the model depending on initialization and the frequency-adjusted learner’s followed it

THEORY-932 — The replicator-plus-entropy model of Boltzmann Q-learning in two-player stateless games is, in the small-step limit, the expected dynamics of a learner that updates every action’s value each step, and so of frequency-adjusted Q-learning wherever every action’s probability is at least its adjustment parameter; it is not that of Q-learning, whose expected update carries an extra factor of each action’s selection probability, so that its policy dynamics depend on the Q-values; in Kaisers and Tuyls’s three 2×2 games at one temperature, compared by eye, Q-learning’s trajectories departed from the model depending on initialization and the frequency-adjusted learner’s followed it (Active)
- THEORY-1234 — The replicator-plus-entropy model of Boltzmann Q-learning in two-player stateless games is exact for frequency-adjusted Q-learning, not for Q-learning, whose expected update carries an extra factor of each action’s selection probability and whose trajectories in 2×2 games depart from the model depending on initialization (Proposed)

From Weight decay on factors is a nuclear-norm penalty on their product: with tied weights λ‖W‖²_F = λ‖WWᵀ‖, and for untied factors the least (‖U‖²_F + ‖V‖²_F)/2 over UVᵀ = M is ‖M‖, so for any loss that sees the weights only through the product, square-penalized training is nuclear-norm-penalized training of a product of bounded rank

THEORY-726 — Weight decay on factors is a nuclear-norm penalty on their product: with tied weights λ‖W‖²_F = λ‖WWᵀ‖*, and for untied factors the least (‖U‖²_F + ‖V‖²_F)/2 over UVᵀ = M is ‖M‖*, so for any loss that sees the weights only through the product, square-penalized training is nuclear-norm-penalized training of a product of bounded rank (Active)
- THEORY-180 — In a tied single-head attention layer trained by ridge-penalized empirical risk minimization on high-dimensional Gaussian sequences, weight decay is a nuclear-norm penalty on the query–key map, and the learned map is a soft-thresholded noisy copy of the target: its outliers are recovered target directions and its bulk is finite-sample noise (Proposed)

From In a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set fixes the generalising representation of addition when its same-answer pairs leave only translation and scale free; for non-modular addition with p = 10 and one-dimensional embeddings, generalisation sets in near the training fraction, about 0.4, at which that becomes likely

THEORY-181 — In a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set fixes the generalising representation of addition when its same-answer pairs leave only translation and scale free; for non-modular addition with p = 10 and one-dimensional embeddings, generalisation sets in near the training fraction, about 0.4, at which that becomes likely (Active)
- THEORY-782 — In networks that learn addition from summed embeddings, the critical training fraction of grokking is the fraction at which the training set’s same-answer pairs come to fix the generalising representation up to translation and scale, at every group size, as Liu, Kitouni, Nolte, Michaud, Tegmark and Williams propose (Proposed)

From For the quartic proxy of word2vec with tied weights, symmetric reweighting and constant weights, the rank-d minimisers are the top d eigenvectors of the relative deviation-from-independence matrix M*, scaled by the square roots of their eigenvalues, up to rotation; and from an initialisation aligned with those eigenvectors, gradient flow acquires each one in an independent logistic rise, in order of eigenvalue

THEORY-739 — For the quartic proxy of word2vec with tied weights, symmetric reweighting and constant weights, the rank-d minimisers are the top d eigenvectors of the relative deviation-from-independence matrix M*, scaled by the square roots of their eigenvalues, up to rotation; and from an initialisation aligned with those eigenvectors, gradient flow acquires each one in an independent logistic rise, in order of eigenvalue (Active)
- THEORY-182 — A rank-limited contrastive word embedding trained from small initialisation learns the top eigenvectors of the co-occurrence matrix’s relative deviation from independence, one at a time in order of eigenvalue, not the best low-rank approximation of its unconstrained optimum (Proposed)

From A two-layer linear network trained by gradient flow on white inputs from a decoupled, balanced start learns each singular mode of its input–output correlation in a logistic rise that waits a time of order its inverse strength times the log of the inverse initial scale, so the modes arrive one at a time in order of strength; a single-layer network on the same data learns every mode on one timescale with no plateau, so under these conditions the stages come from the product of layers and the data’s spectrum

THEORY-740 — A two-layer linear network trained by gradient flow on white inputs from a decoupled, balanced start learns each singular mode of its input–output correlation in a logistic rise that waits a time of order its inverse strength times the log of the inverse initial scale, so the modes arrive one at a time in order of strength; a single-layer network on the same data learns every mode on one timescale with no plateau, so under these conditions the stages come from the product of layers and the data’s spectrum (Active)
- THEORY-186 — A deep linear network trained from small weights learns the singular modes of its input–output correlations one at a time in order of strength, each in a sharp sigmoidal transition, while a shallow network learns them all together; the stages come from the product of layers and the data’s spectrum, not from nonlinearity (Proposed)
- THEORY-481 — Under gradient flow averaged over tasks and data, a linear-attention layer trained on in-context regression learns as a two-layer linear network from an aligned start does: when tasks share the input covariance’s eigenbasis and the weights start and stay aligned with it, then in the long-context limit each input mode is learned on its own logistic curve whose timescale falls with the square of its variance, so learning is staged by the input spectrum, and the layer converges to the estimator preconditioned by the inverse input covariance; at finite context length the modes decouple exactly only when inputs are isotropic and the modes start equal, and otherwise couple at order 1/N (Active)

From In a deep linear network trained by gradient descent on whitened inputs from decoupled, balanced initial weights, depth slows learning through the initial end-to-end strength of each mode: counting iterations as learning time over the stable step size, from strength of order one the number of iterations stays bounded as depth grows, once the step size falls with depth as stability requires, while from small strength ε the plateau lengthens from ln(1/ε) with one hidden layer towards 1/ε at great depth

THEORY-187 — In a deep linear network trained by gradient descent on whitened inputs from decoupled, balanced initial weights, depth slows learning through the initial end-to-end strength of each mode: counting iterations as learning time over the stable step size, from strength of order one the number of iterations stays bounded as depth grows, once the step size falls with depth as stability requires, while from small strength ε the plateau lengthens from ln(1/ε) with one hidden layer towards 1/ε at great depth (Active)
- THEORY-792 — In a deep linear network trained by gradient descent from generic initial weights, not only from decoupled ones, depth slows learning through the initial end-to-end strength of each mode, so that random orthogonal weights, whose product is an isometry, learn in a number of iterations that stays bounded as depth grows, as Saxe, McClelland and Ganguli argue from simulation (Proposed)

From On the Random Hierarchy Model, in the experiments reported (s = 2, L ≤ 4, small v), deep CNNs matched to the tree and trained in a feature-learning regime reach a tenth of chance error from about n_c m^L examples, polynomial in the input dimension, at the training-set size at which their hidden layers become invariant to swapping synonymous tuples; two-layer networks and the same CNNs as fixed kernels need a number proportional to the whole input space, exponential in the input dimension, and deep CNNs stay near chance when every tuple is made equally likely in every class

THEORY-195 — On the Random Hierarchy Model, in the experiments reported (s = 2, L ≤ 4, small v), deep CNNs matched to the tree and trained in a feature-learning regime reach a tenth of chance error from about n_c m^L examples, polynomial in the input dimension, at the training-set size at which their hidden layers become invariant to swapping synonymous tuples; two-layer networks and the same CNNs as fixed kernels need a number proportional to the whole input space, exponential in the input dimension, and deep CNNs stay near chance when every tuple is made equally likely in every class (Active)
- THEORY-786 — Deep networks learn hierarchically generated data from a number of examples set by when the correlations between input patches and the class rise above sampling noise, so detecting those correlations is how they learn the grammar, as Cagnetta and coauthors propose (Proposed)

From In the Random Hierarchy Model with v symbols per level, m rules per symbol and s-tuples, the covariance between two tokens whose lowest common ancestor is ℓ levels up has zero mean over grammars and root-mean-square size about ((1 − m/v^(s−1))/(v³ m^(2ℓ−1)))^(1/2), falling by a factor m per level, while its estimate from P sequences carries noise of about (v²P)^(−1/2), so the two meet at P_ℓ ≈ v m^(2ℓ−1)/(1 − m/v^(s−1)); all to leading order in large v and m

THEORY-778 — In the Random Hierarchy Model with v symbols per level, m rules per symbol and s-tuples, the covariance between two tokens whose lowest common ancestor is ℓ levels up has zero mean over grammars and root-mean-square size about ((1 − m/v^(s−1))/(v³ m^(2ℓ−1)))^(1/2), falling by a factor m per level, while its estimate from P sequences carries noise of about (v²P)^(−1/2), so the two meet at P_ℓ ≈ v m^(2ℓ−1)/(1 − m/v^(s−1)); all to leading order in large v and m (Active)
- THEORY-201 — In next-token prediction on hierarchically generated sequences, the correlation between two tokens falls by about a factor m per level of their common ancestor, so a training set of P sequences resolves correlations only out to a distance t*(P) at which they meet sampling noise; learners come to represent the hidden symbols up to that depth, giving loss steps at about v m^(2ℓ−1) examples, and on character-level text the context at which the loss saturates grows as P^(1/(2β)) with β the measured correlation decay (Proposed)

From On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses

THEORY-204 — On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses (Active)
- THEORY-784 — Language models trained on limited data show a test loss falling as P^(−γ/(2β)) because P training tokens resolve token correlations only out to a horizon growing as P^(1/(2β)), so that more data mainly lengthens the context a model can use, when the model learns within that horizon faster than it grows, as Cagnetta, Raventós, Ganguli and Wyart propose (Proposed)

From In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²))

THEORY-779 — In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²)) (Active)
- THEORY-209 — A converged network trained with SGD is not at a point and does not sample its training loss: under a quadratic approximation it samples a Gibbs distribution on a modified loss, isotropic in the Hessian’s column space, with probability currents that keep it oscillating, so its speed and spread are set by the learning rate, momentum and batch size (Proposed)

From In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise

THEORY-211 — In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise (Active)
- THEORY-790 — In deep networks trained by SGD on real data, minibatch noise drives parameter groups related by a rescaling symmetry toward the point where their gradient noise balances, at a rate set by learning rate over batch size, so that at stationarity they sit neither where gradient flow would leave them nor spread out as under isotropic noise, as Ziyin, Li and Ueda propose (Proposed)

From Every mirror symmetry of a network’s per-sample loss makes its symmetric subspace invariant under gradient descent and SGD, and when the loss’s Hessian is bounded below by −c, weight decay γ‖θ‖² with γ > c/2 puts every minimum in that subspace; rescaling symmetries that hold at λ = −1, rotation symmetries and permutation symmetries contain such mirrors, whose subspaces are zeroed weights, reduced rank and identical neurons

THEORY-213 — Every mirror symmetry of a network’s per-sample loss makes its symmetric subspace invariant under gradient descent and SGD, and when the loss’s Hessian is bounded below by −c, weight decay γ‖θ‖² with γ > c/2 puts every minimum in that subspace; rescaling symmetries that hold at λ = −1, rotation symmetries and permutation symmetries contain such mirrors, whose subspaces are zeroed weights, reduced rank and identical neurons (Active)
- THEORY-781 — Strong gradient noise in SGD, like weight decay above a threshold, makes the subspace fixed by a mirror symmetry of a network’s loss attracting even where it is a saddle, so that networks trained with SGD at large learning rates or small batches reach sparse, low-rank and identical-neuron solutions, as Ziyin proposes (Proposed)

From For a symmetrized Barlow Twins loss on a linear or fixed-kernel model, gradient flow from a small initialization of common scale aligned with the top eigenvectors of the pairs’ symmetrized cross-correlation learns them one at a time, in order of eigenvalue, and ends at a top spectral solution, which is exactly a minimum-norm zero-loss solution: kernel PCA on K_Γ

THEORY-229 — For a symmetrized Barlow Twins loss on a linear or fixed-kernel model, gradient flow from a small initialization of common scale aligned with the top eigenvectors of the pairs’ symmetrized cross-correlation learns them one at a time, in order of eigenvalue, and ends at a top spectral solution, which is exactly a minimum-norm zero-loss solution: kernel PCA on K_Γ (Active)
- THEORY-793 — Gradient flow on a symmetrized Barlow Twins loss with a linear or fixed-kernel model, from generic small initialization and not only from initialization aligned with the eigenvectors, learns the top eigenvectors of the pairs’ symmetrized cross-correlation one at a time in order of eigenvalue and ends at a top spectral solution, as Simon and coauthors derive informally (Proposed)

From In Liu, Liu, Michaud, Gore and Tegmark’s Geometry model of independent skills with orthogonal task vectors, simulation shows skills progressing together under gradient descent, the weaker taking about 1/p of the time, and in sequence under sign descent, a weaker skill progressing fast only once a stronger is done and the ratio of their times approaching 2; on a rotated quadratic the sequence appears under sign descent only when the loss’s eigendirections are misaligned with the coordinates

THEORY-253 — In Liu, Liu, Michaud, Gore and Tegmark’s Geometry model of independent skills with orthogonal task vectors, simulation shows skills progressing together under gradient descent, the weaker taking about 1/p of the time, and in sequence under sign descent, a weaker skill progressing fast only once a stronger is done and the ratio of their times approaching 2; on a rotated quadratic the sequence appears under sign descent only when the loss’s eigendirections are misaligned with the coordinates (Active)
- THEORY-783 — In neural-network training, sign-descent-like optimizers such as Adam make independent skills of unequal frequency compete for coordinate signs, so that they are learned one after another where gradient descent would learn them together; this is one source of sequential skill onset, beside dependence between skills, as Liu, Liu, Michaud, Gore and Tegmark propose (Proposed)

From In Goldwasser, Gruber, Kalai and Paradise’s common-nonsense model, where the source language and the plausibility prior are uniform over sets from which shared random nonsense is removed and translators are one-to-one, maximum likelihood with the prior alone translates without parallel data with error O(log|Θ| / (α·min(m, |T|))) with high probability, falling in the shared nonsense α, in the size of the source language and in the number of samples

THEORY-270 — In Goldwasser, Gruber, Kalai and Paradise’s common-nonsense model, where the source language and the plausibility prior are uniform over sets from which shared random nonsense is removed and translators are one-to-one, maximum likelihood with the prior alone translates without parallel data with error O(log|Θ| / (α·min(m, |T|))) with high probability, falling in the shared nonsense α, in the size of the source language and in the number of samples (Active)
- THEORY-806 — Translation without parallel data, as of an unknown animal communication system into a human language, is learnable from a plausibility prior over the target language alone, with error falling as the common ground between the languages grows and as the source language grows richer (Proposed)

From Under MSE loss, in Han, Papyan and Donoho’s VGG, ResNet and DenseNet runs on five balanced ten-class datasets, the trained classifier stays near the least-squares classifier for its own features from an early epoch; and on that central path, for unconstrained features continually renormalized to identity within-class covariance and without weight decay, gradient flow drives the class-separation singular values to infinity at equal rates, which in that model is neural collapse

THEORY-278 — Under MSE loss, in Han, Papyan and Donoho’s VGG, ResNet and DenseNet runs on five balanced ten-class datasets, the trained classifier stays near the least-squares classifier for its own features from an early epoch; and on that central path, for unconstrained features continually renormalized to identity within-class covariance and without weight decay, gradient flow drives the class-separation singular values to infinity at equal rates, which in that model is neural collapse (Active)
- THEORY-797 — Deep classifiers trained with MSE loss stay near the least-squares classifier for their own features and move along the central path by the dynamics of continually renormalized gradient flow, whose equal-rate growth of the whitened class-separation singular values is how they reach neural collapse, as Han, Papyan and Donoho propose (Proposed)

From In the unconstrained-features max-margin problem with k classes under step imbalance and feature dimension at least k − 1, the unique optimal logits are the simplex-encoded labels, every margin equals one, and the classifier and embedding Gram matrices are the two square-root factors of that matrix’s singular value decomposition; the geometry is the simplex equiangular tight frame exactly when the classes are balanced or there are two of them

THEORY-738 — In the unconstrained-features max-margin problem with k classes under step imbalance and feature dimension at least k − 1, the unique optimal logits are the simplex-encoded labels, every margin equals one, and the classifier and embedding Gram matrices are the two square-root factors of that matrix’s singular value decomposition; the geometry is the simplex equiangular tight frame exactly when the classes are balanced or there are two of them (Active)
- THEORY-286 — Under class imbalance the last-layer geometry that unregularized cross-entropy prefers is not the simplex ETF but simplex-encoded-label interpolation: the logits equal the simplex-encoded labels, the classifier and embedding Gram matrices come from that matrix’s SVD, any finite weight decay changes the geometry, and training approaches it more slowly as imbalance grows (Proposed)

From For an optimal capacity-bounded learner on an orthogonal mixture of equally frequent random facts and a second domain, the facts are learned not at all below a capacity threshold set by the mixing ratio and in full once capacity exceeds it by the facts’ total entropy, and Pythia models trained on synthetic biographies mixed into web text show a threshold in model size and in mixing ratio

THEORY-293 — For an optimal capacity-bounded learner on an orthogonal mixture of equally frequent random facts and a second domain, the facts are learned not at all below a capacity threshold set by the mixing ratio and in full once capacity exceeds it by the facts’ total entropy, and Pythia models trained on synthetic biographies mixed into web text show a threshold in model size and in mixing ratio (Active)
- THEORY-809 — Language models trained by gradient descent on a mixture of random facts and other data reach the capacity-optimal allocation, so the model size and the mixing ratio at which they begin to learn the facts are those of an optimal capacity-bounded learner (Proposed)

From For a wide two-layer ReLU network with zero bias on one-dimensional data whose two classes lie on opposite sides of the origin, the tangent kernel is a constant multiple of the initialization kernel, so lazy training leaves the within-class variability of its features, NC1, where initialization put it

THEORY-294 — For a wide two-layer ReLU network with zero bias on one-dimensional data whose two classes lie on opposite sides of the origin, the tangent kernel is a constant multiple of the initialization kernel, so lazy training leaves the within-class variability of its features, NC1, where initialization put it (Active)
- THEORY-813 — Training a wide network in the lazy, kernel regime does not make its features more collapsed than at initialization, so neural collapse, where it occurs, is something only feature learning produces (Proposed)

From In a spectral contrastive graph that joins labelled known classes to unlabelled data, the least-squares residual of a novel labelling on the top-k eigenspace, with the labelled entries left free, equals exactly the part of the labelling outside that eigenspace that the labelled rows of the discarded eigenvectors cannot absorb

THEORY-737 — In a spectral contrastive graph that joins labelled known classes to unlabelled data, the least-squares residual of a novel labelling on the top-k eigenspace, with the labelled entries left free, equals exactly the part of the labelling outside that eigenspace that the labelled rows of the discarded eigenvectors cannot absorb (Active)
- THEORY-320 — In a spectral contrastive graph that joins labelled known classes to unlabelled data, a novel labelling’s linear-probe residual is exactly its part outside the top eigenspace that the labelled rows of the discarded eigenvectors cannot absorb, so which partition of the unlabelled data is discovered depends on the known classes: ones sharing the target’s attribute can remove the residual, and ones sharing a rival attribute can raise it (Proposed)

THEORY-349 — For binary outcomes and a fixed representation of individuals, a predictor can pass every test in a bounded class without being close to nature: outcome indistinguishability to tests that see the sampled prediction is equivalent, with stated losses, to multicalibration over a closely related class of sets, a predictor meeting it exists at size polynomial in the tests’ size and the tolerance whatever nature’s complexity, and learning one is as hard as weak agnostic learning of the class (Active)
- THEORY-354 — An observer’s power to refute a predictor depends on its access to the predictor as well as on its budget: against tests that see only the sampled prediction a predictor of size O(s/ε²) always passes, but if fine-grained clique counting is hard, or BPP ≠ PSPACE, some cheap tests that may query the predictor adaptively pass only predictors about as costly as nature’s own (Active)

From In a finite partial-monitoring game, the signals of a set D of actions determine the expected loss difference of actions i and j under every outcome distribution iff ℓ_i − ℓ_j lies in the sum of the row spaces of their signal matrices; when it does not, two outcome distributions differing along a vector in the common kernel give every action in D the same signal law and reverse the preference between i and j, so no amount of play restricted to D can learn the comparison

THEORY-962 — In a finite partial-monitoring game, the signals of a set D of actions determine the expected loss difference of actions i and j under every outcome distribution iff ℓ_i − ℓ_j lies in the sum of the row spaces of their signal matrices; when it does not, two outcome distributions differing along a vector in the common kernel give every action in D the same signal law and reverse the preference between i and j, so no amount of play restricted to D can learn the comparison (Active)
- THEORY-961 — In a finite partial-monitoring game, an action optimal wherever two neighbouring actions tie has a loss vector on the segment between theirs, so a degenerate or duplicate action never creates a neighbouring pair or decides the trivial class, yet its signals count toward the pair’s local observability; and the T^2/3 lower bound for a pair that is not locally observable needs the separating direction in the kernel of the signal matrices of all such actions, not only of the two neighbours (Active)

From In small attention-only transformer language models of two to six layers, in-context learning, measured as the loss at token 500 minus the loss at token 50, appears abruptly in one early window of training in which induction heads form, and ablating those heads at test time removes nearly all of it; one-layer models show neither, and letting a single layer attend by the previous token brings the jump to one-layer models and moves it earlier in two-layer ones

THEORY-973 — In small attention-only transformer language models of two to six layers, in-context learning, measured as the loss at token 500 minus the loss at token 50, appears abruptly in one early window of training in which induction heads form, and ablating those heads at test time removes nearly all of it; one-layer models show neither, and letting a single layer attend by the previous token brings the jump to one-layer models and moves it earlier in two-layer ones (Active)
- THEORY-990 — In transformer language models of every size, including large ones with MLPs, induction heads, extended to fuzzy matching of similar tokens and abstract patterns, are the mechanism of most in-context learning, as Olsson et al. propose (Proposed)

linguistics

From As Raviv, Meyer and Lev-Ari report it, in a laboratory game where 12 fully connected groups of eight and 12 of four built a new communication system with total interaction held fixed, the larger groups’ languages grew more systematic faster, ended more systematic and differed less from one another, while communicative success and convergence did not differ by size; input variability, higher in the larger groups, predicted the next round’s gain in structure, an association measured and not manipulated

THEORY-164 — As Raviv, Meyer and Lev-Ari report it, in a laboratory game where 12 fully connected groups of eight and 12 of four built a new communication system with total interaction held fixed, the larger groups’ languages grew more systematic faster, ended more systematic and differed less from one another, while communicative success and convergence did not differ by size; input variability, higher in the larger groups, predicted the next round’s gain in structure, an association measured and not manipulated (Active)
- THEORY-1171 — In groups that build a new communication system, larger groups make it more systematic, and the greater variety of input they face drives the effect (Proposed)

From In Zaslavsky, Kemp, Regier and Tishby’s analysis, the colour-naming systems of the World Color Survey languages and English lie near the information-bottleneck bound for compressing perceptual meanings into words, at a trade-off parameter just above 1 and nearer than hue-rotated variants of themselves, with the need distribution over meanings estimated from the naming data

THEORY-163 — In Zaslavsky, Kemp, Regier and Tishby’s analysis, the colour-naming systems of the World Color Survey languages and English lie near the information-bottleneck bound for compressing perceptual meanings into words, at a trade-off parameter just above 1 and nearer than hue-rotated variants of themselves, with the need distribution over meanings estimated from the naming data (Active)
- THEORY-791 — The colour-naming systems of the world’s languages lie near the information-bottleneck bound for compressing perceptual meanings into words, and differences in one trade-off parameter place each language along it, as Zaslavsky, Kemp, Regier and Tishby propose (Proposed)

From On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses

THEORY-204 — On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses (Active)
- THEORY-784 — Language models trained on limited data show a test loss falling as P^(−γ/(2β)) because P training tokens resolve token correlations only out to a horizon growing as P^(1/(2β)), so that more data mainly lengthens the context a model can use, when the model learns within that horizon faster than it grows, as Cagnetta, Raventós, Ganguli and Wyart propose (Proposed)

logic

From Classical Boolean logic is restored when a presheaf of contextual data acquires a global section or is sheafified

THEORY-024 — Classical Boolean logic is restored when a presheaf of contextual data acquires a global section or is sheafified (Rejected)
- THEORY-516 — For any presheaf of propositions over a small category of contexts, a sieve-valued valuation satisfies the generalized functional-composition condition exactly when it is a subobject of that presheaf, and the construction runs unchanged in classical physics, where every microstate gives a global section yet truth values are still non-Boolean sieves; so the logic of the truth values is fixed by the base category, not by whether a presheaf has global sections (Active)

From If commonsense knowledge includes defaults that later evidence displaces, a monotonic logic, in which an added axiom can never remove a theorem, cannot represent that displacement as it stands; Minsky’s further claim, that no logistic system, one that separates propositions from general laws of inference, can carry commonsense knowledge even once repaired, is asserted, not shown

THEORY-462 — If commonsense knowledge includes defaults that later evidence displaces, a monotonic logic, in which an added axiom can never remove a theorem, cannot represent that displacement as it stands; Minsky’s further claim, that no logistic system, one that separates propositions from general laws of inference, can carry commonsense knowledge even once repaired, is asserted, not shown (Active)
- THEORY-1239 — No logistic system, one that separates propositions from general laws of inference, can carry commonsense knowledge, whose defaults later evidence displaces: attempts to repair its monotonicity will continue to fail (Proposed)

From On the court agenda {P, Q, R, R iff (P and Q)} under universal domain, proposition-wise majority voting is not robustly rational, while the premise-based procedure is, and it makes the group’s judgments a function of the members’ whole judgment sets that the members’ judgments on the conclusion neither fix nor need to enter

THEORY-725 — On the court agenda {P, Q, R, R iff (P and Q)} under universal domain, proposition-wise majority voting is not robustly rational, while the premise-based procedure is, and it makes the group’s judgments a function of the members’ whole judgment sets that the members’ judgments on the conclusion neither fix nor need to enter (Active)
- THEORY-127 — On the court agenda, and on any agenda in which every proposition reaches every other by conditional entailment and some minimally inconsistent triple stays consistent with two of its members negated, a group whose judgments are robustly rational under universal domain and respect unanimity cannot form its judgment on each proposition from its members’ judgments on that proposition alone unless one member dictates; its judgments must supervene on the members’ whole sets of judgments, and so can depart from what the members judge on the same proposition while being wholly fixed by them (Active)

From In Aerts’s subspace model of Hampton’s membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice

THEORY-255 — In Aerts’s subspace model of Hampton’s membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice (Active)
- THEORY-807 — The guppy-type overextension in Hampton’s membership data arises because a combined concept is in part a new, emergent concept and not only a logical combination of its parts, so human concept combination runs in two superposed layers, one classical and logical and one quantum and conceptual (Proposed)

From For a logically closed theory, the contractions satisfying the six basic AGM postulates are exactly the partial meet contractions, and those satisfying the two supplementary postulates as well are exactly the ones whose selection of remainders is generated by a transitive relation; the theory alone does not fix the contraction, and the results concern one change of one fixed theory

THEORY-353 — For a logically closed theory, the contractions satisfying the six basic AGM postulates are exactly the partial meet contractions, and those satisfying the two supplementary postulates as well are exactly the ones whose selection of remainders is generated by a transitive relation; the theory alone does not fix the contraction, and the results concern one change of one fixed theory (Active)
- THEORY-347 — Belief bases with the same logical closure can revise to opposite conclusions on the same input, so a closed belief set does not fix its own revision; partial meet contraction of a base need not satisfy Recovery, and bases keep distinct inconsistent states apart, while a belief-set model reproduces the base difference only by attaching a different selection mechanism to the same set (Active)

From For positive relational algebra on relations whose tuples are annotated from a commutative semiring, every query’s annotated answer equals its answer over provenance polynomials N[X] in the input tuple ids, evaluated at the input annotations, so set, bag, c-table and event-table semantics all factor through the polynomials; revaluing inputs can remove output tuples but never add them, and nothing is shown for negation, difference or aggregation

THEORY-373 — For positive relational algebra on relations whose tuples are annotated from a commutative semiring, every query’s annotated answer equals its answer over provenance polynomials N[X] in the input tuple ids, evaluated at the input annotations, so set, bag, c-table and event-table semantics all factor through the polynomials; revaluing inputs can remove output tuples but never add them, and nothing is shown for negation, difference or aggregation (Active)
- THEORY-380 — For datalog over a finite input instance annotated from a commutative ω-continuous semiring, with an answer’s tag defined as the sum over its derivation trees of the product of its leaves’ tags, the tags are the least fixed point of a finite polynomial system and factor through formal power series N∞⟦X⟧ in the input tuple ids; an answer tuple’s series has an infinite coefficient exactly when one of its derivation trees contains a tuple lying on a cycle of unit rules (Active)

loss-landscapes

THEORY-115 — Linear connectivity from a shared, already-stable start holds only when the objectives the runs optimise share structure, and breaks as that structure is removed (Proposed)
- THEORY-105 — For RoBERTa-base fine-tuned on English classification data, with loss measured by a linear probe refitted on the frozen encoder, points sampled from the convex hull of models fine-tuned on one dataset, one task family or twelve GLUE and SuperGLUE datasets lost no more than any model fine-tuned elsewhere and beat the group’s own members in 88% to 96.7% of comparisons, and the centroid of the general-level models was a better starting point for BitFit than the pretrained model (Active)
  - THEORY-828 — Fine-tuning from one pretrained model typically lands near the edge of a task-specific low-loss region in weight space that contains the convex hull of the fine-tuned models, and averaging their weights, as model soups and stochastic weight averaging do, works by moving inward (Proposed)

From Two ε-dropout-stable solutions of a multilayer ReLU net, under a loss convex in the output, are joined by a piecewise-linear path whose loss never exceeds the worse endpoint’s by more than ε, and joined pairwise paths compose with no higher barrier than their worst

THEORY-736 — Two ε-dropout-stable solutions of a multilayer ReLU net, under a loss convex in the output, are joined by a piecewise-linear path whose loss never exceeds the worse endpoint’s by more than ε, and joined pairwise paths compose with no higher barrier than their worst (Active)
- THEORY-109 — The solutions SGD finds in a deep network lie in one connected low-loss set, joined by simple curves and spanning volumes of measurable dimension, and this is a property of the solutions training finds, not of every minimum (Proposed)
  - THEORY-112 — Aligning independently trained solutions over the architecture’s symmetries removed the linear barrier between them for MNIST MLPs and 32×-wide ResNet-20s on CIFAR-10, two-thirds of it for an ImageNet ResNet-50, and not all of it for VGG-16 or ResNet-20 at 1× width, the barrier after weight matching falling with width; for 6- to 8-layer ViTs and GPT-2s, permutations alone removed 63–90% of it, and adding an orthogonal map on the residual stream, learned against the loss at the path’s midpoint, took it to zero or near zero except on BookCorpus (0.42) (Active)
    - THEORY-110 — A set of trained solutions has a centre that is linearly connected, after alignment, to the rest of them: the solution set is a star domain even where it is not convex (Proposed) — also carries further THEORY-109, THEORY-741
    - THEORY-1188 — Once the architecture’s symmetries are factored out, most of the linear barrier between independently trained solutions disappears; the group that matters is the full one, permutations for MLPs and CNNs but for transformers also an orthogonal map on the residual stream, without which a barrier remains; and the barrier left after alignment falls with width (Proposed)
  - THEORY-114 — In image classification, linear mode connectivity appeared during training and not at initialization: copies of ResNet-20 and VGG-16 on CIFAR-10 and of ResNet-50 and Inception-v3 on ImageNet, trained from a shared state, became stable to SGD noise in test error only after 1.5–20% of training (LeNet on MNIST was stable from the start), and between independently initialized MLPs on MNIST and CIFAR-10 the barrier after weight-matching alignment was high at initialization and fell gradually through training (Active)
    - THEORY-1172 — Linear mode connectivity emerges during training, not at initialization: copies trained from a shared state become stable to SGD noise after 1.5–20% of training, and independently trained networks become linearly connected after alignment only gradually (Proposed)
  - THEORY-557 — Low-loss paths between independently trained solutions are found outside supervised image classifiers: graph neural networks trained from different seeds on one graph are joined by quadratic Bezier curves where the straight line has a barrier, and one pair each of diffusion and contrastive models by chains of low-loss checkpoints (Proposed)
THEORY-741 — A two-layer ReLU student of width at least kM, fitting an orthonormal teacher of M neurons on the sphere, has for any k global minima a global minimum joined to each by a two-piece linear path of minima; and in a deep linear network of depth L and width above 1 + r(L − 1), any r global minima in general position share a global minimum linearly connected to each (Active)

From A linear barrier between two minima that survives the removal of symmetries marks a difference in the input attributes the two models rely on: they compute by different mechanisms

THEORY-116 — A linear barrier between two minima that survives the removal of symmetries marks a difference in the input attributes the two models rely on: they compute by different mechanisms (Proposed)
- THEORY-111 — In image classifiers, between two networks linearly connected by spawning from a shared checkpoint or by permutation alignment (MLPs, and VGG-16, ResNet-20 and ResNet-50 on CIFAR-10 and Tiny-ImageNet), nearly every layer’s features at three points on the line were proportional to the interpolation of the endpoints’ features, and pairs that were not linearly connected were far from it (Active) — also carries further THEORY-732
  - THEORY-1187 — Along a low-loss linear path between two networks, every layer’s features are the interpolation of the endpoints’ features (Proposed)
THEORY-732 — For ReLU networks interpolated linearly between two endpoints, weak additivity and commutativity at every layer make each layer’s features exactly the interpolation of the endpoints’ features, output-layer linearity bounds the path’s error by twice the endpoints’, and for one hidden layer with interpolating minima linear mode connectivity alone gives the exact case (Active)

From Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble’s accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model

THEORY-113 — Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble’s accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model (Active)
- THEORY-1185 — Merging models trained on different tasks from different initialisations fails where their features have no counterpart in the other model, and such features grow more common with depth (Proposed)

From In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²))

THEORY-779 — In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²)) (Active)
- THEORY-209 — A converged network trained with SGD is not at a point and does not sample its training loss: under a quadratic approximation it samples a Gibbs distribution on a modified loss, isotropic in the Hessian’s column space, with probability currents that keep it oscillating, so its speed and spread are set by the learning rate, momentum and batch size (Proposed)

From In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise

THEORY-211 — In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise (Active)
- THEORY-790 — In deep networks trained by SGD on real data, minibatch noise drives parameter groups related by a rescaling symmetry toward the point where their gradient noise balances, at a rate set by learning rate over batch size, so that at stationarity they sit neither where gradient flow would leave them nor spread out as under isotropic noise, as Ziyin, Li and Ueda propose (Proposed)

From Every mirror symmetry of a network’s per-sample loss makes its symmetric subspace invariant under gradient descent and SGD, and when the loss’s Hessian is bounded below by −c, weight decay γ‖θ‖² with γ > c/2 puts every minimum in that subspace; rescaling symmetries that hold at λ = −1, rotation symmetries and permutation symmetries contain such mirrors, whose subspaces are zeroed weights, reduced rank and identical neurons

THEORY-213 — Every mirror symmetry of a network’s per-sample loss makes its symmetric subspace invariant under gradient descent and SGD, and when the loss’s Hessian is bounded below by −c, weight decay γ‖θ‖² with γ > c/2 puts every minimum in that subspace; rescaling symmetries that hold at λ = −1, rotation symmetries and permutation symmetries contain such mirrors, whose subspaces are zeroed weights, reduced rank and identical neurons (Active)
- THEORY-781 — Strong gradient noise in SGD, like weight decay above a threshold, makes the subspace fixed by a mirror symmetry of a network’s loss attracting even where it is a saddle, so that networks trained with SGD at large learning rates or small batches reach sparse, low-rank and identical-neuron solutions, as Ziyin proposes (Proposed)

From Under MSE loss, in Han, Papyan and Donoho’s VGG, ResNet and DenseNet runs on five balanced ten-class datasets, the trained classifier stays near the least-squares classifier for its own features from an early epoch; and on that central path, for unconstrained features continually renormalized to identity within-class covariance and without weight decay, gradient flow drives the class-separation singular values to infinity at equal rates, which in that model is neural collapse

THEORY-278 — Under MSE loss, in Han, Papyan and Donoho’s VGG, ResNet and DenseNet runs on five balanced ten-class datasets, the trained classifier stays near the least-squares classifier for its own features from an early epoch; and on that central path, for unconstrained features continually renormalized to identity within-class covariance and without weight decay, gradient flow drives the class-separation singular values to infinity at equal rates, which in that model is neural collapse (Active)
- THEORY-797 — Deep classifiers trained with MSE loss stay near the least-squares classifier for their own features and move along the central path by the dynamics of continually renormalized gradient flow, whose equal-rate growth of the whitened class-separation singular values is how they reach neural collapse, as Han, Papyan and Donoho propose (Proposed)

mathematical-statistics

From For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix

THEORY-717 — For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix (Active)
- THEORY-156 — For experiments on a finite parameter set whose outcome spaces are standard Borel, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation (Active)
  - THEORY-1096 — Blackwell’s comparison of experiments, its easy direction: an experiment is a family of distributions indexed by a common parameter, and if one experiment is a garbling of another by a Markov kernel, every risk attainable with the garbling in any decision problem with bounded losses is attainable with the original, on any outcome spaces (Active)
  - THEORY-1177 — For experiments on a finite parameter set, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation (Proposed)

From Recovery of a k-sparse vector by ℓ₁ minimization from n random measurements in dimension N switches, as N grows with δ = n/N and ρ = k/n fixed, from near-certain success to near-certain failure across a curve ρ(δ) that, for Gaussian matrices, the authors’ earlier face-counting theorems fix; in 2,948,000 optimizations at N = 200, 400 and 1600, eight non-Gaussian ensembles approached that curve in a way consistent with gaps shrinking as N^(−1/2), an exponent fitted on the same data, while at finite N the location depended on the ensemble

THEORY-391 — Recovery of a k-sparse vector by ℓ₁ minimization from n random measurements in dimension N switches, as N grows with δ = n/N and ρ = k/n fixed, from near-certain success to near-certain failure across a curve ρ(δ) that, for Gaussian matrices, the authors’ earlier face-counting theorems fix; in 2,948,000 optimizations at N = 200, 400 and 1600, eight non-Gaussian ensembles approached that curve in a way consistent with gaps shrinking as N^(−1/2), an exponent fitted on the same data, while at finite N the location depended on the ensemble (Active)
- THEORY-812 — As dimension grows with the undersampling and sparsity ratios fixed, the ℓ₁ phase transition for recovering sparse vectors from random measurements sits at the same curve for a broad class of random-matrix ensembles, those with a kind of orthogonality or weak independence, as for Gaussian matrices, while differing at finite size (Proposed)

mathematics

From A representation is determined by its centred kernel up to a translation and an orthogonal transformation, and the rotation-aligned distance between two representations equals the Bures distance between their kernels

THEORY-004 — A representation is determined by its centred kernel up to a translation and an orthogonal transformation, and the rotation-aligned distance between two representations equals the Bures distance between their kernels (Active)
- THEORY-008 — On a finite set of stimuli, what a regularized linear readout can decode from a representation is a function of its normalized kernel, and CKA, CCA and GULP are averages of readout agreement (Active)
- THEORY-017 — In a Hilbert-space model only unitarily invariant structure is intrinsic; a basis or tensor factorisation, and so any parts or features, is extra data supplied from outside the space, and the record’s Hilbert-space works differ in where they get it (Active) — also carries further THEORY-721
  - THEORY-018 — Softmax training identifies a language model’s final-layer representation only up to an invertible linear map, so no inner product on it is intrinsic, and Park, Choe and Veitch’s causal inner product is fixed by a stipulation (Active)
    - THEORY-234 — Two softmax next-token predictors induce the same distribution exactly when their embeddings and unembeddings are related by an invertible linear map on the subspaces that enter the logits, with arbitrary distortion off them, so a linear relational property is shared by all such models or none when it lies in those subspaces, and can fail to be shared otherwise (Active)
THEORY-721 — On a complex Hilbert space no orthonormal basis or tensor factorisation of given dimensions is distinguished by the space alone, and orthonormal bases correspond exactly to commutative special †-Frobenius algebras; over the reals that correspondence fails (Active)

From Symmetry forces spectral degeneracy but degeneracy does not identify a symmetry: an operator commuting with a group has eigenspaces built from its real irreducibles, so abelian rotation groups force pairs over ℝ, and the spectrum fixes neither the group nor a basis inside a multiplet

THEORY-019 — Symmetry forces spectral degeneracy but degeneracy does not identify a symmetry: an operator commuting with a group has eigenspaces built from its real irreducibles, so abelian rotation groups force pairs over ℝ, and the spectrum fixes neither the group nor a basis inside a multiplet (Active)
- THEORY-084 — Because the Gauss–Newton Fisher JᵀJ and the NTK Gram matrix JJᵀ share their non-zero spectrum, the lazy-regime Fisher of a ReLU network on spherical data inherits the NTK’s harmonic-degree blocks, and a spectral threshold on it is well posed only at gaps between blocks (Proposed)

From For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix

THEORY-717 — For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix (Active)
- THEORY-156 — For experiments on a finite parameter set whose outcome spaces are standard Borel, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation (Active)
  - THEORY-1096 — Blackwell’s comparison of experiments, its easy direction: an experiment is a family of distributions indexed by a common parameter, and if one experiment is a garbling of another by a Markov kernel, every risk attainable with the garbling in any decision problem with bounded losses is attainable with the original, on any outcome spaces (Active)
  - THEORY-1177 — For experiments on a finite parameter set, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation (Proposed)

From Aligning three networks jointly, through one multi-marginal fused Gromov–Wasserstein coupling with an anchor-based random-walk cost and a cluster decomposition, finds held-out correspondences better than pairwise and existing multi-network methods in most cases, as the paper’s plots show, in both pairwise and whole-tuple hits, on benchmarks whose true correspondences are consistent across the three networks by construction; whether the joint coupling itself is the cause is not separated from the new cost and the clustering

THEORY-1159 — Aligning three networks jointly, through one multi-marginal fused Gromov–Wasserstein coupling with an anchor-based random-walk cost and a cluster decomposition, finds held-out correspondences better than pairwise and existing multi-network methods in most cases, as the paper’s plots show, in both pairwise and whole-tuple hits, on benchmarks whose true correspondences are consistent across the three networks by construction; whether the joint coupling itself is the cause is not separated from the new cost and the clustering (Active)
- THEORY-1210 — When several networks are aligned, fitting one joint multi-marginal coupling finds held-out correspondences better than fitting them in pairs, and the gain comes from the joint fit itself, not from the cost or the search space that come with it (Proposed)

From Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned

THEORY-720 — Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned (Active)
- THEORY-174 — Classical simulations between no-signalling empirical models on different scenarios never create contextuality, and a convex map between two scenarios’ model sets is induced by a classical non-adaptive procedure exactly when some non-contextual model of the hom scenario that satisfies its predicate induces it (Active)
  - THEORY-1170 — A convex map between two scenarios’ model sets is induced by a classical adaptive procedure exactly when some non-contextual model of the corresponding adaptive hom scenario induces it (Proposed)

From In Contextuality-by-Default 2.0, whether a system of non-binary variables is contextual depends on which variables represent the measurements: coarse-graining can turn a noncontextual system into a contextual one, while for connections of two variables, and for consistently connected systems, CbD 1.0 and 2.0 agree

THEORY-722 — In Contextuality-by-Default 2.0, whether a system of non-binary variables is contextual depends on which variables represent the measurements: coarse-graining can turn a noncontextual system into a contextual one, while for connections of two variables, and for consistently connected systems, CbD 1.0 and 2.0 agree (Active)
- THEORY-168 — In Contextuality-by-Default, whether a fixed set of measurements is contextual depends on how the system is represented: which couplings are imposed, and which dichotomizations of the variables are included (Active)

From Weight decay on factors is a nuclear-norm penalty on their product: with tied weights λ‖W‖²_F = λ‖WWᵀ‖, and for untied factors the least (‖U‖²_F + ‖V‖²_F)/2 over UVᵀ = M is ‖M‖, so for any loss that sees the weights only through the product, square-penalized training is nuclear-norm-penalized training of a product of bounded rank

THEORY-726 — Weight decay on factors is a nuclear-norm penalty on their product: with tied weights λ‖W‖²_F = λ‖WWᵀ‖*, and for untied factors the least (‖U‖²_F + ‖V‖²_F)/2 over UVᵀ = M is ‖M‖*, so for any loss that sees the weights only through the product, square-penalized training is nuclear-norm-penalized training of a product of bounded rank (Active)
- THEORY-180 — In a tied single-head attention layer trained by ridge-penalized empirical risk minimization on high-dimensional Gaussian sequences, weight decay is a nuclear-norm penalty on the query–key map, and the learned map is a soft-thresholded noisy copy of the target: its outliers are recovered target directions and its bulk is finite-sample noise (Proposed)

From For an embedding whose Gram matrix is the absolute co-occurrence deviation matrix at full rank, words on a periodic lattice whose positive and negative co-occurrence parts each depend only on separation have Fourier modes of the lattice as principal components, in degenerate cos/sin pairs with amplitude the square root of the kernel’s Fourier transform, and the exponential kernel gives a closed loop with integer frequencies on a periodic lattice and an open curve with non-integer frequencies on an interval

THEORY-730 — For an embedding whose Gram matrix is the absolute co-occurrence deviation matrix at full rank, words on a periodic lattice whose positive and negative co-occurrence parts each depend only on separation have Fourier modes of the lattice as principal components, in degenerate cos/sin pairs with amplitude the square root of the kernel’s Fourier transform, and the exponential kernel gives a closed loop with integer frequencies on a periodic lattice and an open curve with non-integer frequencies on an interval (Active)
- THEORY-183 — When the co-occurrence of words on a latent continuum depends only on their separation, a spectral word embedding places them on the Fourier modes of that continuum, so the circles and rippled curves of months, years and places come from a symmetry of the corpus statistics (Proposed)

From In four natural odour sources (strawberry, tomato, blueberry and mouse-urine volatiles), the rank order of correlations between compound concentrations has the Betti-curve signature of points near the boundary of a three-dimensional hyperbolic ball, not of any uniform Euclidean cube tried, and human odour-descriptor profiles for 127 odorants have that of a full three-dimensional hyperbolic ball

THEORY-191 — In four natural odour sources (strawberry, tomato, blueberry and mouse-urine volatiles), the rank order of correlations between compound concentrations has the Betti-curve signature of points near the boundary of a three-dimensional hyperbolic ball, not of any uniform Euclidean cube tried, and human odour-descriptor profiles for 127 odorants have that of a full three-dimensional hyperbolic ball (Active)
- THEORY-618 — Neural circuits that must represent hierarchically organised states in few dimensions use a hyperbolic geometry, three-dimensional where it has been measured, because hyperbolic space holds exponentially many well-separated states within a small radius; the argument from Zipf’s law to this geometry does not hold, and the evidence is two Betti-curve studies tested only against uniform Euclidean nulls (Proposed) — also carries further THEORY-192
THEORY-192 — In twelve sessions of rat dorsal CA1 from three data sets, pairwise correlations among place cells have the Betti-curve signature of points sampled uniformly from a three-dimensional hyperbolic ball, not of a low-dimensional Euclidean cube, and the fitted radius rose with the logarithm of exploration time (Active)

From In the hyperbolic plane, a root carrying deg_max ≥ 15 chains of length m, embedded with worst-case distortion 1 + ε for ε ≤ 1, forces some node to hyperbolic distance at least (ℓ/(4ε))·ln(deg_max/(π√2)) from the origin, so fixed-point Poincaré coordinates need Ω((ℓ/ε) log deg_max) bits

THEORY-734 — In the hyperbolic plane, a root carrying deg_max ≥ 15 chains of length m, embedded with worst-case distortion 1 + ε for ε ≤ 1, forces some node to hyperbolic distance at least (ℓ/(4ε))·ln(deg_max/(π√2)) from the origin, so fixed-point Poincaré coordinates need Ω((ℓ/ε) log deg_max) bits (Active)
- THEORY-196 — A finite tree embeds in the hyperbolic plane with worst-case distortion 1 + ε, but because a hyperbolic distance d takes about d bits to write, its coordinates need Θ((ℓ/ε) log deg_max) bits: logarithmic in the branching, linear in the longest path ℓ, so hyperbolic space is economical for shallow bushy hierarchies and not for deep ones (Proposed)

From For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose’s invariant subspaces, which exist on open sets but not generically at every scale

THEORY-733 — For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose’s invariant subspaces, which exist on open sets but not generically at every scale (Active)
- THEORY-257 — Dynamical independence, zero information flow from the microscopic history into a coarse-graining given its own history, fixes emergent macroscopic variables by the dynamics rather than by a chosen query, but for linear VAR models of order above 1 generically none is exact, and the optima are many (Proposed)

From With a kernel normalised to k(x,x) = 1, the covariance operator of a distribution is a density operator, and its quantum relative entropy is a lower bound on the Kullback–Leibler divergence that is tight for every pair of distributions only for an orthonormal, hence discrete, embedding, lies above the KL of the kernel-smoothed distributions, and on the torus, with Fourier coefficients proportional to e^(−σ‖ω‖₁) and strictly positive Lipschitz densities, is within O(σ²) of KL as the bandwidth σ shrinks

THEORY-313 — With a kernel normalised to k(x,x) = 1, the covariance operator of a distribution is a density operator, and its quantum relative entropy is a lower bound on the Kullback–Leibler divergence that is tight for every pair of distributions only for an orthonormal, hence discrete, embedding, lies above the KL of the kernel-smoothed distributions, and on the torus, with Fourier coefficients proportional to e^(−σ‖ω‖₁) and strictly positive Lipschitz densities, is within O(σ²) of KL as the bandwidth σ shrinks (Active)
- THEORY-811 — With a kernel normalised to unit diagonal, the kernel quantum relative entropy of two distributions is within O(σ²) of their Kullback–Leibler divergence as the bandwidth σ shrinks, for translation-invariant kernels on the torus generally and for kernels on other compact groups (Proposed)

From For a fixed linear operator in an inner-product norm, growth over time t is the top singular value of the propagator, which the eigenvalues bound only from below; it starts at the numerical abscissa, which can be positive when every mode decays; and the forced response is bounded by the condition number of the eigenvectors over the distance to the spectrum

THEORY-727 — For a fixed linear operator in an inner-product norm, growth over time t is the top singular value of the propagator, which the eigenvalues bound only from below; it starts at the numerical abscissa, which can be positive when every mode decays; and the forced response is bounded by the condition number of the eigenvectors over the distance to the spectrum (Active)
- THEORY-323 — Under spatially uncorrelated white noise, a stable non-normal linear system holds more variance than the sum of its pole contributions, and equality holds only when the operator is normal, so its spectrum understates the standing variance (Proposed)

From For a linear system du/dt = A(t)u, non-normality is necessary for time dependence to destabilize an operator that is uniformly stable at every instant, and a non-normal operator neutral at every instant can be asymptotically unstable under switching

THEORY-724 — For a linear system du/dt = A(t)u, non-normality is necessary for time dependence to destabilize an operator that is uniformly stable at every instant, and a non-normal operator neutral at every instant can be asymptotically unstable under switching (Active)
- THEORY-325 — Random variation of a linear operator destabilizes it generically, needing less variation the more non-normal and less damped the mean operator (Proposed)

From Locality-sensitive hashing answers approximate near-neighbour queries over n points with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, where ρ = ln(1/p1)/ln(1/p2) is set by how much more often near pairs collide than far ones and is at most 1/(1+ε) for approximation 1+ε in Hamming space, with constant success probability per query; the paper gives upper bounds only and proves no limit on what organizing can buy

THEORY-388 — Locality-sensitive hashing answers approximate near-neighbour queries over n points with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, where ρ = ln(1/p1)/ln(1/p2) is set by how much more often near pairs collide than far ones and is at most 1/(1+ε) for approximation 1+ε in Hamming space, with constant success probability per query; the paper gives upper bounds only and proves no limit on what organizing can buy (Active)
- THEORY-372 — For c-approximate near-neighbour search over n points in Euclidean space, space n^{1+ρu+o(1)} and query time n^{ρq+o(1)} are achievable on any data set whenever c²√ρq + (c²−1)√ρu ≥ √(2c²−1) (c√ρq + (c−1)√ρu ≥ √(2c−1) in Hamming space), and no list-of-points structure, the model of known data-independent hashing, does better on random instances; without that model the curve is matched only at query time n^{o(1)} by one or two cell probes (Active)

From For positive relational algebra on relations whose tuples are annotated from a commutative semiring, every query’s annotated answer equals its answer over provenance polynomials N[X] in the input tuple ids, evaluated at the input annotations, so set, bag, c-table and event-table semantics all factor through the polynomials; revaluing inputs can remove output tuples but never add them, and nothing is shown for negation, difference or aggregation

THEORY-373 — For positive relational algebra on relations whose tuples are annotated from a commutative semiring, every query’s annotated answer equals its answer over provenance polynomials N[X] in the input tuple ids, evaluated at the input annotations, so set, bag, c-table and event-table semantics all factor through the polynomials; revaluing inputs can remove output tuples but never add them, and nothing is shown for negation, difference or aggregation (Active)
- THEORY-380 — For datalog over a finite input instance annotated from a commutative ω-continuous semiring, with an answer’s tag defined as the sum over its derivation trees of the product of its leaves’ tags, the tags are the least fixed point of a finite polynomial system and factor through formal power series N∞⟦X⟧ in the input tuple ids; an answer tuple’s series has an infinite coefficient exactly when one of its derivation trees contains a tuple lying on a cycle of unit rules (Active)

From Recovery of a k-sparse vector by ℓ₁ minimization from n random measurements in dimension N switches, as N grows with δ = n/N and ρ = k/n fixed, from near-certain success to near-certain failure across a curve ρ(δ) that, for Gaussian matrices, the authors’ earlier face-counting theorems fix; in 2,948,000 optimizations at N = 200, 400 and 1600, eight non-Gaussian ensembles approached that curve in a way consistent with gaps shrinking as N^(−1/2), an exponent fitted on the same data, while at finite N the location depended on the ensemble

THEORY-391 — Recovery of a k-sparse vector by ℓ₁ minimization from n random measurements in dimension N switches, as N grows with δ = n/N and ρ = k/n fixed, from near-certain success to near-certain failure across a curve ρ(δ) that, for Gaussian matrices, the authors’ earlier face-counting theorems fix; in 2,948,000 optimizations at N = 200, 400 and 1600, eight non-Gaussian ensembles approached that curve in a way consistent with gaps shrinking as N^(−1/2), an exponent fitted on the same data, while at finite N the location depended on the ensemble (Active)
- THEORY-812 — As dimension grows with the undersampling and sparsity ratios fixed, the ℓ₁ phase transition for recovering sparse vectors from random measurements sits at the same curve for a broad class of random-matrix ensembles, those with a kind of orthogonality or weak independence, as for Gaussian matrices, while differing at finite size (Proposed)

From A random linear map into k ≥ 4(ε²/2 − ε³/3)⁻¹ ln n dimensions keeps every pairwise squared distance of a fixed set of n points in ℓ₂ within 1 ± ε, because each difference vector’s squared length concentrates with tails exp(−(k/2)(ε²/2 − ε³/3)) and a union bound over the pairs costs only ln n; the proofs cover a random subspace, Gaussian entries and independent zero-mean ±1 or √3·{+1, 0, −1} entries, and no matrix with nonnegative entries

THEORY-397 — A random linear map into k ≥ 4(ε²/2 − ε³/3)⁻¹ ln n dimensions keeps every pairwise squared distance of a fixed set of n points in ℓ₂ within 1 ± ε, because each difference vector’s squared length concentrates with tails exp(−(k/2)(ε²/2 − ε³/3)) and a union bound over the pairs costs only ln n; the proofs cover a random subspace, Gaussian entries and independent zero-mean ±1 or √3·{+1, 0, −1} entries, and no matrix with nonnegative entries (Active)
- THEORY-395 — For ε between about lg^0.5001 n/√min{n, d} and 1, some n-point set in ℝ^d needs Ω(ε⁻² lg(ε²n)) dimensions under any map, linear or not, that keeps its pairwise squared distances within 1 ± ε, so a random projection with a union bound is optimal up to a constant for the worst set; nearly orthogonal sets can need fewer, and the bound says nothing about typical data (Active)
  - THEORY-399 — Storing every inner product of n points of the unit ball of R^k to additive error ε takes Θ(n log n/ε²) bits once k ≥ log n/ε² and Θ(nk log(2 + log n/(ε²k))) below, for 1/n^0.49 ≤ ε ≤ 0.1, and comparing these counts across dimensions shows that some n such points cannot be mapped into fewer than Ω(min{k, log(2 + ε²n)/ε²}) dimensions with additive error ε for every ε ≥ 2/√n; this is a worst-case lower bound with unstated constants, the small-ε bit count is open for k above that dimension, and no one-family embedding that small is proved (Active)

From On a finite data set, a symmetric positive affinity normalized to a Markov operator is conjugate to a symmetric one, and its eigenvectors scaled by powers of their eigenvalues embed the data so that Euclidean distance equals the diffusion distance weighted by the inverse stationary law

THEORY-728 — On a finite data set, a symmetric positive affinity normalized to a Markov operator is conjugate to a symmetric one, and its eigenvectors scaled by powers of their eigenvalues embed the data so that Euclidean distance equals the diffusion distance weighted by the inverse stationary law (Active)
- THEORY-565 — From a symmetric, positivity-preserving, positive semi-definite affinity, the normalized Markov operator is conjugate to a symmetric one, so its leading eigenfunctions scaled by powers of their eigenvalues embed the data with Euclidean distance equal to diffusion distance, and a density-corrected normalization recovers the Laplace–Beltrami operator independently of how the data were sampled (Proposed) — also carries further THEORY-744
THEORY-744 — As the kernel width ε goes to 0, Coifman and Lafon’s α-normalized diffusion operator (I − P_{ε,α})/ε, for data sampled with density q = e^{−U} from a compact manifold without boundary, converges uniformly on smooth functions, at rate O(ε), to Δf + 2(1 − α)⟨∇U, ∇f⟩ with Δ the non-negative Laplace–Beltrami operator, which is minus the generator of a reversible diffusion with drift −2(1 − α)∇U and invariant density proportional to q^{2(1−α)}; α = 0 gives potential 2U and invariant density q², α = 1/2 gives the Langevin dynamics whose invariant density is the sampling density itself, and α = 1 gives Brownian motion on the manifold, with no trace of the density (Active)

From An adjunction between a category and the opposite of another restricts to a dual equivalence between the full subcategories on which its unit and counit are isomorphisms

THEORY-729 — An adjunction between a category and the opposite of another restricts to a dual equivalence between the full subcategories on which its unit and counit are isomorphisms (Active)
- THEORY-604 — Most mathematical dualities are adjunctions between one category and the opposite of another, which become dual equivalences on their fixed points; what makes them dualities rather than equivalences is the reversal of arrows, so many of the ‘dualities’ physicists name are, in the mathematician’s sense, equivalences or symmetries (Proposed)

metaphysics

From A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape

THEORY-040 — A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape (Proposed)
- THEORY-1118 — McKenna and Coates argue, with Dryden and with Buss and Westlund, that a condition on free agency stated only in the agent’s present psychological structure is met by an agent whose structure a manipulator installed, so a mesh account must count that agent free or add a historical condition; Frankfurt, read first-hand, takes the first option (Active)

From In relational quantum mechanics as Rovelli first stated it, the relation between two observers’ descriptions is itself a quantum fact relative to a third system, so no observer’s view can be deduced from another’s and there is no description in toto, only a net of partial descriptions; the views are held consistent only in that the third system predicts correlated outcomes, which is shown for a single measurement and asserted in general

THEORY-1080 — In relational quantum mechanics as Rovelli first stated it, the relation between two observers’ descriptions is itself a quantum fact relative to a third system, so no observer’s view can be deduced from another’s and there is no description in toto, only a net of partial descriptions; the views are held consistent only in that the third system predicts correlated outcomes, which is shown for a single measurement and asserted in general (Active)
- THEORY-1222 — In relational quantum mechanics as stated in Rovelli (1996), rules (i) and (ii), that there is no state except within a further observer’s information and no information without interaction, exclude any contradiction between observers’ views: what a third system predicts of two observers’ information always agrees with what it finds on interacting with them (Proposed)

From Di Biagio and Rovelli locate the Frauchiger–Renner contradiction and the violation of the Local Friendliness inequalities in one step, a superobserver using facts relative to a friend he holds in full quantum control as if they were his own, which is the classical sum rule applied where those facts are not stable for him; they base quantum ontology on facts relative to systems rather than on stable facts, arguing that stability is itself relational and only approximate and counting the Local Friendliness violation as evidence, with an absolute macroreality emerging only approximately and only relative to systems for which decoherence is strong

THEORY-1081 — Di Biagio and Rovelli locate the Frauchiger–Renner contradiction and the violation of the Local Friendliness inequalities in one step, a superobserver using facts relative to a friend he holds in full quantum control as if they were his own, which is the classical sum rule applied where those facts are not stable for him; they base quantum ontology on facts relative to systems rather than on stable facts, arguing that stability is itself relational and only approximate and counting the Local Friendliness violation as evidence, with an absolute macroreality emerging only approximately and only relative to systems for which decoherence is strong (Active)
- THEORY-1230 — In relational quantum mechanics, with facts fixed by interaction and the classical sum rule used only within one context or where facts are stable, facts relative to different systems never yield contradictory predictions in extended Wigner’s-friend scenarios, and a superobserver’s later facts agree with his friend’s (Proposed)

model-comparison

From In a linear-in-parameters model with Gaussian noise and a Gaussian prior, the evidence is exactly the best-fit likelihood times an Occam factor set by the posterior curvature, and when the design grows linearly with n that factor costs (k/2) log n

THEORY-718 — In a linear-in-parameters model with Gaussian noise and a Gaussian prior, the evidence is exactly the best-fit likelihood times an Occam factor set by the posterior curvature, and when the design grows linearly with n that factor costs (k/2) log n (Active)
- THEORY-087 — The Bayesian complexity penalty grows as (d/2) log n in a regular model, where the curvature at the optimum sets the Occam factor, and as λ log n in a singular one, where λ is the real log canonical threshold; with the prior positive on the optimal set, λ falls below d/2 when that set is infinite or has a point of degenerate curvature, and equals d/2 when it is finitely many points of non-degenerate curvature (Active)
  - THEORY-080 — In a linearised network with a Gaussian prior on its parameters, the Occam factor is a sum over the NTK’s eigenvalues, so on spherical data each harmonic pays ½ log(1 + βnµ_k/α) and the kernel’s eigenvalue decay schedules the complexity penalty across frequencies (Proposed)

From Orseau, McGill and Legg cast whether an identified system is an agent or a device as Bayesian model comparison between descriptions of the function it optimises and descriptions of its input-output mapping, so that the verdict is relative to the observer’s prior; on seven toy gridworld trajectories goal-seeking comes out agent-like and wall-following device-like, and they argue that under a speed prior everything may look like an agent

THEORY-624 — Orseau, McGill and Legg cast whether an identified system is an agent or a device as Bayesian model comparison between descriptions of the function it optimises and descriptions of its input-output mapping, so that the verdict is relative to the observer’s prior; on seven toy gridworld trajectories goal-seeking comes out agent-like and wall-following device-like, and they argue that under a speed prior everything may look like an agent (Active)
- THEORY-1236 — Agency is relative to an observer: an identified system is an agent, for that observer, to the degree that a description of the function it optimises explains its behaviour better under the observer’s prior than any description of its input-output mapping (Proposed)

moral-psychology

From Moral judgment draws on several innately prepared intuitive foundations, at least Care, Fairness, Loyalty, Authority and Sanctity, each a response to a different adaptive challenge, and Sanctity descends from pathogen avoidance rather than from cooperation

THEORY-539 — Moral judgment draws on several innately prepared intuitive foundations, at least Care, Fairness, Loyalty, Authority and Sanctity, each a response to a different adaptive challenge, and Sanctity descends from pathogen avoidance rather than from cooperation (Proposed)
- THEORY-613 — The Moral Foundations Questionnaire-2’s six factors, Care, Equality, Proportionality, Loyalty, Authority and Purity, fit across 19 nations, Equality and Proportionality are nearly uncorrelated in WEIRD nations, and the US split between individualizing and binding foundations appears in only three national networks (Proposed)

motivation

From A regulation is autonomous to the degree it is integrated with one’s other values and needs: autonomy is graded, and is not set by the order of the attitude that endorses it

THEORY-054 — A regulation is autonomous to the degree it is integrated with one’s other values and needs: autonomy is graded, and is not set by the order of the attitude that endorses it (Proposed)
- THEORY-047 — A regulation taken in under controlling conditions is introjected rather than integrated, so the social history of how a value was acquired shapes how autonomous acting on it is (Proposed)

From Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one

THEORY-067 — Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one (Proposed)
- THEORY-1117 — Ainslie proposes that self-control over time needs no faculty of will: hyperbolic discounting makes preferences that dominate at different delays, these grow into interests that bargain with no governor above them, and resolve is the recursive staking of expected future choices on the present one (Active)

From How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control

THEORY-070 — How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control (Proposed)
- THEORY-1110 — Shenhav, Botvinick and Cohen propose that control is allocated top-down by a dedicated specifying system: dACC computes the expected value of a control signal, its probability-weighted payoff less an intrinsic cost rising with intensity, and specifies the signal that lateral prefrontal and subcortical structures implement (Active)

From Between tests, uncorrected for publication bias, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); tests in which the plan was rehearsed had larger effects (.45 against .33, with no gain beyond one rehearsal), tests in which it was recorded had smaller ones (.31 against .44), and adding how or for how long to a time and place went with about half the effect (.24 against .46)

THEORY-908 — Between tests, uncorrected for publication bias, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); tests in which the plan was rehearsed had larger effects (.45 against .33, with no gain beyond one rehearsal), tests in which it was recorded had smaller ones (.31 against .44), and adding how or for how long to a time and place went with about half the effect (.24 against .46) (Active)
- THEORY-1175 — Between tests, implementation intentions in the contingent if-then format had larger effects than plans that schedule when, where and what without linking cue to response (d = .43 against .29); rehearsing the plan once raised the effect, keeping a copy to revisit lowered it (.31 against .44), and adding how or for how long to a time-and-place cue roughly halved it (Proposed)

From Gollwitzer and Sheeran (2006) conclude, narratively from a handful of studies that measured or manipulated the goal intention, that implementation intentions affect performance only when the goal intention they serve is strong and activated, so the planned response is not elicited mechanically by its cue; in 2024, across eight tests, plan effects were larger with strong, activated or self-concordant goal intentions than with weak, unactivated or self-discordant ones (d = .79 for the difference), and in 31 physical-activity tests split at the median of intention, planning helped with strong intentions (d = .31) and had no effect with weak ones (d ≈ 0)

THEORY-904 — Gollwitzer and Sheeran (2006) conclude, narratively from a handful of studies that measured or manipulated the goal intention, that implementation intentions affect performance only when the goal intention they serve is strong and activated, so the planned response is not elicited mechanically by its cue; in 2024, across eight tests, plan effects were larger with strong, activated or self-concordant goal intentions than with weak, unactivated or self-discordant ones (d = .79 for the difference), and in 31 physical-activity tests split at the median of intention, planning helped with strong intentions (d = .31) and had no effect with weak ones (d ≈ 0) (Active)
- THEORY-1176 — Implementation intentions affect performance only when the goal intention they serve is strong and activated, so the cue-triggered response is gated by the goal and does not fire on the cue alone (Proposed)

THEORY-918 — Across 642 experimental tests to September 2019, forming implementation intentions improved outcomes, with uncorrected effects on behaviour (d = .27), cognitive performance (.47), affect regulation (.66) and social judgement (.52), but publication bias was substantial: the pooled d = .36 falls to a model-averaged d = .15 under robust Bayesian meta-analysis, which still found extreme evidence of an effect (BF10 = 319), with large heterogeneity (Active)
- THEORY-1192 — Across 642 experimental tests to September 2019, forming implementation intentions reliably improved behaviour, cognitive performance, affect regulation and social judgement, but publication bias was substantial: the pooled d = .36 falls to a model-averaged d = .15 under robust Bayesian meta-analysis, with large heterogeneity (Proposed)

myth-and-folklore

THEORY-167 — The Russian wondertale is built from a small fixed set of character functions in a fixed order, so that all wondertales share one composition and differ in who performs the functions and how (Proposed)
- THEORY-1164 — Propp proposes, from his analysis of Afanasyev’s tales 50–151, that the Russian wondertale is built from 31 character functions, each defined by its consequence for the action, occurring in one order and falling to seven roles, so that wondertales share one composition and differ in who performs the functions and how (Active)

natural-sciences

From For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient

THEORY-026 — For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient (Active)
- THEORY-817 — Once an agent’s actions affect the percepts it receives, maximal prediction of those percepts is generally not what thermodynamic efficiency requires: in percept-action loops with feedback, maximally predictive agents generally extract less work than the most efficient agent (Proposed)

From On McCutcheon and colleagues’ survey, bacterial endosymbionts lose their own cell-envelope and envelope-transport genes together below about 400 kb, and begin to lose translation genes (ribosomal proteins, tRNAs, aminoacyl-tRNA synthetases) below about 200 kb, so that the smallest-genome symbionts cannot be self-sufficient and their membranes, transport and protein synthesis are presumably supplied by the host, as for organelles

THEORY-1049 — On McCutcheon and colleagues’ survey, bacterial endosymbionts lose their own cell-envelope and envelope-transport genes together below about 400 kb, and begin to lose translation genes (ribosomal proteins, tRNAs, aminoacyl-tRNA synthetases) below about 200 kb, so that the smallest-genome symbionts cannot be self-sufficient and their membranes, transport and protein synthesis are presumably supplied by the host, as for organelles (Active)
- THEORY-1208 — Bacterial endosymbionts lose their cell-envelope and envelope-transport genes together below about 400 kb and begin to lose translation genes below about 200 kb, and the host supplies the membranes, transport and protein synthesis that the smallest-genome symbionts no longer encode, as for organelles (Proposed)

From On Karnkowska and colleagues’ genome analysis, the oxymonad Monocercomonoides exilis lost its mitochondrion only after the organelle’s essential functions had been removed one by one, Fe–S cluster assembly to a bacterial-type SUF system shared with mitochondriate relatives and then the glycine cleavage system by loss, and the loss left the rest of the cell canonical, with no compensating changes found

THEORY-1098 — On Karnkowska and colleagues’ genome analysis, the oxymonad Monocercomonoides exilis lost its mitochondrion only after the organelle’s essential functions had been removed one by one, Fe–S cluster assembly to a bacterial-type SUF system shared with mitochondriate relatives and then the glycine cleavage system by loss, and the loss left the rest of the cell canonical, with no compensating changes found (Active)
- THEORY-1223 — Monocercomonoides exilis lost its mitochondrion only after the organelle’s essential functions had been removed in steps, Fe–S cluster assembly to a cytosolic SUF system and then the glycine cleavage system by loss, and the loss required no compensating change elsewhere in the cell (Proposed)

From On Kay and colleagues’ relaxed-clock dating of pre-LECA gene duplications, most duplications in eukaryotic gene families of archaeal origin, including actin, tubulin, vesicle-trafficking and nuclear-targeted families, predate the onset of duplications in genes of alphaproteobacterial origin at about 2.2 billion years ago, which the authors take as the start of mitochondrial endosymbiosis, so the host was elaborating its cytoskeleton, endomembranes and nuclear machinery before the mitochondrion; each date is a maximum age for its paralogues’ functions, many intervals reach below 2.2 billion years, and the order of phagocytosis and the mitochondrion is left open

THEORY-1099 — On Kay and colleagues’ relaxed-clock dating of pre-LECA gene duplications, most duplications in eukaryotic gene families of archaeal origin, including actin, tubulin, vesicle-trafficking and nuclear-targeted families, predate the onset of duplications in genes of alphaproteobacterial origin at about 2.2 billion years ago, which the authors take as the start of mitochondrial endosymbiosis, so the host was elaborating its cytoskeleton, endomembranes and nuclear machinery before the mitochondrion; each date is a maximum age for its paralogues’ functions, many intervals reach below 2.2 billion years, and the order of phagocytosis and the mitochondrion is left open (Active)
- THEORY-1203 — The host lineage elaborated its cytoskeleton, endomembrane system and nuclear machinery before mitochondrial endosymbiosis began, about 2.2 billion years ago: most pre-LECA duplications in eukaryotic gene families of archaeal origin predate it (Proposed)

From On Sørensen and colleagues’ review, a bacterial endosymbiont becomes genetically integrated, the authors’ mark of an organelle, when nuclear genes whose products are imported into it functionally replace essential genes it has lost, and this does not require gene transfer from the endosymbiont: in the Paulinella chromatophore and the nitroplast, the two systems with hundreds of imported proteins, many of which fill gaps left by lost genes, the imported proteins come mostly from the host’s own genes or from other bacteria, and in the nitroplast no gene transfer from the endosymbiont has been found

THEORY-1100 — On Sørensen and colleagues’ review, a bacterial endosymbiont becomes genetically integrated, the authors’ mark of an organelle, when nuclear genes whose products are imported into it functionally replace essential genes it has lost, and this does not require gene transfer from the endosymbiont: in the Paulinella chromatophore and the nitroplast, the two systems with hundreds of imported proteins, many of which fill gaps left by lost genes, the imported proteins come mostly from the host’s own genes or from other bacteria, and in the nitroplast no gene transfer from the endosymbiont has been found (Active)
- THEORY-1228 — A bacterial endosymbiont becomes genetically integrated, an organelle, when host proteins imported into it replace genes it has lost, and this does not require gene transfer from the endosymbiont: in the Paulinella chromatophore and the nitroplast, the two systems with hundreds of imported proteins, the replacing genes are mostly the host’s own or come from other bacteria (Proposed)

From Tobiasson, Luo, Wolf and Koonin find, on 5,850 eukaryotic gene stems, that normalized stem length does not behave as a clock of acquisition time: alphaproteobacterial stems are longer than Asgard ones for the 70% of stems below 0.3 and shorter only above 0.35, oxidative-phosphorylation genes presumed acquired together vary by more than two orders of magnitude, and information-processing genes have short stems; they conclude that even normalized stem lengths are set primarily by factors other than acquisition time, at least over the long timescales analysed, and suggest that adaptation after acquisition lengthens stems

THEORY-1101 — Tobiasson, Luo, Wolf and Koonin find, on 5,850 eukaryotic gene stems, that normalized stem length does not behave as a clock of acquisition time: alphaproteobacterial stems are longer than Asgard ones for the 70% of stems below 0.3 and shorter only above 0.35, oxidative-phosphorylation genes presumed acquired together vary by more than two orders of magnitude, and information-processing genes have short stems; they conclude that even normalized stem lengths are set primarily by factors other than acquisition time, at least over the long timescales analysed, and suggest that adaptation after acquisition lengthens stems (Active)
- THEORY-1201 — Normalized stem length is not a clock of gene acquisition time: it is set mainly by factors other than when a gene was acquired, such as adaptation to the new cell after acquisition, so relative timings of eukaryogenesis drawn from stem lengths alone are not supported (Proposed)

From On Pittis and Gabaldón’s stem-length analysis of the LECA proteome, families of alphaproteobacterial origin and mitochondrial location sit closer to their prokaryotic relatives, after correction for each family’s rate, than families of other bacterial origin (P = 0.044), which sit closer than archaeal families, and the authors read this as late acquisition of the mitochondrion by a host whose proteome was already chimaeric, under their stated assumption that rates before and after LECA are correlated

THEORY-1102 — On Pittis and Gabaldón’s stem-length analysis of the LECA proteome, families of alphaproteobacterial origin and mitochondrial location sit closer to their prokaryotic relatives, after correction for each family’s rate, than families of other bacterial origin (P = 0.044), which sit closer than archaeal families, and the authors read this as late acquisition of the mitochondrion by a host whose proteome was already chimaeric, under their stated assumption that rates before and after LECA are correlated (Active)
- THEORY-1206 — The mitochondrion was acquired late in eukaryogenesis, by a host whose proteome was already chimaeric, with genes from several bacterial sources acquired before it (Proposed)

From On Shigenobu and colleagues’ genome of Buchnera sp. APS, the aphid symbiont keeps the genes for the amino acids its host cannot make and has lost almost all of those its host can, uses host-made glutamate and aspartate as precursors, and lacks phospholipid synthesis and nearly all its own transcriptional regulators, so that its metabolism interlocks with the host’s and it presumably lives only inside the bacteriocyte, a state reached by gene loss from an E. coli-like ancestor

THEORY-1103 — On Shigenobu and colleagues’ genome of Buchnera sp. APS, the aphid symbiont keeps the genes for the amino acids its host cannot make and has lost almost all of those its host can, uses host-made glutamate and aspartate as precursors, and lacks phospholipid synthesis and nearly all its own transcriptional regulators, so that its metabolism interlocks with the host’s and it presumably lives only inside the bacteriocyte, a state reached by gene loss from an E. coli-like ancestor (Active)
- THEORY-1219 — Buchnera’s metabolism interlocks with its aphid host’s: it makes the essential amino acids its host cannot from host-supplied glutamate and aspartate, depends on the host for the non-essential amino acids and its membrane phospholipids, and cannot live outside the bacteriocyte (Proposed)

From In Tobiasson, Luo, Wolf and Koonin’s constrained-tree survey of widespread eukaryotic protein families, Asgard archaea are the largest single prokaryotic source, with about half the likelihood weight in the best-resolved families and the lead in nearly every functional system, while Alphaproteobacteria contribute a small, coherent set centred on the mitochondrion and other bacteria about a third of families, scattered across functions with no pathway led by one donor except a narrow Myxococcota signal in nucleotide-sugar and nicotinate metabolism; the authors read this as one Asgard host lineage, one endosymbiont and piecemeal gene transfer rather than further symbioses

THEORY-1104 — In Tobiasson, Luo, Wolf and Koonin’s constrained-tree survey of widespread eukaryotic protein families, Asgard archaea are the largest single prokaryotic source, with about half the likelihood weight in the best-resolved families and the lead in nearly every functional system, while Alphaproteobacteria contribute a small, coherent set centred on the mitochondrion and other bacteria about a third of families, scattered across functions with no pathway led by one donor except a narrow Myxococcota signal in nucleotide-sugar and nicotinate metabolism; the authors read this as one Asgard host lineage, one endosymbiont and piecemeal gene transfer rather than further symbioses (Active)
- THEORY-1197 — Eukaryogenesis involved one archaeal host lineage, of Asgard descent, and one bacterial endosymbiont, the alphaproteobacterial ancestor of mitochondria; the eukaryotic genes from other bacteria came by piecemeal gene transfer, not from further symbiotic partners (Proposed)

THEORY-1105 — In Bernabeu, Manzano-Morales, Marcet-Houben and Gabaldón’s reconstruction of the last eukaryotic common ancestor’s proteome, two bacterial phyla besides the mitochondrial ancestor, Myxococcota and Planctomycetota, each account for a share of the proteome comparable to that of Alphaproteobacteria and Asgard archaea under thresholds calibrated on those accepted partners, and stem-length distributions order the acquisitions as waves, Asgard first, then Planctomycetota, then the mitochondrial ancestor and Myxococcota together; the authors read this as serial associations with several partners, while conceding that their data cannot say whether any donor was a symbiont (Active)
- THEORY-1215 — Besides Asgard archaea and the alphaproteobacterial ancestor of mitochondria, Myxococcota and Planctomycetota were major donors of genes to the eukaryotic lineage before its last common ancestor, in waves: Asgard first, then Planctomycetota before the mitochondrion, then the mitochondrial ancestor and Myxococcota together (Proposed)

From Campbell and colleagues report, from four cicada species with five or six eggs each, that Hodgkinia cells per egg range from about 4,900 in a species with one Hodgkinia lineage to about 30,000, some six times as many, in Magicicada septendecim, whose Hodgkinia has split into many interdependent lineages and the only species that differs significantly, while Sulcia numbers are not raised, the transmission route is the same, and every egg sampled carries every lineage tracked by marker genes; the authors take the raised titre as a host adaptation that compensates for the symbiont’s fragmentation, and a cost of it

THEORY-1106 — Campbell and colleagues report, from four cicada species with five or six eggs each, that Hodgkinia cells per egg range from about 4,900 in a species with one Hodgkinia lineage to about 30,000, some six times as many, in Magicicada septendecim, whose Hodgkinia has split into many interdependent lineages and the only species that differs significantly, while Sulcia numbers are not raised, the transmission route is the same, and every egg sampled carries every lineage tracked by marker genes; the authors take the raised titre as a host adaptation that compensates for the symbiont’s fragmentation, and a cost of it (Active)
- THEORY-1214 — Cicada hosts whose endosymbiont Hodgkinia has split into many interdependent lineages compensate by sending more Hodgkinia cells to each egg, with Sulcia numbers and the transmission route unchanged, so that every egg still receives every lineage; the raised titre is a host adaptation driven by the symbiont’s fragmentation, and a cost of it (Proposed)

From On Sørensen and colleagues’ scenario, an endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations, and host effector control without vertical transmission, as in rhizobia, does not lead to it

THEORY-1107 — On Sørensen and colleagues’ scenario, an endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations, and host effector control without vertical transmission, as in rhizobia, does not lead to it (Active)
- THEORY-1211 — An endosymbiont becomes an organelle in steps: the host first targets effector proteins into it; an imported host protein redundant with one of its functions then lets the endosymbiont’s own gene decay, making it dependent; and a general import pathway, once evolved, brings in hundreds of proteins; this genetic integration arises only in permanent, vertically transmitted associations (Proposed)

From On Imachi and colleagues’ cultivation of Candidatus Prometheoarchaeum syntrophicum MK-D1, the first Asgard archaeon grown in the laboratory is a small, internally simple coccus with long branching protrusions that degrades amino acids only in syntrophy, handing hydrogen and formate to partners that can be exchanged, and from its genome and other Asgard genomes the authors infer that the Asgard ancestor was likewise dependent on metabolic partners; the isolate is a modern relative separated from the host lineage by about two billion years, grown only in co-culture, and the entangle–engulf–endogenize model built on it is a hypothesis the authors call not definitive

THEORY-1108 — On Imachi and colleagues’ cultivation of Candidatus Prometheoarchaeum syntrophicum MK-D1, the first Asgard archaeon grown in the laboratory is a small, internally simple coccus with long branching protrusions that degrades amino acids only in syntrophy, handing hydrogen and formate to partners that can be exchanged, and from its genome and other Asgard genomes the authors infer that the Asgard ancestor was likewise dependent on metabolic partners; the isolate is a modern relative separated from the host lineage by about two billion years, grown only in co-culture, and the entangle–engulf–endogenize model built on it is a hypothesis the authors call not definitive (Active)
- THEORY-1200 — The Asgard archaeal ancestor of eukaryotes depended on metabolic partners: like Candidatus Prometheoarchaeum syntrophicum, it degraded amino acids in syntrophy, handing hydrogen and formate to partners that removed them (Proposed)

From Santana-Molina, Williams, Snel and Spang find that the last eukaryotic common ancestor’s central carbon metabolism was chimeric: a few Asgard-archaeal enzymes in cytosolic glycolysis and the pentose phosphate pathway, four placed with Asgard archaea at bootstrap 39–97% and two ambiguously, most alphaproteobacterial ones in the mitochondrial TCA cycle and pyruvate conversion, and others from various bacteria with no third dominant donor, since remodelled by loss, horizontal replacement and retargeting so that compartment no longer follows origin; the authors take the cytosol–mitochondrion split as support for syntrophy

THEORY-1109 — Santana-Molina, Williams, Snel and Spang find that the last eukaryotic common ancestor’s central carbon metabolism was chimeric: a few Asgard-archaeal enzymes in cytosolic glycolysis and the pentose phosphate pathway, four placed with Asgard archaea at bootstrap 39–97% and two ambiguously, most alphaproteobacterial ones in the mitochondrial TCA cycle and pyruvate conversion, and others from various bacteria with no third dominant donor, since remodelled by loss, horizontal replacement and retargeting so that compartment no longer follows origin; the authors take the cytosol–mitochondrion split as support for syntrophy (Active)
- THEORY-1205 — Eukaryotes began in a syntrophic partnership: an archaeal host fermented sugars and passed reducing equivalents to an alphaproteobacterial endosymbiont that contributed a TCA cycle and an electron transport chain, which is why the last eukaryotic common ancestor’s central carbon metabolism is archaeal in cytosolic glycolysis and the pentose phosphate pathway and alphaproteobacterial in the mitochondrial TCA cycle (Proposed)

From The breakup of the Eulalia parent body beside the Jupiter 3:1 resonance, 785–965 Myr ago, can plausibly have supplied the cluster of large lunar craters dated near 800 Ma, but shown only for the most favourable of five model solutions, the one carried through, with an average largest lunar impactor too small for Copernicus, and with the cluster’s absolute date inherited from Copernicus alone

THEORY-499 — The breakup of the Eulalia parent body beside the Jupiter 3:1 resonance, 785–965 Myr ago, can plausibly have supplied the cluster of large lunar craters dated near 800 Ma, but shown only for the most favourable of five model solutions, the one carried through, with an average largest lunar impactor too small for Copernicus, and with the cluster’s absolute date inherited from Copernicus alone (Active)
- THEORY-1195 — About 800 Myr ago the breakup of the Eulalia parent body beside the Jupiter 3:1 resonance sent an impact shower to the terrestrial planets and the Moon that made the cluster of large lunar craters dated near 800 Ma (Proposed)

From In Bhattacharyya et al.’s single-cell swarm assays of E. coli MG1655, the daughters of one mother swarmed alike at the fourth generation sampled and varied again at the seventh; swarm size correlated with a plasmid fepA-promoter reporter of Fur derepression by low iron (Spearman r = 0.71), and an iron chelator or added iron shifted swarming and held the shared state longer; a three-class switching model fitted to two time points matched the lineages only with a delay of three to four generations before switching, which the authors read as ruling out a bistable switch

THEORY-179 — In Bhattacharyya et al.’s single-cell swarm assays of E. coli MG1655, the daughters of one mother swarmed alike at the fourth generation sampled and varied again at the seventh; swarm size correlated with a plasmid fepA-promoter reporter of Fur derepression by low iron (Spearman r = 0.71), and an iron chelator or added iron shifted swarming and held the shared state longer; a three-class switching model fitted to two time points matched the lineages only with a delay of three to four generations before switching, which the authors read as ruling out a bistable switch (Active)
- THEORY-1209 — In genetically identical E. coli, a cell’s swarming potential is a transient inherited state: its descendants share it for about four generations and lose it by about seven, and it tracks the cell’s iron status rather than a bistable switch (Proposed)

From In an ensemble of sign-consistent Boolean rules on the 87-gene core of E. coli’s regulatory network, a transient knockout or overexpression of one gene can leave the network in a different attractor after release, and every gene that does so reaches a strongly connected component containing a positive circuit

THEORY-178 — In an ensemble of sign-consistent Boolean rules on the 87-gene core of E. coli’s regulatory network, a transient knockout or overexpression of one gene can leave the network in a different attractor after release, and every gene that does so reaches a strongly connected component containing a positive circuit (Active)
- THEORY-780 — Transcriptional regulation alone can give a bacterium’s gene expression a memory: in E. coli, a transient knockdown or overexpression of one regulator, with the environment held fixed, can leave other genes in a different self-maintaining expression state after it is released, as Zhao, Wytock, Reynolds and Motter predict from a Boolean-network model of its regulatory core (Proposed)

From On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers’ low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead

THEORY-194 — On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers’ low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead (Active)
- THEORY-789 — The block coarse-graining that maximizes mutual information with the system beyond a buffer picks out the renormalization-group-relevant variables in general, so those variables can be found from samples by information alone, without being known beforehand, as Koch-Janusz and Ringel propose (Proposed)

From In a one-locus model with non-overlapping generations, random mating and weak selection, gene frequencies change as if each individual maximized its inclusive fitness, so a trait that costs its bearer spreads only if its effect on relatives exceeds 1/r times the cost; the increase of mean inclusive fitness is proved only for traits whose total effect on others is not negative

THEORY-408 — In a one-locus model with non-overlapping generations, random mating and weak selection, gene frequencies change as if each individual maximized its inclusive fitness, so a trait that costs its bearer spreads only if its effect on relatives exceeds 1/r times the cost; the increase of mean inclusive fitness is proved only for traits whose total effect on others is not negative (Active)
- THEORY-567 — In the donation game, where a cooperator pays c for another to get b, defectors win in a well-mixed population, and each of five mechanisms makes cooperators evolutionarily stable above a benefit-to-cost threshold: kin selection r > c/b, direct reciprocity w > c/b, indirect reciprocity q > c/b, network reciprocity b/c > k and group selection b/c > 1 + n/m; the rules cover altruism only, not mutualism, coordination or conflict (Active) — also carries further THEORY-672
- THEORY-655 — Social behaviour evolves as if each individual valued a neighbour’s fitness against its own by their coefficient of relationship, and its discriminations evolve to make that coefficient determinate; so kin are favoured and strangers met with hostility, relatives fight less, and haplodiploid sisters, related by 3/4, are predisposed to worker altruism (Proposed)
THEORY-672 — In the iterated Prisoner’s Dilemma with a probability w of meeting again, no strategy does better against an established TIT FOR TAT than TIT FOR TAT does against itself if and only if w ≥ (T−R)/(T−P) and w ≥ (T−R)/(R−S); always defecting is stable for every w, a cluster of reciprocators can invade it, and a nice stable strategy resists clusters, but the stability is only neutral, since always cooperating ties (Active)

network-science

From A graph network that learns its sheaf restriction maps from node features at each layer reaches the top, mostly within one standard deviation, of the standard small heterophilic node-classification benchmarks and stays within about one point of the best on homophilic ones; on a synthetic balanced bipartite graph it learns the negative transport the theory prescribes, but whether the maps it learns on real graphs have the geometry the theory credits for its success has not been examined

THEORY-1139 — A graph network that learns its sheaf restriction maps from node features at each layer reaches the top, mostly within one standard deviation, of the standard small heterophilic node-classification benchmarks and stays within about one point of the best on homophilic ones; on a synthetic balanced bipartite graph it learns the negative transport the theory prescribes, but whether the maps it learns on real graphs have the geometry the theory credits for its success has not been examined (Active)
- THEORY-1220 — A graph network that learns its sheaf restriction maps from node features succeeds on heterophilic graphs because it learns the class-aligned transports its separation theory prescribes: sign flips or rotations across classes, identity within them (Proposed)

From Aligning three networks jointly, through one multi-marginal fused Gromov–Wasserstein coupling with an anchor-based random-walk cost and a cluster decomposition, finds held-out correspondences better than pairwise and existing multi-network methods in most cases, as the paper’s plots show, in both pairwise and whole-tuple hits, on benchmarks whose true correspondences are consistent across the three networks by construction; whether the joint coupling itself is the cause is not separated from the new cost and the clustering

THEORY-1159 — Aligning three networks jointly, through one multi-marginal fused Gromov–Wasserstein coupling with an anchor-based random-walk cost and a cluster decomposition, finds held-out correspondences better than pairwise and existing multi-network methods in most cases, as the paper’s plots show, in both pairwise and whole-tuple hits, on benchmarks whose true correspondences are consistent across the three networks by construction; whether the joint coupling itself is the cause is not separated from the new cost and the clustering (Active)
- THEORY-1210 — When several networks are aligned, fitting one joint multi-marginal coupling finds held-out correspondences better than fitting them in pairs, and the gain comes from the joint fit itself, not from the cost or the search space that come with it (Proposed)

From In Friedkin’s simulations of reflected appraisal with Friedkin–Johnsen anchoring on initial positions, 99.0–100% of 100,000 random strongly connected networks of 5 to 25 members, complete or with random zero weights, evolve to a single leader and the rest to two

THEORY-1161 — In Friedkin’s simulations of reflected appraisal with Friedkin–Johnsen anchoring on initial positions, 99.0–100% of 100,000 random strongly connected networks of 5 to 25 members, complete or with random zero weights, evolve to a single leader and the rest to two (Active)
- THEORY-1232 — Under reflected appraisal across issues, whether power concentrates in one member depends on the single-issue step: with Friedkin–Johnsen anchoring on initial positions almost every random strongly connected group evolves to a single leader, and with DeGroot consensus the same random complete networks keep power spread (Proposed)

THEORY-146 — In Holman and Bruner’s bandit model of honest Bayesian researchers whose beliefs funding cannot touch, who differ in method and whose influence and training of newcomers follow productivity, industry funding of those whose methods already favour its drug lowers, in the runs their figures report, the share of the community on the better treatment, and has no effect without methodological diversity; at the one setting reported, an independent funder that counts industry-funded trials toward its merit threshold lowers that share further, and one that excludes them raises it (Active)
- THEORY-1174 — Industry can bias a scientific community toward a worse treatment without corrupting any individual, by selectively funding researchers whose methods already favour its product, provided researchers differ in method and both influence and the training of newcomers follow productivity; an independent funder that discounts industry-funded work counteracts this, and one that funds on productivity alone makes it worse (Proposed)

From In Friedkin, Jia and Bullo’s study of 30 four-person student groups discussing 15 choice dilemmas, a member’s self-reported closure to influence on the next issue rose with their influence centrality on the previous issue, an effect constant along the sequence, and with their average centrality over all prior issues, an effect that grew along it; influence networks thinned and discussions shortened while the rate of consensus did not change. Closure and centrality both come from the members’ own chip allocations, and prior closure is not controlled

THEORY-902 — In Friedkin, Jia and Bullo’s study of 30 four-person student groups discussing 15 choice dilemmas, a member’s self-reported closure to influence on the next issue rose with their influence centrality on the previous issue, an effect constant along the sequence, and with their average centrality over all prior issues, an effect that grew along it; influence networks thinned and discussions shortened while the rate of consensus did not change. Closure and centrality both come from the members’ own chip allocations, and prior closure is not controlled (Active)
- THEORY-1212 — In 30 four-person student groups discussing 15 choice dilemmas, a member’s self-reported closure to influence on the next issue rose with their influence over prior outcomes, increasingly so along the sequence, and the groups’ influence networks thinned and their discussions shortened while the rate of consensus did not change: evidence for a reflected-appraisal step by which influence concentrates in deliberating groups (Proposed)

From When a group learns only about the options its members pursue and good evidence is scarce, sparser communication makes it more likely to settle on the better option, at a large cost in speed; when evidence is plentiful, sparsity only slows learning

THEORY-143 — When a group learns only about the options its members pursue and good evidence is scarce, sparser communication makes it more likely to settle on the better option, at a large cost in speed; when evidence is plentiful, sparsity only slows learning (Proposed)
- THEORY-148 — A group learning by trying needs diversity in what its members pursue that lasts long enough to survive misleading early evidence and then gives way to consensus; limited communication and stubborn priors each supply it, and together they make it permanent, so the group never settles (Proposed)
- THEORY-422 — In the standard formal models of group learning, no communication network is best for every epistemic problem: costly transmission favours minimally connected or star networks, pooling of estimates by averaging favours regular networks, and learning by trying in bandit problems can favour sparse ones, so a group’s network cannot be judged until its learning problem is known (Active)

From In an ensemble of sign-consistent Boolean rules on the 87-gene core of E. coli’s regulatory network, a transient knockout or overexpression of one gene can leave the network in a different attractor after release, and every gene that does so reaches a strongly connected component containing a positive circuit

THEORY-178 — In an ensemble of sign-consistent Boolean rules on the 87-gene core of E. coli’s regulatory network, a transient knockout or overexpression of one gene can leave the network in a different attractor after release, and every gene that does so reaches a strongly connected component containing a positive circuit (Active)
- THEORY-780 — Transcriptional regulation alone can give a bacterium’s gene expression a memory: in E. coli, a transient knockdown or overexpression of one regulator, with the environment held fixed, can leave other genes in a different self-maintaining expression state after it is released, as Zhao, Wytock, Reynolds and Motter predict from a Boolean-network model of its regulatory core (Proposed)

From Under geometric coarse-graining of the S¹ hidden-metric network model, which merges blocks of r angular neighbours into supernodes linked if any of their members were, the supernodes’ link probability approaches the S¹ form with β kept, κ′ = (Σκ^β)^{1/β}, μ′ = μ/r and R′ = R/r, its ratio to that form tending to 1 as their separation grows large against the block width and the coupling μκ′κ″; between nearby supernodes it exceeds that form

THEORY-190 — Under geometric coarse-graining of the S¹ hidden-metric network model, which merges blocks of r angular neighbours into supernodes linked if any of their members were, the supernodes’ link probability approaches the S¹ form with β kept, κ′ = (Σκ^β)^{1/β}, μ′ = μ/r and R′ = R/r, its ratio to that form tending to 1 as their separation grows large against the block width and the coupling μκ′κ″; between nearby supernodes it exceeds that form (Active)
- THEORY-799 — Under geometric coarse-graining of the S¹ hidden-metric network model the average degree flows as r^ν, with ν = 2/(γ − 1) − 1 when the degree exponent γ sets it and 2/β − 1 when β does, so the coarse-grained networks run toward a complete graph for γ < 3 or β < 2 and toward a ring otherwise (Proposed)
- THEORY-803 — Real complex networks embedded in the hyperbolic plane are self-similar under geometric renormalization, their coarse-grained layers keeping the degree distribution, clustering and degree–degree correlations of the original, and this self-similarity is evidence that a hidden metric space underlies them (Proposed)

From In one study of a stack of proximity graphs on nested random subsets of the data, each element’s top layer drawn geometrically, greedy search on uniform random data in 8 dimensions needed a number of distance computations that grew no worse than logarithmically with the number of items and stayed below a single-layer navigable graph’s at every size tested, and in 4 dimensions the beam width needed for a fixed recall saturated; the logarithmic bound is derived only for exact Delaunay graphs of bounded degree, which the index does not build, and on 200 million SIFT vectors the measured growth departs from it

THEORY-365 — In one study of a stack of proximity graphs on nested random subsets of the data, each element’s top layer drawn geometrically, greedy search on uniform random data in 8 dimensions needed a number of distance computations that grew no worse than logarithmically with the number of items and stayed below a single-layer navigable graph’s at every size tested, and in 4 dimensions the beam width needed for a fixed recall saturated; the logarithmic bound is derived only for exact Delaunay graphs of bounded degree, which the index does not build, and on 200 million SIFT vectors the measured growth departs from it (Active)
- THEORY-815 — Layering a navigable small-world graph on nested random subsets of the data, with links separated by length scale and each layer’s degree capped, makes greedy nearest-neighbour search cost grow logarithmically with the number of items on data of low intrinsic dimension, where the single-layer graph’s cost grows polylogarithmically (Proposed)

From In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary

THEORY-421 — In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary (Active)
- THEORY-825 — Hyperbolic graph embedding models trained on task losses that never mention the hierarchy do not place a tree’s root nearest the origin, and their distance from the origin orders hierarchical levels only partly (Proposed)

neuroscience

From Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled

THEORY-045 — Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled (Proposed)
- THEORY-090 — A basic, affective form of consciousness is realised subcortically, in the upper brainstem and thalamus, before and without cortex, and its first function is alarm: survival behaviour, care for the body and generalised one-shot learning, with no choice between options (Proposed)
- THEORY-093 — A content of mind is conscious when a continuous flow of homeostatic feelings identifies it as the organism’s own; the feelings are conscious in themselves and hybrid, partly made of the body state they map, because interoceptive pathways let the body act directly on the neurons that map it (Proposed)

From How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control

THEORY-070 — How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control (Proposed)
- THEORY-1110 — Shenhav, Botvinick and Cohen propose that control is allocated top-down by a dedicated specifying system: dACC computes the expected value of a control signal, its probability-weighted payoff less an intrinsic cost rising with intensity, and specifies the signal that lateral prefrontal and subcortical structures implement (Active)

From In Vishne, Gerber, Knight and Deouell’s intracranial recordings from ten epilepsy patients viewing clearly visible images for up to 1.5 s, the pattern of high-frequency activity across ventral temporal cortex carried image category, and exemplar identity among 60 images, in a time-invariant code for as long as the image stayed on while amplitude fell about 80%; at the sampled prefrontal and parietal sites, category content was a burst about 150 to 600 ms after onset that did not track duration

THEORY-102 — In Vishne, Gerber, Knight and Deouell’s intracranial recordings from ten epilepsy patients viewing clearly visible images for up to 1.5 s, the pattern of high-frequency activity across ventral temporal cortex carried image category, and exemplar identity among 60 images, in a time-invariant code for as long as the image stayed on while amplitude fell about 80%; at the sampled prefrontal and parietal sites, category content was a burst about 150 to 600 ms after onset that did not track duration (Active)
- THEORY-830 — In human visual cortex the content of a sustained seen image is carried by a stable population code that lasts as long as the image, not by a sustained level of activity, and frontoparietal cortex marks only its onset; so a sustained experience rests on sensory representation and its discrete updates on frontoparietal representation (Proposed)

From As reviewed by Gerstner et al. (2018) from five papers, mostly in slices and one laboratory per region, pairing pre- and postsynaptic activity changed the synapse when a neuromodulator or plateau potential followed within about 1 s in striatum, within 5 s (noradrenaline, LTP after pre-before-post pairing) or 2.5 s (serotonin, LTD after post-before-pre pairing) in visual and prefrontal cortex, and within about 2 s in CA1, and not at longer delays; the reviewers read this as a decaying synaptic eligibility trace and as experimental support for three-factor learning rules on behavioural time scales

THEORY-909 — As reviewed by Gerstner et al. (2018) from five papers, mostly in slices and one laboratory per region, pairing pre- and postsynaptic activity changed the synapse when a neuromodulator or plateau potential followed within about 1 s in striatum, within 5 s (noradrenaline, LTP after pre-before-post pairing) or 2.5 s (serotonin, LTD after post-before-pre pairing) in visual and prefrontal cortex, and within about 2 s in CA1, and not at longer delays; the reviewers read this as a decaying synaptic eligibility trace and as experimental support for three-factor learning rules on behavioural time scales (Active)
- THEORY-1178 — In slices of striatum, visual and prefrontal cortex and hippocampus, Hebbian co-activation leaves a synaptic eligibility trace, lasting about 1 s in striatum, 3 to 10 s in cortex and about 2 s in CA1, that a later neuromodulator or plateau potential converts into a weight change and whose sign it can set: experimental support for three-factor learning rules on behavioural time scales (Proposed)

From In an illustrative active-inference simulation of post-traumatic context inference, run once per phenotype, a prior that one is in a harmful context, made overly precise by a run of harmful trials under the “vulnerable” learning and forgetting weights, makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision makes the attenuation policy less vigorous without correcting the prior, and the policy returns when precision is restored

THEORY-285 — In an illustrative active-inference simulation of post-traumatic context inference, run once per phenotype, a prior that one is in a harmful context, made overly precise by a run of harmful trials under the “vulnerable” learning and forgetting weights, makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision makes the attenuation policy less vigorous without correcting the prior, and the policy returns when precision is restored (Active)
- THEORY-1179 — In an active-inference model of post-traumatic context inference, an overly precise prior that one is in a harmful context makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision removes the symptoms without correcting the prior (Proposed)

From In one simulated T-maze task with hand-set parameters, run once per condition, an active inference agent’s habit, learned by counting its own state transitions and policy choices, emerged from goal-directed behaviour, and once acquired it persisted, and reinforced itself, after the reward was devalued, while an agent devalued before acquisition switched at once

THEORY-923 — In one simulated T-maze task with hand-set parameters, run once per condition, an active inference agent’s habit, learned by counting its own state transitions and policy choices, emerged from goal-directed behaviour, and once acquired it persisted, and reinforced itself, after the reward was devalued, while an agent devalued before acquisition switched at once (Active)
- THEORY-1191 — In a simulated active inference agent, a habit learned by counting the agent’s own state transitions and policy choices emerges from goal-directed behaviour, and once acquired it persists, and reinforces itself, after the reward is devalued, while an agent devalued before acquisition switches at once (Proposed)

From In Posani et al.’s analysis of the IBL Brainwide Map, recorded while trained mice performed one visual decision task, the eight-variable selectivity profiles of neurons within one cortical area clustered beyond a covariance-matched Gaussian null only in VISp, AUDp and SSp-ul of about 20 areas tested, while profiles pooled across areas clustered along anatomical lines; and over the task conditions each area distinguishes, at least 95% of random balanced dichotomies were linearly decodable above a label-shuffled null in 15 of 16 areas, a lenient criterion

THEORY-124 — In Posani et al.’s analysis of the IBL Brainwide Map, recorded while trained mice performed one visual decision task, the eight-variable selectivity profiles of neurons within one cortical area clustered beyond a covariance-matched Gaussian null only in VISp, AUDp and SSp-ul of about 20 areas tested, while profiles pooled across areas clustered along anatomical lines; and over the task conditions each area distinguishes, at least 95% of random balanced dichotomies were linearly decodable above a label-shuffled null in 15 of 16 areas, a lenient criterion (Active)
- THEORY-1196 — During a visual decision task, neurons within a single area of mouse cortex rarely form discrete functional types, yet the task conditions the area distinguishes are linearly separable in nearly every way (Proposed)

From In two midlife community samples (n = 182 and 176), instructed reappraisal of aversive pictures, mostly by reinterpretation, lowered reported negative affect, while the BOLD responses of the amygdala, periaqueductal gray, hypothalamus and the other subcortical regions that responded to the pictures showed evidence of no change (Bayes factors below 1/10); the emotion-responsive regions it reduced were all cortical, mainly higher visual and parietal

THEORY-871 — In two midlife community samples (n = 182 and 176), instructed reappraisal of aversive pictures, mostly by reinterpretation, lowered reported negative affect, while the BOLD responses of the amygdala, periaqueductal gray, hypothalamus and the other subcortical regions that responded to the pictures showed evidence of no change (Bayes factors below 1/10); the emotion-responsive regions it reduced were all cortical, mainly higher visual and parietal (Active)
- THEORY-1237 — Instructed reappraisal lowers reported negative affect to aversive pictures while the responses of the amygdala, periaqueductal gray, hypothalamus and other subcortical regions to the same pictures show evidence of no change; the regions it reduces are cortical, mainly higher visual and parietal (Proposed)

From In one study of 33 experienced Transcendental Meditation practitioners and 33 matched controls who counted instead, the practitioners rated their session higher on a retrospective pure-awareness scale whose top was defined for them as free from content (d = 0.77), and a classifier on EEG separated meditation from counting at 65% balanced accuracy on held-out subjects and, within practitioners, meditation from rest with 88% sensitivity; the authors found no robust association between the ratings and the EEG features

THEORY-870 — In one study of 33 experienced Transcendental Meditation practitioners and 33 matched controls who counted instead, the practitioners rated their session higher on a retrospective pure-awareness scale whose top was defined for them as free from content (d = 0.77), and a classifier on EEG separated meditation from counting at 65% balanced accuracy on held-out subjects and, within practitioners, meditation from rest with 88% sensitivity; the authors found no robust association between the ratings and the EEG features (Active)
- THEORY-1241 — Experienced Transcendental Meditation practitioners report more pure awareness than matched counting controls, and multivariate EEG separates the meditation period from counting and from rest, but the reports are not shown to track any EEG feature (Proposed)

From An ideal observer is optimal relative to the information it is given, so a constrained ideal is the optimal decoder of a degraded input, not a restricted decoder of the full one; comparing real observers with ideals taken at successive stages of the visual system places losses by stage, and in early-vision tasks human performance is reported often to parallel the retinal-stage ideal across conditions while falling well below it

THEORY-336 — An ideal observer is optimal relative to the information it is given, so a constrained ideal is the optimal decoder of a degraded input, not a restricted decoder of the full one; comparing real observers with ideals taken at successive stages of the visual system places losses by stage, and in early-vision tasks human performance is reported often to parallel the retinal-stage ideal across conditions while falling well below it (Active)
- THEORY-1242 — In early-vision tasks, human performance varies across conditions as the information available at the retinal output does, while falling well below the retinal-stage ideal, so substantial losses lie in central factors (Proposed)

From In four natural odour sources (strawberry, tomato, blueberry and mouse-urine volatiles), the rank order of correlations between compound concentrations has the Betti-curve signature of points near the boundary of a three-dimensional hyperbolic ball, not of any uniform Euclidean cube tried, and human odour-descriptor profiles for 127 odorants have that of a full three-dimensional hyperbolic ball

THEORY-191 — In four natural odour sources (strawberry, tomato, blueberry and mouse-urine volatiles), the rank order of correlations between compound concentrations has the Betti-curve signature of points near the boundary of a three-dimensional hyperbolic ball, not of any uniform Euclidean cube tried, and human odour-descriptor profiles for 127 odorants have that of a full three-dimensional hyperbolic ball (Active)
- THEORY-618 — Neural circuits that must represent hierarchically organised states in few dimensions use a hyperbolic geometry, three-dimensional where it has been measured, because hyperbolic space holds exponentially many well-separated states within a small radius; the argument from Zipf’s law to this geometry does not hold, and the evidence is two Betti-curve studies tested only against uniform Euclidean nulls (Proposed) — also carries further THEORY-192
THEORY-192 — In twelve sessions of rat dorsal CA1 from three data sets, pairwise correlations among place cells have the Betti-curve signature of points sampled uniformly from a three-dimensional hyperbolic ball, not of a low-dimensional Euclidean cube, and the fitted radius rose with the logarithm of exploration time (Active)

From In Hopfield’s 1982 simulations at N = 30 and 100, a network of threshold units with symmetric Hebbian couplings acts as a content-addressable memory whose stored random patterns are attractors: recall degrades severely near 0.15N patterns, a pattern placed 10 bits from another at N = 100 often fuses with it, and at 25-fold overload no stored pattern stays stable

THEORY-272 — In Hopfield’s 1982 simulations at N = 30 and 100, a network of threshold units with symmetric Hebbian couplings acts as a content-addressable memory whose stored random patterns are attractors: recall degrades severely near 0.15N patterns, a pattern placed 10 bits from another at N = 100 often fuses with it, and at 25-fold overload no stored pattern stays stable (Active)
- THEORY-800 — A network of N threshold units with symmetric Hebbian couplings stores random patterns as attractors up to a load proportional to N, about 0.15N for all large N, past which recall fails for every pattern at once, and a provision for forgetting old memories, such as couplings that saturate, keeps the most recent ones retrievable however many are added (Proposed)

From Brushing a seen rubber hand in synchrony with the hidden real hand makes touch be felt at the rubber hand, makes it be reported as one’s own, and shifts the felt position of the real hand toward it; a small asynchrony removes these effects

THEORY-765 — Brushing a seen rubber hand in synchrony with the hidden real hand makes touch be felt at the rubber hand, makes it be reported as one’s own, and shifts the felt position of the real hand toward it; a small asynchrony removes these effects (Rejected)
- THEORY-766 — Brushing a seen rubber hand in synchrony with the hidden real hand, against asynchronous brushing, raises reports that touch is felt at the rubber hand and, more weakly and less reliably, that the rubber hand is one’s own; the proprioceptive drift toward the rubber hand that accompanies these reports is not a measure of them; the standard comparison conditions do not equalise participants’ expectations, so whether the stimulation or the expectation produces the report difference is open (Active)
  - THEORY-827 — The rubber hand illusion is a perceptual illusion driven by multisensory integration: the difference in reports between synchronous and asynchronous brushing is produced by the stimulation, not by participants’ expectations (Proposed)

operations-research

From When stored views are never updated, greedily storing the view of largest marginal saving in query cost gets at least 1 − 1/e of the optimal saving on a data-cube lattice or any OR or AND view graph, against the optimum for the same number of views or for the space the greedy ends up using, which may exceed the budget by one view

THEORY-356 — When stored views are never updated, greedily storing the view of largest marginal saving in query cost gets at least 1 − 1/e of the optimal saving on a data-cube lattice or any OR or AND view graph, against the optimum for the same number of views or for the space the greedy ends up using, which may exceed the budget by one view (Active)
- THEORY-350 — Under a budget on maintenance time, a stored view’s upkeep depends on which other views are stored, so upkeep is not additive and adding a view can lower it, and greedily adding the single view of best saving per unit of upkeep can be arbitrarily bad; on OR view graphs a greedy over inverted tree sets gets 1 − 1/e of the saving of an optimum given as much upkeep, using up to twice the budget, if no tree-piece of the optimum alone exceeds the budget (Active)

From When a cached interval of width W is kept valid by its source, which pushes a refresh at cost C_vr whenever the value leaves it, and queries fetch the exact value at cost C_qr whenever it is too wide, and the chances of the two per step go as 1/W² and as W, the expected refresh cost is least at W* = (ρK₁/K₂)^(1/3) with ρ = 2C_vr/C_qr, where ρ times the chance of a pushed refresh equals the chance of a fetch

THEORY-386 — When a cached interval of width W is kept valid by its source, which pushes a refresh at cost C_vr whenever the value leaves it, and queries fetch the exact value at cost C_qr whenever it is too wide, and the chances of the two per step go as 1/W² and as W, the expected refresh cost is least at W* = (ρK₁/K₂)^(1/3) with ρ = 2C_vr/C_qr, where ρ times the chance of a pushed refresh equals the chance of a fetch (Active)
- THEORY-377 — In Olston, Loo and Widom’s simulations, a rule that multiplies a cached interval’s width by 1 + α on a pushed refresh with probability min{ρ, 1}, and divides it by 1 + α on a query’s exact fetch with probability min{1/ρ, 1}, settled within 1% of the best fixed width on one random walk and within 5% over eight walk settings; on a 50-host network-traffic trace, restricted to exact copies, it almost matched a tuned adaptive cache-or-not rule, and with a 50-value cache, where queries tolerated imprecision, it cost less; that it converges to the best width is argued, not proved (Active)

phenomenology

From In one study of 33 experienced Transcendental Meditation practitioners and 33 matched controls who counted instead, the practitioners rated their session higher on a retrospective pure-awareness scale whose top was defined for them as free from content (d = 0.77), and a classifier on EEG separated meditation from counting at 65% balanced accuracy on held-out subjects and, within practitioners, meditation from rest with 88% sensitivity; the authors found no robust association between the ratings and the EEG features

THEORY-870 — In one study of 33 experienced Transcendental Meditation practitioners and 33 matched controls who counted instead, the practitioners rated their session higher on a retrospective pure-awareness scale whose top was defined for them as free from content (d = 0.77), and a classifier on EEG separated meditation from counting at 65% balanced accuracy on held-out subjects and, within practitioners, meditation from rest with 88% sensitivity; the authors found no robust association between the ratings and the EEG features (Active)
- THEORY-1241 — Experienced Transcendental Meditation practitioners report more pure awareness than matched counting controls, and multivariate EEG separates the meditation period from counting and from rest, but the reports are not shown to track any EEG feature (Proposed)

From Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness

THEORY-866 — Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness (Active)
- THEORY-1246 — The so-called contentless goal-states of Shamatha, Transcendental and Stillness Meditation are neither truly contentless nor identical: they lack thoughts, perceptions, images and memories but keep wakefulness, calm, stillness, a non-conceptual knowing and the like, and they differ across the practices in attentional stability (Proposed)

philosophy-of-biology

From Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency

THEORY-071 — Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency (Proposed)
- THEORY-1119 — Barandiaran, Di Paolo and Rohde propose that minimal agency requires individuality, interactional asymmetry and normativity set by the system’s own organisation, each necessary and together, they say, sufficient; that individuality without the other two is not agency; and that a system optimizing an externally fixed function is not a model of agency (Active)

From Self-production alone gives a system only an all-or-nothing norm, to go on or to die, and that distinction is available to an observer, not to the system; a graded norm the system itself has, by which viable states are better or worse for it, needs adaptivity, a further capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them, and sense-making and agency need adaptivity

THEORY-628 — Self-production alone gives a system only an all-or-nothing norm, to go on or to die, and that distinction is available to an observer, not to the system; a graded norm the system itself has, by which viable states are better or worse for it, needs adaptivity, a further capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them, and sense-making and agency need adaptivity (Proposed)
- THEORY-1166 — Di Paolo argues that autopoiesis as originally defined entails conservation and not active homeostasis, so it gives a system only an all-or-nothing norm, to go on or to die, which an observer and not the system can use; that a graded norm the system itself has needs adaptivity, a capacity to detect tendencies toward the boundary of its viability and to act so as to reverse them; and that sense-making and agency need adaptivity (Active)

From Griesemer and Shavit propose, as a first and rough sketch, that a collective becomes a new level of individuality only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it, some core interactions have become generatively entrenched, and it has organizational closure, and that such a community can be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone

THEORY-1247 — Griesemer and Shavit propose, as a first and rough sketch, that a collective becomes a new level of individuality only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it, some core interactions have become generatively entrenched, and it has organizational closure, and that such a community can be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone (Active)
- THEORY-418 — A collective becomes a new level of individuality, on this sketch, only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it and some core interactions have become generatively entrenched and, as the paper asserts without deriving it from entrenchment, it has organizational closure; such a community can then be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone (Proposed)

From In a closed population, any mapping of ancestors to descendants gives the Price equation: mean fitness times the change in mean character equals the covariance of fitness and character plus the fitness-weighted mean of the change in character, over any time interval and with no assumption beyond closure

THEORY-770 — In a closed population, any mapping of ancestors to descendants gives the Price equation: mean fitness times the change in mean character equals the covariance of fitness and character plus the fitness-weighted mean of the change in character, over any time interval and with no assumption beyond closure (Active)
- THEORY-673 — The Price equation follows from any ancestor–descendant mapping in a closed population and is, like f = ma, a generalization-sketch: a schema that is empty of predictions until a specific model supplies entities, characters, time step and causes, whose value lies in unifying and guiding model construction, so the charges that it is a tautology or lacks predictive power and dynamic sufficiency ask of it what only its specialisations can supply (Proposed)

philosophy-of-language

From A language model idealized as continuing any text with a plausible next word from the collection of all plausible texts follows narrative coherence and not truth, so its false statements are confabulations within the story the context sets up; whether trained and aligned chat models keep that nature is an open empirical question the idealization does not settle

THEORY-547 — A language model idealized as continuing any text with a plausible next word from the collection of all plausible texts follows narrative coherence and not truth, so its false statements are confabulations within the story the context sets up; whether trained and aligned chat models keep that nature is an open empirical question the idealization does not settle (Active)
- THEORY-1240 — Trained, instruction-tuned and preference-tuned chat models keep the nature of the idealized fiction machine: what they assert follows the narrative coherence of the frame, role or story the conversation sets up, not a representation of truth that stays fixed across frames (Proposed)

From To mean something by an utterance, in the non-natural sense, is to intend it to produce an effect in an audience by means of the audience’s recognition of that intention, where the recognition is meant to be a reason for the effect and not merely its cause; an intention to produce the effect, or to have it recognised, is not enough

THEORY-444 — To mean something by an utterance, in the non-natural sense, is to intend it to produce an effect in an audience by means of the audience’s recognition of that intention, where the recognition is meant to be a reason for the effect and not merely its cause; an intention to produce the effect, or to have it recognised, is not enough (Proposed)
- THEORY-511 — Illocutionary acts that are not essentially conventional, such as warning, entreating and objecting, are individuated by a wholly overt, audience-directed intention, Grice’s three intentions with a fourth that the audience recognise the second, and are performed when that intention is understood; acts such as verdicts and umpires’ calls exist only within rule-governed procedures and fall outside the scheme (Proposed)

philosophy-of-mathematics

From An adjunction between a category and the opposite of another restricts to a dual equivalence between the full subcategories on which its unit and counit are isomorphisms

THEORY-729 — An adjunction between a category and the opposite of another restricts to a dual equivalence between the full subcategories on which its unit and counit are isomorphisms (Active)
- THEORY-604 — Most mathematical dualities are adjunctions between one category and the opposite of another, which become dual equivalences on their fixed points; what makes them dualities rather than equivalences is the reversal of arrows, so many of the ‘dualities’ physicists name are, in the mathematician’s sense, equivalences or symmetries (Proposed)

philosophy-of-science

From A Markov-blanket partition is defined relative to a chosen internal set, so a graph has one around almost any set of nodes, and the formalism alone does not say which set is the system

THEORY-066 — A Markov-blanket partition is defined relative to a chosen internal set, so a graph has one around almost any set of nodes, and the formalism alone does not say which set is the system (Active)
- THEORY-073 — In Friston’s 2013 formulation and the applications the record holds, a Markov blanket does not individuate a system: where it falls is fixed by modelling choices made before it is traced (the coupling graph, the internal set and the time at which steady state is assumed), and a blanket read from a fixed graph is the same whether the system produced the boundary or it was imposed (Active)
  - THEORY-1097 — A Markov blanket, in any formulation including sparse coupling at nonequilibrium steady state and time-varying blanket states, does not individuate a system: where it falls is fixed by modelling choices made before it is found, so it presupposes the boundary it is used to find, and it cannot represent a boundary the system produces (Proposed)

THEORY-146 — In Holman and Bruner’s bandit model of honest Bayesian researchers whose beliefs funding cannot touch, who differ in method and whose influence and training of newcomers follow productivity, industry funding of those whose methods already favour its drug lowers, in the runs their figures report, the share of the community on the better treatment, and has no effect without methodological diversity; at the one setting reported, an independent funder that counts industry-funded trials toward its merit threshold lowers that share further, and one that excludes them raises it (Active)
- THEORY-1174 — Industry can bias a scientific community toward a worse treatment without corrupting any individual, by selectively funding researchers whose methods already favour its product, provided researchers differ in method and both influence and the training of newcomers follow productivity; an independent funder that discounts industry-funded work counteracts this, and one that funds on productivity alone makes it worse (Proposed)

THEORY-953 — On Isaac’s structuralist account, the strongest evidence that a quality-space model targets phenomenal experience, and not only behaviour or neural activity, is convergence between models of the same space derived through different behavioural procedures, since experience is what those procedures share; he concedes that the convergence observed is coarse, with discrepancies that in many cases exceed each method’s own uncertainty, and that because the models average over subjects what they converge on is a standard observer (Active)
- THEORY-1204 — The strongest evidence that a quality-space model targets phenomenal experience, and not only behaviour or neural activity, is convergence between models of the same space derived through different behavioural procedures, since experience is what those procedures share (Proposed)

From When a group learns only about the options its members pursue and good evidence is scarce, sparser communication makes it more likely to settle on the better option, at a large cost in speed; when evidence is plentiful, sparsity only slows learning

THEORY-143 — When a group learns only about the options its members pursue and good evidence is scarce, sparser communication makes it more likely to settle on the better option, at a large cost in speed; when evidence is plentiful, sparsity only slows learning (Proposed)
- THEORY-148 — A group learning by trying needs diversity in what its members pursue that lasts long enough to survive misleading early evidence and then gives way to consensus; limited communication and stubborn priors each supply it, and together they make it permanent, so the group never settles (Proposed)
- THEORY-422 — In the standard formal models of group learning, no communication network is best for every epistemic problem: costly transmission favours minimally connected or star networks, pooling of estimates by averaging favours regular networks, and learning by trying in bandit problems can favour sparse ones, so a group’s network cannot be judged until its learning problem is known (Active)

THEORY-243 — In the three-unit identity network x ↦ x(1, 0, 1) ↦ (0, 2, 1)·h, aligning the input with the third unit and aligning it with the direction (h₁ + h₂)/√2, two alignments related by an orthogonal change of basis, pass every interchange intervention on the input equally, so interchange interventions on that variable alone do not fix which internal direction mediates it (Active)
- THEORY-802 — In trained networks as in constructed ones, interchange interventions on a fixed set of variables can be satisfied equally by alignments related by a change of basis, so which internal direction mediates a high-level variable is fixed only relative to a chosen decomposition into units, and complex models will generally have several distinct, fully correct abstractions (Proposed)

From In a community whose consensus model is replaced whenever a single sound pairwise model comparison favours the challenger, which models get proposed can hold the consensus on a false model while results reproduce almost perfectly, so reproducibility need not track convergence to truth

THEORY-295 — In a community whose consensus model is replaced whenever a single sound pairwise model comparison favours the challenger, which models get proposed can hold the consensus on a false model while results reproduce almost perfectly, so reproducibility need not track convergence to truth (Active)
- THEORY-814 — In scientific communities that settle on a consensus model by sound model comparisons, high reproducibility of results is not evidence that the consensus is true, because which alternatives get proposed can hold a false consensus whose results reproduce almost perfectly (Proposed)

THEORY-674 — In Garg and Fetzer’s GPT-4o-mini extraction from NBER and CEPR working papers, edges kept in at least 4 of 9 passes and never checked against human labels, the mean per-paper share of edges labelled as backed by difference-in-differences, instrumental variables, an RCT, regression discontinuity or synthetic control rose from 7.7% in 1990 to 31.7% in 2020, at every overlap threshold; the paper’s per-decade slope fell from +0.0187 to +0.0011 once method mix and graph volume were controlled (Active)
- THEORY-824 — Economics working papers increasingly back their claims with design-based identification: in NBER and CEPR working papers the share of claims supported by difference-in-differences, instrumental variables, an RCT, regression discontinuity or synthetic control rose about fourfold from 1990 to 2020 (Proposed)

pragmatics

From A speaker who chooses words to minimise the information lost about a single intended referent, with words’ literal meanings uniform over their extensions, picks each true word with probability inversely proportional to a power of its extension, so the size principle follows from informative communication; in one betting experiment, learners who knew the referent bet on a novel word’s meaning in line with the inversion of that speaker (r = .93 on condition means), and in one free-description study speakers’ word use followed it only weakly (r = .19)

THEORY-472 — A speaker who chooses words to minimise the information lost about a single intended referent, with words’ literal meanings uniform over their extensions, picks each true word with probability inversely proportional to a power of its extension, so the size principle follows from informative communication; in one betting experiment, learners who knew the referent bet on a novel word’s meaning in line with the inversion of that speaker (r = .93 on condition means), and in one free-description study speakers’ word use followed it only weakly (r = .19) (Active)
- THEORY-172 — In Frank and Goodman’s one-shot reference games, listeners’ mean bets matched a parameter-free listener inverting a speaker who chooses among true utterances by informativeness, with a separately measured salience prior, and not salience alone; and in Bergen, Goodman and Levy’s signalling games with novel symbols, listeners read a signal as excluding what the available alternative would have conveyed (Active)
  - THEORY-785 — Listeners interpret an utterance by inverting a model of a speaker who chooses among alternatives by informativeness, so what a fixed form conveys depends on what else the speaker could have said, as the rational speech act account of Frank, Goodman and coauthors proposes (Proposed)
- THEORY-816 — Learners inferring what a novel word means from its use reason about a speaker who chose the word to be informative about the referent, so the size principle in word learning is a consequence of that inference and not a separate rule of generalisation (Proposed)

From To mean something by an utterance, in the non-natural sense, is to intend it to produce an effect in an audience by means of the audience’s recognition of that intention, where the recognition is meant to be a reason for the effect and not merely its cause; an intention to produce the effect, or to have it recognised, is not enough

THEORY-444 — To mean something by an utterance, in the non-natural sense, is to intend it to produce an effect in an audience by means of the audience’s recognition of that intention, where the recognition is meant to be a reason for the effect and not merely its cause; an intention to produce the effect, or to have it recognised, is not enough (Proposed)
- THEORY-511 — Illocutionary acts that are not essentially conventional, such as warning, entreating and objecting, are individuated by a wholly overt, audience-directed intention, Grice’s three intentions with a fourth that the audience recognise the second, and are performed when that intention is understood; acts such as verdicts and umpires’ calls exist only within rule-governed procedures and fall outside the scheme (Proposed)

probabilistic-modeling

From A Markov-blanket partition is defined relative to a chosen internal set, so a graph has one around almost any set of nodes, and the formalism alone does not say which set is the system

THEORY-066 — A Markov-blanket partition is defined relative to a chosen internal set, so a graph has one around almost any set of nodes, and the formalism alone does not say which set is the system (Active)
- THEORY-073 — In Friston’s 2013 formulation and the applications the record holds, a Markov blanket does not individuate a system: where it falls is fixed by modelling choices made before it is traced (the coupling graph, the internal set and the time at which steady state is assumed), and a blanket read from a fixed graph is the same whether the system produced the boundary or it was imposed (Active)
  - THEORY-1097 — A Markov blanket, in any formulation including sparse coupling at nonequilibrium steady state and time-varying blanket states, does not individuate a system: where it falls is fixed by modelling choices made before it is found, so it presupposes the boundary it is used to find, and it cannot represent a boundary the system produces (Proposed)

From In one simulated T-maze task with hand-set parameters, run once per condition, an active inference agent’s habit, learned by counting its own state transitions and policy choices, emerged from goal-directed behaviour, and once acquired it persisted, and reinforced itself, after the reward was devalued, while an agent devalued before acquisition switched at once

THEORY-923 — In one simulated T-maze task with hand-set parameters, run once per condition, an active inference agent’s habit, learned by counting its own state transitions and policy choices, emerged from goal-directed behaviour, and once acquired it persisted, and reinforced itself, after the reward was devalued, while an agent devalued before acquisition switched at once (Active)
- THEORY-1191 — In a simulated active inference agent, a habit learned by counting the agent’s own state transitions and policy choices emerges from goal-directed behaviour, and once acquired it persists, and reinforces itself, after the reward is devalued, while an agent devalued before acquisition switches at once (Proposed)

From An ideal observer is optimal relative to the information it is given, so a constrained ideal is the optimal decoder of a degraded input, not a restricted decoder of the full one; comparing real observers with ideals taken at successive stages of the visual system places losses by stage, and in early-vision tasks human performance is reported often to parallel the retinal-stage ideal across conditions while falling well below it

THEORY-336 — An ideal observer is optimal relative to the information it is given, so a constrained ideal is the optimal decoder of a degraded input, not a restricted decoder of the full one; comparing real observers with ideals taken at successive stages of the visual system places losses by stage, and in early-vision tasks human performance is reported often to parallel the retinal-stage ideal across conditions while falling well below it (Active)
- THEORY-1242 — In early-vision tasks, human performance varies across conditions as the information available at the retinal output does, while falling well below the retinal-stage ideal, so substantial losses lie in central factors (Proposed)

From Evaluating expected free energy recursively over the beliefs an agent would hold after each plausible outcome makes active inference a pruned tree search over belief states, formally analogous to Bellman recursion; in single deterministic runs on a T-maze and an 8 × 8 maze it planned through short-term costs that agents with a shorter horizon did not; that the pruning keeps deep search tractable beyond toy problems is argued, not shown

THEORY-426 — Evaluating expected free energy recursively over the beliefs an agent would hold after each plausible outcome makes active inference a pruned tree search over belief states, formally analogous to Bellman recursion; in single deterministic runs on a T-maze and an 8 × 8 maze it planned through short-term costs that agents with a shorter horizon did not; that the pruning keeps deep search tractable beyond toy problems is argued, not shown (Active)
- THEORY-1245 — Evaluating expected free energy recursively over the beliefs an agent would hold after each plausible outcome makes active inference a pruned tree search over belief states, formally analogous to Bellman recursion, which lets the agent plan through short-term costs that one-step agents cannot; that the pruning keeps deep search tractable beyond toy problems is argued, not shown (Proposed)

From In the 66 Pew surveys of 2001–2011 that varied the order of two back-to-back binary questions, question order changed the answers but not the probability of giving the same answer to both, the equality a Lüders projection model predicts for any projectors; a classical model with order events can meet the same equality, so the regularity does not by itself favour quantum over classical probability

THEORY-166 — In the 66 Pew surveys of 2001–2011 that varied the order of two back-to-back binary questions, question order changed the answers but not the probability of giving the same answer to both, the equality a Lüders projection model predicts for any projectors; a classical model with order events can meet the same equality, so the regularity does not by itself favour quantum over classical probability (Active)
- THEORY-796 — Question-order effects in surveys generally leave the probability of giving the same answer to both questions unchanged, the QQ equality that a projection model of judgment predicts, as Wang, Solloway, Shiffrin and Busemeyer propose (Proposed)

From A speaker who chooses words to minimise the information lost about a single intended referent, with words’ literal meanings uniform over their extensions, picks each true word with probability inversely proportional to a power of its extension, so the size principle follows from informative communication; in one betting experiment, learners who knew the referent bet on a novel word’s meaning in line with the inversion of that speaker (r = .93 on condition means), and in one free-description study speakers’ word use followed it only weakly (r = .19)

THEORY-472 — A speaker who chooses words to minimise the information lost about a single intended referent, with words’ literal meanings uniform over their extensions, picks each true word with probability inversely proportional to a power of its extension, so the size principle follows from informative communication; in one betting experiment, learners who knew the referent bet on a novel word’s meaning in line with the inversion of that speaker (r = .93 on condition means), and in one free-description study speakers’ word use followed it only weakly (r = .19) (Active)
- THEORY-172 — In Frank and Goodman’s one-shot reference games, listeners’ mean bets matched a parameter-free listener inverting a speaker who chooses among true utterances by informativeness, with a separately measured salience prior, and not salience alone; and in Bergen, Goodman and Levy’s signalling games with novel symbols, listeners read a signal as excluding what the available alternative would have conveyed (Active)
  - THEORY-785 — Listeners interpret an utterance by inverting a model of a speaker who chooses among alternatives by informativeness, so what a fixed form conveys depends on what else the speaker could have said, as the rational speech act account of Frank, Goodman and coauthors proposes (Proposed)
- THEORY-816 — Learners inferring what a novel word means from its use reason about a speaker who chose the word to be informative about the referent, so the size principle in word learning is a consequence of that inference and not a separate rule of generalisation (Proposed)

From When a sequence generator’s latent dynamics factor as a tensor product, its predictive state embeds losslessly and linearly in orthogonal per-factor subspaces of total dimension Σ(d_n − 1); 4-layer GPT-2-style transformers trained by next-token prediction on five Mess3 and Bloch Walk factors settle near that dimension, in per-factor subspaces that are linearly independent and approximately orthogonal in their leading components, and under the token noise tested reach it first even when noise makes it lossy

THEORY-256 — When a sequence generator’s latent dynamics factor as a tensor product, its predictive state embeds losslessly and linearly in orthogonal per-factor subspaces of total dimension Σ(d_n − 1); 4-layer GPT-2-style transformers trained by next-token prediction on five Mess3 and Bloch Walk factors settle near that dimension, in per-factor subspaces that are linearly independent and approximately orthogonal in their leading components, and under the token noise tested reach it first even when noise makes it lossy (Active)
- THEORY-808 — Transformers trained by next-token prediction on sequence processes whose latent dynamics factor as a tensor product learn to represent the belief state in approximately orthogonal per-factor subspaces of about the factored dimension rather than the joint one, and reach that factored representation first even when noise makes it lossy (Proposed)

From MSN’s objective penalizes unequal cluster sizes explicitly and SwAV’s enforces equal sizes, and in one study with mini-batches of two to eight ImageNet classes, SimCLR, MSN and VICReg lost 11 to 18 points of ImageNet linear accuracy while MAE and data2vec lost under one

THEORY-306 — MSN’s objective penalizes unequal cluster sizes explicitly and SwAV’s enforces equal sizes, and in one study with mini-batches of two to eight ImageNet classes, SimCLR, MSN and VICReg lost 11 to 18 points of ImageNet linear accuracy while MAE and data2vec lost under one (Active)
- THEORY-805 — Volume-maximizing joint-embedding objectives as a class, VICReg and SimCLR as well as MSN and SwAV, carry a prior toward equal-sized clusters, and that prior costs them semantic transfer when the pretraining data are class-imbalanced, including on naturally long-tailed data (Proposed)

From With a kernel normalised to k(x,x) = 1, the covariance operator of a distribution is a density operator, and its quantum relative entropy is a lower bound on the Kullback–Leibler divergence that is tight for every pair of distributions only for an orthonormal, hence discrete, embedding, lies above the KL of the kernel-smoothed distributions, and on the torus, with Fourier coefficients proportional to e^(−σ‖ω‖₁) and strictly positive Lipschitz densities, is within O(σ²) of KL as the bandwidth σ shrinks

THEORY-313 — With a kernel normalised to k(x,x) = 1, the covariance operator of a distribution is a density operator, and its quantum relative entropy is a lower bound on the Kullback–Leibler divergence that is tight for every pair of distributions only for an orthonormal, hence discrete, embedding, lies above the KL of the kernel-smoothed distributions, and on the torus, with Fourier coefficients proportional to e^(−σ‖ω‖₁) and strictly positive Lipschitz densities, is within O(σ²) of KL as the bandwidth σ shrinks (Active)
- THEORY-811 — With a kernel normalised to unit diagonal, the kernel quantum relative entropy of two distributions is within O(σ²) of their Kullback–Leibler divergence as the bandwidth σ shrinks, for translation-invariant kernels on the torus generally and for kernels on other compact groups (Proposed)

THEORY-349 — For binary outcomes and a fixed representation of individuals, a predictor can pass every test in a bounded class without being close to nature: outcome indistinguishability to tests that see the sampled prediction is equivalent, with stated losses, to multicalibration over a closely related class of sets, a predictor meeting it exists at size polynomial in the tests’ size and the tolerance whatever nature’s complexity, and learning one is as hard as weak agnostic learning of the class (Active)
- THEORY-354 — An observer’s power to refute a predictor depends on its access to the predictor as well as on its budget: against tests that see only the sampled prediction a predictor of size O(s/ε²) always passes, but if fine-grained clique counting is hard, or BPP ≠ PSPACE, some cheap tests that may query the predictor adaptively pass only predictors about as costly as nature’s own (Active)

psychopathology-and-treatment

From In an illustrative active-inference simulation of post-traumatic context inference, run once per phenotype, a prior that one is in a harmful context, made overly precise by a run of harmful trials under the “vulnerable” learning and forgetting weights, makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision makes the attenuation policy less vigorous without correcting the prior, and the policy returns when precision is restored

THEORY-285 — In an illustrative active-inference simulation of post-traumatic context inference, run once per phenotype, a prior that one is in a harmful context, made overly precise by a run of harmful trials under the “vulnerable” learning and forgetting weights, makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision makes the attenuation policy less vigorous without correcting the prior, and the policy returns when precision is restored (Active)
- THEORY-1179 — In an active-inference model of post-traumatic context inference, an overly precise prior that one is in a harmful context makes exploratory sampling look uninformative, so the agent attenuates the periphery, never receives the evidence that would correct the prior, and the belief sustains itself; lowering precision removes the symptoms without correcting the prior (Proposed)

From As Chou, Wilson and Smith review them, in about six foraging studies, mostly sub-clinical, higher stress or anxiety went with less exploration, and in one task with a predator to check for (Trier et al. 2023), people higher in trait somatic anxiety checked less and did worse, a reduction the study’s further analyses associated with greater aversion to detecting the predator

THEORY-907 — As Chou, Wilson and Smith review them, in about six foraging studies, mostly sub-clinical, higher stress or anxiety went with less exploration, and in one task with a predator to check for (Trier et al. 2023), people higher in trait somatic anxiety checked less and did worse, a reduction the study’s further analyses associated with greater aversion to detecting the predator (Active)
- THEORY-1180 — In foraging tasks, higher stress or anxiety goes consistently with less exploration, and in one task with a predator to check for, people higher in somatic anxiety checked less and did worse, a reduction associated with greater aversion to detecting the predator (Proposed)

From In Brown, Price and Dombrovski’s reading of seven computational learning studies they selected without a systematic search, with mixed directions, anxiety goes with altered adjustment of learning rate to uncertainty, mostly for aversive outcomes, while overall learning rate, choice consistency and outcome valuation show little relation to it

THEORY-897 — In Brown, Price and Dombrovski’s reading of seven computational learning studies they selected without a systematic search, with mixed directions, anxiety goes with altered adjustment of learning rate to uncertainty, mostly for aversive outcomes, while overall learning rate, choice consistency and outcome valuation show little relation to it (Active)
- THEORY-1182 — Across seven computational learning studies selected without a systematic search, anxiety goes with altered adjustment of learning rate to uncertainty, mostly for aversive outcomes, while overall learning rate, choice consistency and outcome valuation show little relation to it (Proposed)

From As Mancinelli and Bach’s systematic review reports it, rating it B, in three laboratory studies of healthy humans a signalled, costless avoidance response offered for a stimulus never paired with the aversive outcome was used, and withdrawing it raised expectancy of the outcome for that stimulus, though Xia et al. (2019) found no change in skin conductance or startle; every study signalled the response’s availability, which may itself have read as a danger cue

THEORY-920 — As Mancinelli and Bach’s systematic review reports it, rating it B, in three laboratory studies of healthy humans a signalled, costless avoidance response offered for a stimulus never paired with the aversive outcome was used, and withdrawing it raised expectancy of the outcome for that stimulus, though Xia et al. (2019) found no change in skin conductance or startle; every study signalled the response’s availability, which may itself have read as a danger cue (Active)
- THEORY-1184 — In three laboratory studies of healthy humans, offering a costless avoidance response for a stimulus never paired with the aversive outcome led people to use it, and withdrawing the response then raised their expectancy of the outcome for that stimulus; signalling of the response is a confound (Proposed)

From Chou, Wilson and Smith read about eight computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples, with mixed results and none replicated, as suggesting that directed exploration rises with trait or cognitive anxiety and falls with state or somatic anxiety; none they review links random exploration to anxiety

THEORY-917 — Chou, Wilson and Smith read about eight computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples, with mixed results and none replicated, as suggesting that directed exploration rises with trait or cognitive anxiety and falls with state or somatic anxiety; none they review links random exploration to anxiety (Active)
- THEORY-1186 — Across computational studies of exploration in reward-based bandit tasks, mostly in sub-clinical samples and not yet replicated, directed exploration tends to rise with trait or cognitive anxiety and to fall with state or somatic anxiety (Proposed)

From As Mancinelli and Bach’s systematic review of avoidance learning in healthy humans reports it, rating both A, an avoidance response left available during extinction persists, in some participants for hundreds of unpunished trials, and keeps expectancy of the aversive outcome higher than in people denied it (nine studies); and less effective or costlier responses extinguish faster (six studies, one exception)

THEORY-901 — As Mancinelli and Bach’s systematic review of avoidance learning in healthy humans reports it, rating both A, an avoidance response left available during extinction persists, in some participants for hundreds of unpunished trials, and keeps expectancy of the aversive outcome higher than in people denied it (nine studies); and less effective or costlier responses extinguish faster (six studies, one exception) (Active)
- THEORY-1189 — In laboratory avoidance learning in healthy humans, an avoidance response that remains available during extinction persists, in some participants for hundreds of unpunished trials, and keeps expectancy of the aversive outcome higher than in people denied it; the more effective and the cheaper the response, the slower it extinguishes (Proposed)

From In Zorowitz, Momennejad and Daw’s simulations of a Markov decision process, an agent that evaluates actions as if it will fail to avoid danger at later steps, while in fact choosing optimally at each step, spreads threat value to states far from the danger and avoids early and widely, which reproduces qualitatively exaggerated threat appraisal, fear generalization and several laboratory findings associated with anxiety

THEORY-915 — In Zorowitz, Momennejad and Daw’s simulations of a Markov decision process, an agent that evaluates actions as if it will fail to avoid danger at later steps, while in fact choosing optimally at each step, spreads threat value to states far from the danger and avoids early and widely, which reproduces qualitatively exaggerated threat appraisal, fear generalization and several laboratory findings associated with anxiety (Active)
- THEORY-1190 — In a Markov decision process, an agent that evaluates actions as if it will fail to avoid danger at later steps, while in fact choosing optimally at each step, spreads threat value to states far from the danger and avoids early and widely, which reproduces qualitatively exaggerated threat appraisal, fear generalization and several laboratory findings associated with anxiety (Proposed)

From As Craske et al.’s 2014 review reports it, citing six studies and reporting no data of its own, how far fear falls during an extinction or exposure session, or across sessions, does not predict how much fear is expressed at follow-up, in animals, human laboratory samples and fearful, mostly analog, human samples

THEORY-921 — As Craske et al.’s 2014 review reports it, citing six studies and reporting no data of its own, how far fear falls during an extinction or exposure session, or across sessions, does not predict how much fear is expressed at follow-up, in animals, human laboratory samples and fearful, mostly analog, human samples (Active)
- THEORY-1193 — How far fear falls during an extinction or exposure session, or across sessions, does not predict how much fear is expressed at follow-up, in animals, human laboratory samples and fearful human samples (Proposed)

quantum-foundations

From An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality

THEORY-012 — An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality (Active)
- THEORY-011 — Generalized contextuality is strictly broader than Kochen–Specker contextuality: a qubit already has no preparation-noncontextual model, though no Kochen–Specker argument with sharp measurements exists in dimension two (Active)
- THEORY-014 — In both formalisms of contextuality, classicality is the existence of a nonnegative version of a linear representation that always exists over the reals, so negativity is universal and only its unavoidability is nonclassical (Active) — also carries further THEORY-016
- THEORY-015 — Kochen–Specker noncontextuality is measurement noncontextuality plus outcome determinism for sharp measurements, and preparation noncontextuality implies that determinism, so every Kochen–Specker proof refutes universal generalized noncontextuality while measurement noncontextuality alone is consistent with quantum theory (Active) — also carries further THEORY-016
- THEORY-526 — The Čech cohomology obstruction of a support section is a sound but incomplete witness of contextuality: it vanishes exactly when the section lies in a compatible family of integer combinations of support sections, so it is non-zero on every section of every All-vs-Nothing model, over any coefficient ring, on a connected cover, yet it misses the Hardy model entirely, part of a strongly contextual Kochen–Specker cover, and every section of a strongly contextual model on the fully symmetric (2,2,4) Bell cover (Active)
  - THEORY-524 — On cyclic measurement scenarios, where the contexts’ overlaps form one chordless N-cycle, the ℤ/2 Čech obstruction computed on the (N−1)-th iterated joint model is exact for logical and strong contextuality, the bound N − 1 being tight; beyond cycles no such result is established, the published extension is false on its own example, and the completeness conjecture fails on a Kochen–Specker cover that is never detected at any level (Active)
THEORY-016 — An operational theory admits a generalized-noncontextual model exactly when its GPT admits a positive quasiprobability representation, and for a tomographically local theory any diagram-preserving such model is an exact frame with exactly as many ontic states as the GPT’s dimension (Active)

From Classical Boolean logic is restored when a presheaf of contextual data acquires a global section or is sheafified

THEORY-024 — Classical Boolean logic is restored when a presheaf of contextual data acquires a global section or is sheafified (Rejected)
- THEORY-516 — For any presheaf of propositions over a small category of contexts, a sieve-valued valuation satisfies the generalized functional-composition condition exactly when it is a subobject of that presheaf, and the construction runs unchanged in classical physics, where every microstate gives a global section yet truth values are still non-Boolean sieves; so the logic of the truth values is fixed by the base category, not by whether a presheaf has global sections (Active)

From In Di Biagio and Rovelli’s relational account two classical subsystems’ perspectives merge, agreeing on a variable’s value and on each other’s, when they couple to that variable or to each other, and a coupling to a complementary variable can remove a relative fact from a perspective that did not interact; that this answers the combination problem and keeps perspectives consistent is shown only in worked examples computed relative to a single outside system

THEORY-1074 — In Di Biagio and Rovelli’s relational account two classical subsystems’ perspectives merge, agreeing on a variable’s value and on each other’s, when they couple to that variable or to each other, and a coupling to a complementary variable can remove a relative fact from a perspective that did not interact; that this answers the combination problem and keeps perspectives consistent is shown only in worked examples computed relative to a single outside system (Active)
- THEORY-1213 — In relational quantum mechanics as Di Biagio and Rovelli reformulate it, classical subsystems whose variables couple properly, so as to get correlated, come to agree, so that the merging of perspectives answers the combination problem and the facts held relative to different outside systems never conflict about the same variables (Proposed)

From In relational quantum mechanics as Rovelli first stated it, the relation between two observers’ descriptions is itself a quantum fact relative to a third system, so no observer’s view can be deduced from another’s and there is no description in toto, only a net of partial descriptions; the views are held consistent only in that the third system predicts correlated outcomes, which is shown for a single measurement and asserted in general

THEORY-1080 — In relational quantum mechanics as Rovelli first stated it, the relation between two observers’ descriptions is itself a quantum fact relative to a third system, so no observer’s view can be deduced from another’s and there is no description in toto, only a net of partial descriptions; the views are held consistent only in that the third system predicts correlated outcomes, which is shown for a single measurement and asserted in general (Active)
- THEORY-1222 — In relational quantum mechanics as stated in Rovelli (1996), rules (i) and (ii), that there is no state except within a further observer’s information and no information without interaction, exclude any contradiction between observers’ views: what a third system predicts of two observers’ information always agrees with what it finds on interacting with them (Proposed)

From Di Biagio and Rovelli locate the Frauchiger–Renner contradiction and the violation of the Local Friendliness inequalities in one step, a superobserver using facts relative to a friend he holds in full quantum control as if they were his own, which is the classical sum rule applied where those facts are not stable for him; they base quantum ontology on facts relative to systems rather than on stable facts, arguing that stability is itself relational and only approximate and counting the Local Friendliness violation as evidence, with an absolute macroreality emerging only approximately and only relative to systems for which decoherence is strong

THEORY-1081 — Di Biagio and Rovelli locate the Frauchiger–Renner contradiction and the violation of the Local Friendliness inequalities in one step, a superobserver using facts relative to a friend he holds in full quantum control as if they were his own, which is the classical sum rule applied where those facts are not stable for him; they base quantum ontology on facts relative to systems rather than on stable facts, arguing that stability is itself relational and only approximate and counting the Local Friendliness violation as evidence, with an absolute macroreality emerging only approximately and only relative to systems for which decoherence is strong (Active)
- THEORY-1230 — In relational quantum mechanics, with facts fixed by interaction and the classical sum rule used only within one context or where facts are stable, facts relative to different systems never yield contradictory predictions in extended Wigner’s-friend scenarios, and a superobserver’s later facts agree with his friend’s (Proposed)

From Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned

THEORY-720 — Classical simulations between no-signalling empirical models on finite scenarios, which may change the cover, never create contextuality: a model is non-contextual exactly when the empty model simulates it, the non-contextual fraction cannot fall along a simulation, strong contextuality is reflected backwards, and only non-contextual models can be cloned (Active)
- THEORY-174 — Classical simulations between no-signalling empirical models on different scenarios never create contextuality, and a convex map between two scenarios’ model sets is induced by a classical non-adaptive procedure exactly when some non-contextual model of the hom scenario that satisfies its predicate induces it (Active)
  - THEORY-1170 — A convex map between two scenarios’ model sets is induced by a classical adaptive procedure exactly when some non-contextual model of the corresponding adaptive hom scenario induces it (Proposed)

From In Aerts’s subspace model of Hampton’s membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice

THEORY-255 — In Aerts’s subspace model of Hampton’s membership data, and in any lattice whose membership weight is order-preserving, whatever its negation, taking concept conjunction and disjunction as meet and join cannot produce overextension or underextension, so a model of the guppy-type deviations must take conjunction as some operation other than the meet or drop the monotone weight, while the Kolmogorov-type deviations can be had from the meet of a non-Boolean subspace lattice (Active)
- THEORY-807 — The guppy-type overextension in Hampton’s membership data arises because a combined concept is in part a new, emergent concept and not only a logical combination of its parts, so human concept combination runs in two superposed layers, one classical and logical and one quantum and conceptual (Proposed)

religion

From Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness

THEORY-866 — Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness (Active)
- THEORY-1246 — The so-called contentless goal-states of Shamatha, Transcendental and Stillness Meditation are neither truly contentless nor identical: they lack thoughts, perceptions, images and memories but keep wakefulness, calm, stillness, a non-conceptual knowing and the like, and they differ across the practices in attentional stability (Proposed)

representation-learning

From On a finite augmentation space, over unconstrained functions of the two views, the spectral contrastive and NT-Logistic losses are both minimized at the positive-pair density ratio

THEORY-719 — On a finite augmentation space, over unconstrained functions of the two views, the spectral contrastive and NT-Logistic losses are both minimized at the positive-pair density ratio (Active)
- THEORY-001 — On a finite augmentation space, InfoNCE, logistic and spectral contrastive losses share one population optimum: the positive-pair density ratio (Proposed)
  - THEORY-007 — Kernel PCA under the positive-pair density ratio recovers the eigenfunctions of the positive-pair Markov chain, and their top span is minimax-optimal for linear prediction of approximately view-invariant targets (Active)
    - THEORY-005 — The positive-pair density ratio is the kernel of the conditional-expectation operator on L²(p), so spectral representations are that operator’s eigenfunctions, well defined when the positive-pair χ²-divergence is finite (Proposed)
    - THEORY-009 — In the positive-pair spectral self-supervised papers, self-adjointness comes from the symmetry of the positive-pair distribution, not from the Riesz representation theorem; elsewhere in the family it comes from a symmetrized loss, a precondition on the kernel, or composing the conditional-expectation operator with its adjoint (Active)

From For the final embedding and unembedding of a softmax model, the output distribution fixes the representations only up to an invertible linear map, and whether closeness in distribution bounds representational dissimilarity depends on the distance: not for KL in the saturation regime, but yes for the log-likelihood variance distance

THEORY-723 — For the final embedding and unembedding of a softmax model, the output distribution fixes the representations only up to an invertible linear map, and whether closeness in distribution bounds representational dissimilarity depends on the distance: not for KL in the saturation regime, but yes for the log-likelihood variance distance (Active)
- THEORY-002 — Convergence of representations, in the Platonic hypothesis’s sense, is convergence of kernels, which fixes representations only up to the symmetry group of what is observed; whether closeness in distribution implies closeness of representation depends on the distance used (Proposed)

From In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet

THEORY-083 — In Papyan, Han and Donoho’s runs of VGG, ResNet and DenseNet trained with cross-entropy on seven balanced datasets without augmentation, training past zero error shrinks last-layer within-class variability against between-class variability, and moves the centred class means and the classifier toward a common simplex equiangular tight frame and the decision toward nearest class mean, nearly to completion on the four easier datasets and only partly on ImageNet (Active)
- THEORY-788 — Training a deep classifier past zero training error generally collapses its last-layer features onto their class means, drives the centred means toward a simplex equiangular tight frame aligned with the classifier, and makes the decision nearest class mean, as Papyan, Han and Donoho propose from their runs (Proposed)

From On Pythia language models from 410M to 12B parameters, in three single prompt pairs and two small prompt distributions, verifying attention or neuron nodes by activation patching in the order given by attribution patching, a first-order gradient estimate of each node’s noising effect, found the nodes with the largest effects at about 2 to 50 times the cost of an oracle ordering, below unordered patching, subsampling and block search in every setting; its misses were concentrated in queries and keys behind saturated attention and in nodes whose direct and indirect effects nearly cancel

THEORY-996 — On Pythia language models from 410M to 12B parameters, in three single prompt pairs and two small prompt distributions, verifying attention or neuron nodes by activation patching in the order given by attribution patching, a first-order gradient estimate of each node’s noising effect, found the nodes with the largest effects at about 2 to 50 times the cost of an oracle ordering, below unordered patching, subsampling and block search in every setting; its misses were concentrated in queries and keys behind saturated attention and in nodes whose direct and indirect effects nearly cancel (Active)
- THEORY-1000 — For fine-grained nodes of transformer language models generally, attribution patching corrected for attention saturation is a reliable prefilter for activation patching: nodes it ranks low rarely have large single-node patching effects, so verifying nodes in its order finds the important ones at a small multiple of the cost of an oracle, as Kramár et al. expect (Proposed)

From On four subject–verb agreement tasks in Pythia-70M and Gemma-2-2B, circuits of sparse-autoencoder features and autoencoder error nodes, selected by linear estimates of indirect effect, carried half or more of the behaviour on held-out pairs with about 100 and 500 nodes, where circuits of neurons needed about 1,500 and more than 30,000; with the error nodes removed, the same feature circuits carried little of it, under about a quarter in Pythia and almost none in Gemma

THEORY-1007 — On four subject–verb agreement tasks in Pythia-70M and Gemma-2-2B, circuits of sparse-autoencoder features and autoencoder error nodes, selected by linear estimates of indirect effect, carried half or more of the behaviour on held-out pairs with about 100 and 500 nodes, where circuits of neurons needed about 1,500 and more than 30,000; with the error nodes removed, the same feature circuits carried little of it, under about a quarter in Pythia and almost none in Gemma (Active)
- THEORY-1014 — In a language model, a behaviour that looks like one mechanism, whether picked out by one contrast or by one cluster of next-token predictions, is often carried by several distinct pathways of sparse features that converge on the output without fully intersecting, as in Pythia-70M’s agreement across a relative clause, its prediction of “to” before an infinitival object, and its incrementing of sequences (Proposed)

From In a one-layer transformer whose MLP has 512 ReLU neurons, a sparse autoencoder with 4,096 features trained on that layer’s activations yields features that a blinded rater and an automated pipeline judged far more interpretable than the neurons, that for the four features examined in detail correspond to no neuron, and whose effects on the output, which the autoencoder never saw, agree with their activations; splicing in the reconstruction recovers 79% of the loss reduction the MLP provides, measured against zero-ablating it

THEORY-991 — In a one-layer transformer whose MLP has 512 ReLU neurons, a sparse autoencoder with 4,096 features trained on that layer’s activations yields features that a blinded rater and an automated pipeline judged far more interpretable than the neurons, that for the four features examined in detail correspond to no neuron, and whose effects on the output, which the autoencoder never saw, agree with their activations; splicing in the reconstruction recovers 79% of the loss reduction the MLP provides, measured against zero-ablating it (Active)
- THEORY-1016 — The features a sparse autoencoder recovers from a language model are a resolution set by the dictionary’s width and sparsity penalty, not a unique set of the model’s units: as the dictionary widens, features split into families of finer features, and splits also merge, so that the number of features is not well-posed (Proposed)

From In Claude 3.5 Haiku, writing the second line of a rhyming couplet whose first line ends “grab it”, features for the candidate rhymes “rabbit” and “habit” are active on the newline token before the line, steering them changes the line’s final word only when applied at that token, and suppressing “rabbit” turns the line towards “habit”; injecting a “rabbit” or “green” word feature at that point in 25 other poems ended the line with the injected word in 70% of cases, and such planned-word features were found in about half of the poems the authors examined

THEORY-1017 — In Claude 3.5 Haiku, writing the second line of a rhyming couplet whose first line ends “grab it”, features for the candidate rhymes “rabbit” and “habit” are active on the newline token before the line, steering them changes the line’s final word only when applied at that token, and suppressing “rabbit” turns the line towards “habit”; injecting a “rabbit” or “green” word feature at that point in 25 other poems ended the line with the injected word in 70% of cases, and such planned-word features were found in about half of the poems the authors examined (Active)
- THEORY-982 — Claude 3.5 Haiku routinely computes intermediate steps within a single forward pass and plans its outputs ahead, representing a target before the text that leads to it and writing towards it, as Lindsey et al. propose from their case studies (Proposed) — also carries further THEORY-984
THEORY-984 — In Claude 3.5 Haiku, on the prompt “Fact: the capital of the state containing Dallas is”, which it completes with “Austin”, an attribution graph shows Dallas features activating Texas features that, with “say a capital” features, promote Austin, beside a direct edge from Dallas to “say Austin”; inhibiting the Dallas features lowers the Texas features and leaves “say a capital” largely intact, and replacing the Texas features with those of California, Georgia, British Columbia, China or the Byzantine Empire makes the model name that place’s capital (Active)

From Attribution patching estimates the effect of patching an activation from a clean run into a corrupted run as the difference between the two activations dotted with the gradient of the metric on the corrupted run, so the estimates for every activation come from two forward passes and one backward pass and add over sub-components; the estimate is exact when the metric is affine in the activation, gives no effect for a nonlinearity in saturation whatever the true change, and through LayerNorm ignores the component of a patch along the vector being normalized

THEORY-1023 — Attribution patching estimates the effect of patching an activation from a clean run into a corrupted run as the difference between the two activations dotted with the gradient of the metric on the corrupted run, so the estimates for every activation come from two forward passes and one backward pass and add over sub-components; the estimate is exact when the metric is affine in the activation, gives no effect for a nonlinearity in saturation whatever the true change, and through LayerNorm ignores the component of a patch along the vector being normalized (Active)
- THEORY-1021 — In transformer language models, attribution patching agrees well with activation patching for small patches of single heads and their queries, keys, values and attention patterns, for nodes late in a circuit relative to the metric, and for circuits that route information through attention, and agrees badly for large patches such as whole residual streams, for the first MLP layer and the early layers of deep circuits, where nonlinearities move out of saturation, and where an effect needs several features at once, as Nanda proposes (Proposed)

THEORY-1028 — In InceptionV1 trained on ImageNet, the ten curve units of layer mixed3b each respond most to curves of one orientation, with responses invariant to fill and colour and jointly spanning all orientations under rotation of real images; for one of them, 3b:379, which fires on about 11% of dataset positions, activation level separated one author’s blind four-class labels of over 800 activation-binned images, and curves and imperfect curves made up 55% of its contribution to expected value (Active)
- THEORY-1022 — The curve units of InceptionV1’s mixed3b detect curves as such: curves, not correlated stimuli, cause their firing, they respond to curves in general whatever their colour, fill or texture, and they have no meaningful secondary function, their weak responses to edges and angles being part of the same curve-detecting algorithm, as Cammarata, Goh, Carter, Schubert, Petrov and Olah claim (Proposed)

From In GPT-J, for 652 CounterFact facts the model knows, the per-fact causal-tracing effect at an MLP layer adds at most 3.2 points of explained variance in the success of editing that fact at that layer, beyond what the choice of layer alone explains, across ROME, MEMIT and constrained fine-tuning and five editing objectives; ROME rewrites the fact at over 96% at every layer but the last, including layers where tracing does not place it, and the only reliable link appears when the edit is made on the noised input that tracing uses

THEORY-1024 — In GPT-J, for 652 CounterFact facts the model knows, the per-fact causal-tracing effect at an MLP layer adds at most 3.2 points of explained variance in the success of editing that fact at that layer, beyond what the choice of layer alone explains, across ROME, MEMIT and constrained fine-tuning and five editing objectives; ROME rewrites the fact at over 96% at every layer but the last, including layers where tracing does not place it, and the only reliable link appears when the edit is made on the noised input that tracing uses (Active)
- THEORY-1026 — In transformer language models, locating where a forward pass carries information about a fact and finding where a weight edit best changes the expressed fact are different questions, because an edit at one layer can override information accumulated across other layers, so that many layers could hold a fact and some happen to, as Hase, Bansal, Kim and Ghandeharioun propose (Proposed)

THEORY-1033 — In a 16-layer transformer language model trained on WikiText-103, the training prefixes that most activate a sampled feed-forward hidden unit share patterns that human annotators recognize, mostly shallow in layers 1–9 and semantic in layers 10–16; projected through the output embedding, a unit’s output weights put the next token of its top prefix first in up to about 3.5% of units in the upper layers and almost none below; and a layer’s top prediction differs from every unit’s in at least about 68% of examples (Active)
- THEORY-1030 — In transformer language models, each hidden unit of a feed-forward layer acts as a key–value memory, its input weights detecting patterns in the prefix and its output weights promoting the tokens that follow them, and a layer’s output composes many such memories while the residual stream refines the prediction from layer to layer, as Geva, Schuster, Berant and Levy propose (Proposed)

From In a model that fits y = Σ v_i x_i² with ỹ = Σ_r (W_r·x)² + b, on independent inputs of zero mean, unit variance and common fourth moment k, the expected squared error at the optimal bias is (k − 1)Σ(n_i − v_i)² − 2Σn_i² + 2Σn_i²/C_i, in the squared embedding lengths n_i and capacities C_i = (w_i·w_i)²/Σ_j(w_i·w_j)², and its minimum with the capacities summing to the number of neurons D gives C_i = clip((k − 1)v_i/((k − 3)λ) − 2/(k − 3), 0, 1): for k ≤ 3 each feature is given a whole dimension or none, and for k > 3 a feature is shared between neurons exactly when 2λ/(k − 1) < v_i < λ

THEORY-1036 — In a model that fits y = Σ v_i x_i² with ỹ = Σ_r (W_r·x)² + b, on independent inputs of zero mean, unit variance and common fourth moment k, the expected squared error at the optimal bias is (k − 1)Σ(n_i − v_i)² − 2Σn_i² + 2Σn_i²/C_i, in the squared embedding lengths n_i and capacities C_i = (w_i·w_i)²/Σ_j(w_i·w_j)², and its minimum with the capacities summing to the number of neurons D gives C_i = clip((k − 1)v_i/((k − 3)λ) − 2/(k − 3), 0, 1): for k ≤ 3 each feature is given a whole dimension or none, and for k > 3 a feature is shared between neurons exactly when 2λ/(k − 1) < v_i < λ (Active)
- THEORY-1035 — In trained neural networks, polysemantic neurons arise from the loss-optimal allocation of a layer’s limited capacity among features: a feature is shared between neurons when its marginal loss reduction from capacity equals that of the features it shares with, so that sparser, higher-kurtosis inputs, which make returns to capacity diminish, produce more polysemanticity, and the block structure of the embedding, which architecture shapes, sets how many features each neuron mixes, as Scherlis, Sachan, Jermyn, Benton and Shlegeris propose (Proposed)

From In InceptionV1, the ten curve units of mixed3b, which respond to curves of one orientation and not to the opposite one, regain the 180° echo of earlier curve units when their input is restricted to mixed3a early curves with the negative weights removed, and not when the negative weights are kept; a network one author wrote by hand from the described circuit, without looking at the original weights and tuned afterwards, has grayscale curve units that the authors show, by figures only, approximating the natural ones on synthetic curves across radii and orientations; and substituting them for the ten units recovered 26 of the 46 top-5 classifications lost by deleting the units, on 2,000 validation images, a result the authors call preliminary

THEORY-1038 — In InceptionV1, the ten curve units of mixed3b, which respond to curves of one orientation and not to the opposite one, regain the 180° echo of earlier curve units when their input is restricted to mixed3a early curves with the negative weights removed, and not when the negative weights are kept; a network one author wrote by hand from the described circuit, without looking at the original weights and tuned afterwards, has grayscale curve units that the authors show, by figures only, approximating the natural ones on synthetic curves across radii and orientations; and substituting them for the ten units recovered 26 of the 46 top-5 classifications lost by deleting the units, on 2,000 validation images, a result the authors call preliminary (Active)
- THEORY-1047 — The curve units of InceptionV1’s mixed3b implement a simple algorithm that can be read off their weights and written down in a few sentences: families of Gabors, lines and early curves, each a set of rotated copies of one unit, tiled along the curve’s tangent, with negative weights against similar shapes of opposing orientation, rotational and scale equivariance reducing the circuit’s more than 50,000 parameters about fifty-fold, so that a network written from this description reimplements them and the description accurately reflects the circuit, as Cammarata, Goh, Carter, Voss, Schubert and Olah claim (Proposed)

From In the Overview of Early Vision’s taxonomy of InceptionV1, every one of the 1,056 units of conv2d0 to mixed3b is assigned by the authors to exactly one of 77 groups, and 391 of them (37%), from 7% of conv2d2 to 48% of conv2d1 and 45% of mixed3b, are in groups whose own captions call them catch-all, broad and not well defined, low-confidence, unstudied or not understood; the mixed3b curve family is 10 of that layer’s 480 units

THEORY-1042 — In the Overview of Early Vision’s taxonomy of InceptionV1, every one of the 1,056 units of conv2d0 to mixed3b is assigned by the authors to exactly one of 77 groups, and 391 of them (37%), from 7% of conv2d2 to 48% of conv2d1 and 45% of mixed3b, are in groups whose own captions call them catch-all, broad and not well defined, low-confidence, unstudied or not understood; the mixed3b curve family is 10 of that layer’s 480 units (Active)
- THEORY-1040 — The units of the early layers of trained vision networks fall largely into neuron families, sets of variants of one feature across orientation, colour or scale, which reflect structure in the network rather than a convenience of the person sorting them, and the same families form across models, as Olah, Cammarata, Schubert, Goh, Petrov and Carter propose (Proposed)

From A linear barrier between two minima that survives the removal of symmetries marks a difference in the input attributes the two models rely on: they compute by different mechanisms

THEORY-116 — A linear barrier between two minima that survives the removal of symmetries marks a difference in the input attributes the two models rely on: they compute by different mechanisms (Proposed)
- THEORY-111 — In image classifiers, between two networks linearly connected by spawning from a shared checkpoint or by permutation alignment (MLPs, and VGG-16, ResNet-20 and ResNet-50 on CIFAR-10 and Tiny-ImageNet), nearly every layer’s features at three points on the line were proportional to the interpolation of the endpoints’ features, and pairs that were not linearly connected were far from it (Active) — also carries further THEORY-732
  - THEORY-1187 — Along a low-loss linear path between two networks, every layer’s features are the interpolation of the endpoints’ features (Proposed)
THEORY-732 — For ReLU networks interpolated linearly between two endpoints, weak additivity and commutativity at every layer make each layer’s features exactly the interpolation of the endpoints’ features, output-layer linearity bounds the path’s error by twice the endpoints’, and for one hidden layer with interpolating minima linear mode connectivity alone gives the exact case (Active)

From Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble’s accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model

THEORY-113 — Merging ResNets trained from different initialisations on disjoint halves of a label set by permuting one onto the other and averaging fell far short of the ensemble; their cross-model feature correlation fell with depth, merging only the early layers recovered most of the ensemble’s accuracy, and merging features within each model as well as across both helped at 4× width and above; Stoica et al. attribute the failure to features with no counterpart in the other model (Active)
- THEORY-1185 — Merging models trained on different tasks from different initialisations fails where their features have no counterpart in the other model, and such features grow more common with depth (Proposed)

From A graph network that learns its sheaf restriction maps from node features at each layer reaches the top, mostly within one standard deviation, of the standard small heterophilic node-classification benchmarks and stays within about one point of the best on homophilic ones; on a synthetic balanced bipartite graph it learns the negative transport the theory prescribes, but whether the maps it learns on real graphs have the geometry the theory credits for its success has not been examined

THEORY-1139 — A graph network that learns its sheaf restriction maps from node features at each layer reaches the top, mostly within one standard deviation, of the standard small heterophilic node-classification benchmarks and stays within about one point of the best on homophilic ones; on a synthetic balanced bipartite graph it learns the negative transport the theory prescribes, but whether the maps it learns on real graphs have the geometry the theory credits for its success has not been examined (Active)
- THEORY-1220 — A graph network that learns its sheaf restriction maps from node features succeeds on heterophilic graphs because it learns the class-aligned transports its separation theory prescribes: sign flips or rotations across classes, identity within them (Proposed)

THEORY-117 — In Uselis, Dittadi and Oh’s two-concept grids, ResNet-50s trained from scratch, with checkpoints chosen on the unseen pairs, reach unseen combinations as the share of combinations seen grows and not as up to four times more data of the same combinations is added; and when embeddings are exactly linearly factored, seen pairs that connect every value of both concepts, as two per value arranged in one cycle do, determine the embedding of every unseen pair (Active)
- THEORY-794 — In vision models, compositional generalization to unseen combinations of concepts is driven by the diversity of the combinations seen in training rather than by data volume, and arrives with representations that are linearly factored by concept, as Uselis, Dittadi and Oh propose (Proposed)

From In Claude Sonnet 4.5, Opus 4.5 and Opus 4.6, Gurnee, Sofroniew et al. found that projecting out the ten most active J-space directions in layers 38 to 54 of 100 lowered an experiential-language score, graded by Opus 4.6-based model graders on five or six prompts per condition, both in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions left it near baseline and the text stayed fluent; on Haiku 4.5 coherence failed first

THEORY-869 — In Claude Sonnet 4.5, Opus 4.5 and Opus 4.6, Gurnee, Sofroniew et al. found that projecting out the ten most active J-space directions in layers 38 to 54 of 100 lowered an experiential-language score, graded by Opus 4.6-based model graders on five or six prompts per condition, both in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions left it near baseline and the text stayed fluent; on Haiku 4.5 coherence failed first (Active)
- THEORY-1224 — In Claude language models, projecting out the most active J-space directions in early workspace layers lowers the rate of felt, first-person, sensory language in the models’ narrations of their own processing and in their descriptions of another person’s experience, while norm-matched perturbations along other directions do not and the text stays coherent; so in these models the experiential register of report depends on the access-like workspace and is not specific to self-report (Proposed)

THEORY-865 — In Gurnee, Sofroniew et al.’s experiments on Claude Sonnet 4.5, Haiku 4.5, Opus 4.5 and Opus 4.6, the residual-stream directions a corpus-averaged Jacobian lens marks as verbalizable (the J-space) carried most of the causal effect on verbal report and on unspoken two-hop intermediates while holding a small share of their variance (swaps along them succeeding on 59% and 61% of trials, against 5% and 28% for the remainder, which fell to 0% and 6% with the J-space clamped), held instructed concepts, and passed one swapped argument to several functions (76 of 192 swaps); in two small task sets (n = 8 and 11), report and flexible inference followed a swap while continuation, anomaly detection and line wrapping did not, a split the authors partly define by this result; and by the lens’s own statistics the J-space works in a middle band of layers, holds about 25 active directions and under 10% of activation variance, and is amplified and relayed by the weights more than other directions (Active)
- THEORY-1226 — In Claude language models, a sparse set of residual-stream directions that a corpus-averaged Jacobian lens identifies as verbalizable (the J-space) is causally privileged for verbal report, for holding instructed concepts, for unspoken intermediate steps of reasoning and for passing one argument to many functions, while continuation, anomaly detection and line wrapping use the same information without routing through it; it carries workspace-like content only in a middle band of layers, holds on the order of tens of active directions and under a tenth of activation variance, and is amplified and relayed by the weights more than other directions (Proposed)

From Cosine similarities to matched anchor points are unchanged by exactly the maps that preserve each sample’s cosines to the anchors, among them every orthogonal map times a positive scalar and every per-sample positive rescaling but not translations; in Moschella et al.’s experiments independently trained encoders agreed in them only approximately, enough for a decoder reading them to transfer untrained between seeds, architectures and languages with loss, while their nearest neighbours mostly differed

THEORY-175 — Cosine similarities to matched anchor points are unchanged by exactly the maps that preserve each sample’s cosines to the anchors, among them every orthogonal map times a positive scalar and every per-sample positive rescaling but not translations; in Moschella et al.’s experiments independently trained encoders agreed in them only approximately, enough for a decoder reading them to transfer untrained between seeds, architectures and languages with loss, while their nearest neighbours mostly differed (Active)
- THEORY-798 — Independently trained neural encoders of the same data learn latent spaces that differ mainly by an angle-preserving map, a rotation or reflection times a rescaling, so their cosine similarities to shared anchor samples agree across seeds, architectures and languages, as Moschella et al. propose (Proposed)

From In a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set fixes the generalising representation of addition when its same-answer pairs leave only translation and scale free; for non-modular addition with p = 10 and one-dimensional embeddings, generalisation sets in near the training fraction, about 0.4, at which that becomes likely

THEORY-181 — In a model that decodes the sum of two learned embeddings, at zero training loss with an injective decoder, a training set fixes the generalising representation of addition when its same-answer pairs leave only translation and scale free; for non-modular addition with p = 10 and one-dimensional embeddings, generalisation sets in near the training fraction, about 0.4, at which that becomes likely (Active)
- THEORY-782 — In networks that learn addition from summed embeddings, the critical training fraction of grokking is the fraction at which the training set’s same-answer pairs come to fix the generalising representation up to translation and scale, at every group size, as Liu, Kitouni, Nolte, Michaud, Tegmark and Williams propose (Proposed)

From For the quartic proxy of word2vec with tied weights, symmetric reweighting and constant weights, the rank-d minimisers are the top d eigenvectors of the relative deviation-from-independence matrix M*, scaled by the square roots of their eigenvalues, up to rotation; and from an initialisation aligned with those eigenvectors, gradient flow acquires each one in an independent logistic rise, in order of eigenvalue

THEORY-739 — For the quartic proxy of word2vec with tied weights, symmetric reweighting and constant weights, the rank-d minimisers are the top d eigenvectors of the relative deviation-from-independence matrix M*, scaled by the square roots of their eigenvalues, up to rotation; and from an initialisation aligned with those eigenvectors, gradient flow acquires each one in an independent logistic rise, in order of eigenvalue (Active)
- THEORY-182 — A rank-limited contrastive word embedding trained from small initialisation learns the top eigenvectors of the co-occurrence matrix’s relative deviation from independence, one at a time in order of eigenvalue, not the best low-rank approximation of its unconstrained optimum (Proposed)

From For an embedding whose Gram matrix is the absolute co-occurrence deviation matrix at full rank, words on a periodic lattice whose positive and negative co-occurrence parts each depend only on separation have Fourier modes of the lattice as principal components, in degenerate cos/sin pairs with amplitude the square root of the kernel’s Fourier transform, and the exponential kernel gives a closed loop with integer frequencies on a periodic lattice and an open curve with non-integer frequencies on an interval

THEORY-730 — For an embedding whose Gram matrix is the absolute co-occurrence deviation matrix at full rank, words on a periodic lattice whose positive and negative co-occurrence parts each depend only on separation have Fourier modes of the lattice as principal components, in degenerate cos/sin pairs with amplitude the square root of the kernel’s Fourier transform, and the exponential kernel gives a closed loop with integer frequencies on a periodic lattice and an open curve with non-integer frequencies on an interval (Active)
- THEORY-183 — When the co-occurrence of words on a latent continuum depends only on their separation, a spectral word embedding places them on the Fourier modes of that continuum, so the circles and rippled curves of months, years and places come from a symmetry of the corpus statistics (Proposed)

From A two-layer linear network trained by gradient flow on white inputs from a decoupled, balanced start learns each singular mode of its input–output correlation in a logistic rise that waits a time of order its inverse strength times the log of the inverse initial scale, so the modes arrive one at a time in order of strength; a single-layer network on the same data learns every mode on one timescale with no plateau, so under these conditions the stages come from the product of layers and the data’s spectrum

THEORY-740 — A two-layer linear network trained by gradient flow on white inputs from a decoupled, balanced start learns each singular mode of its input–output correlation in a logistic rise that waits a time of order its inverse strength times the log of the inverse initial scale, so the modes arrive one at a time in order of strength; a single-layer network on the same data learns every mode on one timescale with no plateau, so under these conditions the stages come from the product of layers and the data’s spectrum (Active)
- THEORY-186 — A deep linear network trained from small weights learns the singular modes of its input–output correlations one at a time in order of strength, each in a sharp sigmoidal transition, while a shallow network learns them all together; the stages come from the product of layers and the data’s spectrum, not from nonlinearity (Proposed)
- THEORY-481 — Under gradient flow averaged over tasks and data, a linear-attention layer trained on in-context regression learns as a two-layer linear network from an aligned start does: when tasks share the input covariance’s eigenbasis and the weights start and stay aligned with it, then in the long-context limit each input mode is learned on its own logistic curve whose timescale falls with the square of its variance, so learning is staged by the input spectrum, and the layer converges to the estimator preconditioned by the inverse input covariance; at finite context length the modes decouple exactly only when inputs are isotropic and the modes start equal, and otherwise couple at order 1/N (Active)

From On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers’ low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead

THEORY-194 — On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers’ low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead (Active)
- THEORY-789 — The block coarse-graining that maximizes mutual information with the system beyond a buffer picks out the renormalization-group-relevant variables in general, so those variables can be found from samples by information alone, without being known beforehand, as Koch-Janusz and Ringel propose (Proposed)

From On the Random Hierarchy Model, in the experiments reported (s = 2, L ≤ 4, small v), deep CNNs matched to the tree and trained in a feature-learning regime reach a tenth of chance error from about n_c m^L examples, polynomial in the input dimension, at the training-set size at which their hidden layers become invariant to swapping synonymous tuples; two-layer networks and the same CNNs as fixed kernels need a number proportional to the whole input space, exponential in the input dimension, and deep CNNs stay near chance when every tuple is made equally likely in every class

THEORY-195 — On the Random Hierarchy Model, in the experiments reported (s = 2, L ≤ 4, small v), deep CNNs matched to the tree and trained in a feature-learning regime reach a tenth of chance error from about n_c m^L examples, polynomial in the input dimension, at the training-set size at which their hidden layers become invariant to swapping synonymous tuples; two-layer networks and the same CNNs as fixed kernels need a number proportional to the whole input space, exponential in the input dimension, and deep CNNs stay near chance when every tuple is made equally likely in every class (Active)
- THEORY-786 — Deep networks learn hierarchically generated data from a number of examples set by when the correlations between input patches and the class rise above sampling noise, so detecting those correlations is how they learn the grammar, as Cagnetta and coauthors propose (Proposed)

From In the hyperbolic plane, a root carrying deg_max ≥ 15 chains of length m, embedded with worst-case distortion 1 + ε for ε ≤ 1, forces some node to hyperbolic distance at least (ℓ/(4ε))·ln(deg_max/(π√2)) from the origin, so fixed-point Poincaré coordinates need Ω((ℓ/ε) log deg_max) bits

THEORY-734 — In the hyperbolic plane, a root carrying deg_max ≥ 15 chains of length m, embedded with worst-case distortion 1 + ε for ε ≤ 1, forces some node to hyperbolic distance at least (ℓ/(4ε))·ln(deg_max/(π√2)) from the origin, so fixed-point Poincaré coordinates need Ω((ℓ/ε) log deg_max) bits (Active)
- THEORY-196 — A finite tree embeds in the hyperbolic plane with worst-case distortion 1 + ε, but because a hyperbolic distance d takes about d bits to write, its coordinates need Θ((ℓ/ε) log deg_max) bits: logarithmic in the branching, linear in the longest path ℓ, so hyperbolic space is economical for shallow bushy hierarchies and not for deep ones (Proposed)

From For a symmetrized Barlow Twins loss on a linear or fixed-kernel model, gradient flow from a small initialization of common scale aligned with the top eigenvectors of the pairs’ symmetrized cross-correlation learns them one at a time, in order of eigenvalue, and ends at a top spectral solution, which is exactly a minimum-norm zero-loss solution: kernel PCA on K_Γ

THEORY-229 — For a symmetrized Barlow Twins loss on a linear or fixed-kernel model, gradient flow from a small initialization of common scale aligned with the top eigenvectors of the pairs’ symmetrized cross-correlation learns them one at a time, in order of eigenvalue, and ends at a top spectral solution, which is exactly a minimum-norm zero-loss solution: kernel PCA on K_Γ (Active)
- THEORY-793 — Gradient flow on a symmetrized Barlow Twins loss with a linear or fixed-kernel model, from generic small initialization and not only from initialization aligned with the eigenvectors, learns the top eigenvectors of the pairs’ symmetrized cross-correlation one at a time in order of eigenvalue and ends at a top spectral solution, as Simon and coauthors derive informally (Proposed)

THEORY-243 — In the three-unit identity network x ↦ x(1, 0, 1) ↦ (0, 2, 1)·h, aligning the input with the third unit and aligning it with the direction (h₁ + h₂)/√2, two alignments related by an orthogonal change of basis, pass every interchange intervention on the input equally, so interchange interventions on that variable alone do not fix which internal direction mediates it (Active)
- THEORY-802 — In trained networks as in constructed ones, interchange interventions on a fixed set of variables can be satisfied equally by alignments related by a change of basis, so which internal direction mediates a high-level variable is fixed only relative to a chosen decomposition into units, and complex models will generally have several distinct, fully correct abstractions (Proposed)

From When a sequence generator’s latent dynamics factor as a tensor product, its predictive state embeds losslessly and linearly in orthogonal per-factor subspaces of total dimension Σ(d_n − 1); 4-layer GPT-2-style transformers trained by next-token prediction on five Mess3 and Bloch Walk factors settle near that dimension, in per-factor subspaces that are linearly independent and approximately orthogonal in their leading components, and under the token noise tested reach it first even when noise makes it lossy

THEORY-256 — When a sequence generator’s latent dynamics factor as a tensor product, its predictive state embeds losslessly and linearly in orthogonal per-factor subspaces of total dimension Σ(d_n − 1); 4-layer GPT-2-style transformers trained by next-token prediction on five Mess3 and Bloch Walk factors settle near that dimension, in per-factor subspaces that are linearly independent and approximately orthogonal in their leading components, and under the token noise tested reach it first even when noise makes it lossy (Active)
- THEORY-808 — Transformers trained by next-token prediction on sequence processes whose latent dynamics factor as a tensor product learn to represent the belief state in approximately orthogonal per-factor subspaces of about the factored dimension rather than the joint one, and reach that factored representation first even when noise makes it lossy (Proposed)

From Under MSE loss, in Han, Papyan and Donoho’s VGG, ResNet and DenseNet runs on five balanced ten-class datasets, the trained classifier stays near the least-squares classifier for its own features from an early epoch; and on that central path, for unconstrained features continually renormalized to identity within-class covariance and without weight decay, gradient flow drives the class-separation singular values to infinity at equal rates, which in that model is neural collapse

THEORY-278 — Under MSE loss, in Han, Papyan and Donoho’s VGG, ResNet and DenseNet runs on five balanced ten-class datasets, the trained classifier stays near the least-squares classifier for its own features from an early epoch; and on that central path, for unconstrained features continually renormalized to identity within-class covariance and without weight decay, gradient flow drives the class-separation singular values to infinity at equal rates, which in that model is neural collapse (Active)
- THEORY-797 — Deep classifiers trained with MSE loss stay near the least-squares classifier for their own features and move along the central path by the dynamics of continually renormalized gradient flow, whose equal-rate growth of the whitened class-separation singular values is how they reach neural collapse, as Han, Papyan and Donoho propose (Proposed)

From In the unconstrained-features max-margin problem with k classes under step imbalance and feature dimension at least k − 1, the unique optimal logits are the simplex-encoded labels, every margin equals one, and the classifier and embedding Gram matrices are the two square-root factors of that matrix’s singular value decomposition; the geometry is the simplex equiangular tight frame exactly when the classes are balanced or there are two of them

THEORY-738 — In the unconstrained-features max-margin problem with k classes under step imbalance and feature dimension at least k − 1, the unique optimal logits are the simplex-encoded labels, every margin equals one, and the classifier and embedding Gram matrices are the two square-root factors of that matrix’s singular value decomposition; the geometry is the simplex equiangular tight frame exactly when the classes are balanced or there are two of them (Active)
- THEORY-286 — Under class imbalance the last-layer geometry that unregularized cross-entropy prefers is not the simplex ETF but simplex-encoded-label interpolation: the logits equal the simplex-encoded labels, the classifier and embedding Gram matrices come from that matrix’s SVD, any finite weight decay changes the geometry, and training approaches it more slowly as imbalance grows (Proposed)

From For a wide two-layer ReLU network with zero bias on one-dimensional data whose two classes lie on opposite sides of the origin, the tangent kernel is a constant multiple of the initialization kernel, so lazy training leaves the within-class variability of its features, NC1, where initialization put it

THEORY-294 — For a wide two-layer ReLU network with zero bias on one-dimensional data whose two classes lie on opposite sides of the origin, the tangent kernel is a constant multiple of the initialization kernel, so lazy training leaves the within-class variability of its features, NC1, where initialization put it (Active)
- THEORY-813 — Training a wide network in the lazy, kernel regime does not make its features more collapsed than at initialization, so neural collapse, where it occurs, is something only feature learning produces (Proposed)

From MSN’s objective penalizes unequal cluster sizes explicitly and SwAV’s enforces equal sizes, and in one study with mini-batches of two to eight ImageNet classes, SimCLR, MSN and VICReg lost 11 to 18 points of ImageNet linear accuracy while MAE and data2vec lost under one

THEORY-306 — MSN’s objective penalizes unequal cluster sizes explicitly and SwAV’s enforces equal sizes, and in one study with mini-batches of two to eight ImageNet classes, SimCLR, MSN and VICReg lost 11 to 18 points of ImageNet linear accuracy while MAE and data2vec lost under one (Active)
- THEORY-805 — Volume-maximizing joint-embedding objectives as a class, VICReg and SimCLR as well as MSN and SwAV, carry a prior toward equal-sized clusters, and that prior costs them semantic transfer when the pretraining data are class-imbalanced, including on naturally long-tailed data (Proposed)

From In a spectral contrastive graph that joins labelled known classes to unlabelled data, the least-squares residual of a novel labelling on the top-k eigenspace, with the labelled entries left free, equals exactly the part of the labelling outside that eigenspace that the labelled rows of the discarded eigenvectors cannot absorb

THEORY-737 — In a spectral contrastive graph that joins labelled known classes to unlabelled data, the least-squares residual of a novel labelling on the top-k eigenspace, with the labelled entries left free, equals exactly the part of the labelling outside that eigenspace that the labelled rows of the discarded eigenvectors cannot absorb (Active)
- THEORY-320 — In a spectral contrastive graph that joins labelled known classes to unlabelled data, a novel labelling’s linear-probe residual is exactly its part outside the top eigenspace that the labelled rows of the discarded eigenvectors cannot absorb, so which partition of the unlabelled data is discovered depends on the known classes: ones sharing the target’s attribute can remove the residual, and ones sharing a rival attribute can raise it (Proposed)

From In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary

THEORY-421 — In HGCN embeddings of two synthetic 3-ary trees trained for link prediction or node classification, distance from the origin is only a partial reading of hierarchical level: the root is not innermost (distance 1.8–3.3 against a minimum of 1.2–2.4) and level order is recovered for 69–75% of node pairs; a loss built only on pairwise distances leaves the origin arbitrary (Active)
- THEORY-825 — Hyperbolic graph embedding models trained on task losses that never mention the hierarchy do not place a tree’s root nearest the origin, and their distance from the origin orders hierarchical levels only partly (Proposed)

From In Claude 3 Sonnet’s middle-layer residual stream, on two prompts for which every active sparse-autoencoder feature was ablated, how strongly a feature was active predicted its ablation effect on the answer poorly (correlation about 0.12) while its gradient attribution predicted it well (about 0.81), and on the multi-step prompt only three of the ten most active features were among the ten with the largest ablation effect

THEORY-987 — In Claude 3 Sonnet’s middle-layer residual stream, on two prompts for which every active sparse-autoencoder feature was ablated, how strongly a feature was active predicted its ablation effect on the answer poorly (correlation about 0.12) while its gradient attribution predicted it well (about 0.81), and on the multi-step prompt only three of the ten most active features were among the ten with the largest ablation effect (Active)
- THEORY-965 — Sparse-autoencoder features of a large language model’s residual stream are units the model computes with: they track abstract concepts across languages, modalities and concrete or abstract instances, and intervening on one changes the model’s behaviour in the way its label predicts, better than directions found from a few labelled examples (Proposed) — also carries further THEORY-992
THEORY-992 — In sparse autoencoders of about 1M, 4M and 34M features on Claude 3 Sonnet’s middle-layer residual stream, whether a concept among chemical elements, cities, animals and foods had a dedicated feature depended on how often it occurs in the training data: the frequency at which a concept became more than 50% likely to have one was slightly below one over the number of live features, and the three dictionaries’ curves roughly coincided when frequency was rescaled by that number (Active)

From In small attention-only transformer language models of two to six layers, in-context learning, measured as the loss at token 500 minus the loss at token 50, appears abruptly in one early window of training in which induction heads form, and ablating those heads at test time removes nearly all of it; one-layer models show neither, and letting a single layer attend by the previous token brings the jump to one-layer models and moves it earlier in two-layer ones

THEORY-973 — In small attention-only transformer language models of two to six layers, in-context learning, measured as the loss at token 500 minus the loss at token 50, appears abruptly in one early window of training in which induction heads form, and ablating those heads at test time removes nearly all of it; one-layer models show neither, and letting a single layer attend by the previous token brings the jump to one-layer models and moves it earlier in two-layer ones (Active)
- THEORY-990 — In transformer language models of every size, including large ones with MLPs, induction heads, extended to fuzzy matching of similar tokens and abstract patterns, are the mechanism of most in-context learning, as Olsson et al. propose (Proposed)

self

From Brushing a seen rubber hand in synchrony with the hidden real hand makes touch be felt at the rubber hand, makes it be reported as one’s own, and shifts the felt position of the real hand toward it; a small asynchrony removes these effects

THEORY-765 — Brushing a seen rubber hand in synchrony with the hidden real hand makes touch be felt at the rubber hand, makes it be reported as one’s own, and shifts the felt position of the real hand toward it; a small asynchrony removes these effects (Rejected)
- THEORY-766 — Brushing a seen rubber hand in synchrony with the hidden real hand, against asynchronous brushing, raises reports that touch is felt at the rubber hand and, more weakly and less reliably, that the rubber hand is one’s own; the proprioceptive drift toward the rubber hand that accompanies these reports is not a measure of them; the standard comparison conditions do not equalise participants’ expectations, so whether the stimulation or the expectation produces the report difference is open (Active)
  - THEORY-827 — The rubber hand illusion is a perceptual illusion driven by multisensory integration: the difference in reports between synchronous and asynchronous brushing is produced by the stimulation, not by participants’ expectations (Proposed)

self-governance

From A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape

THEORY-040 — A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape (Proposed)
- THEORY-1118 — McKenna and Coates argue, with Dryden and with Buss and Westlund, that a condition on free agency stated only in the agent’s present psychological structure is met by an agent whose structure a manipulator installed, so a mesh account must count that agent free or add a historical condition; Frankfurt, read first-hand, takes the first option (Active)

From A regulation is autonomous to the degree it is integrated with one’s other values and needs: autonomy is graded, and is not set by the order of the attitude that endorses it

THEORY-054 — A regulation is autonomous to the degree it is integrated with one’s other values and needs: autonomy is graded, and is not set by the order of the attitude that endorses it (Proposed)
- THEORY-047 — A regulation taken in under controlling conditions is introjected rather than integrated, so the social history of how a value was acquired shapes how autonomous acting on it is (Proposed)

From Agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, and planning, autonomy, memory and coordination do not supply one

THEORY-219 — Agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, and planning, autonomy, memory and coordination do not supply one (Proposed)
- THEORY-1165 — Kahl proposes that agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, which is absent where revision is imposed from outside, escaped by rollback or chosen by a fixed higher-order rule; that allocating authority among a system’s own components coordinates execution without authoring its governing frame; and that planning, autonomy, memory and coordination do not supply that standpoint (Active)

From Gollwitzer and Sheeran (2006) conclude, narratively from a handful of studies that measured or manipulated the goal intention, that implementation intentions affect performance only when the goal intention they serve is strong and activated, so the planned response is not elicited mechanically by its cue; in 2024, across eight tests, plan effects were larger with strong, activated or self-concordant goal intentions than with weak, unactivated or self-discordant ones (d = .79 for the difference), and in 31 physical-activity tests split at the median of intention, planning helped with strong intentions (d = .31) and had no effect with weak ones (d ≈ 0)

THEORY-904 — Gollwitzer and Sheeran (2006) conclude, narratively from a handful of studies that measured or manipulated the goal intention, that implementation intentions affect performance only when the goal intention they serve is strong and activated, so the planned response is not elicited mechanically by its cue; in 2024, across eight tests, plan effects were larger with strong, activated or self-concordant goal intentions than with weak, unactivated or self-discordant ones (d = .79 for the difference), and in 31 physical-activity tests split at the median of intention, planning helped with strong intentions (d = .31) and had no effect with weak ones (d ≈ 0) (Active)
- THEORY-1176 — Implementation intentions affect performance only when the goal intention they serve is strong and activated, so the cue-triggered response is gated by the goal and does not fire on the cue alone (Proposed)

From Griesemer and Shavit propose, as a first and rough sketch, that a collective becomes a new level of individuality only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it, some core interactions have become generatively entrenched, and it has organizational closure, and that such a community can be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone

THEORY-1247 — Griesemer and Shavit propose, as a first and rough sketch, that a collective becomes a new level of individuality only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it, some core interactions have become generatively entrenched, and it has organizational closure, and that such a community can be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone (Active)
- THEORY-418 — A collective becomes a new level of individuality, on this sketch, only when collaboration toward a shared goal, rather than coordination or cooperation, organizes it and some core interactions have become generatively entrenched and, as the paper asserts without deriving it from entrenchment, it has organizational closure; such a community can then be reproduced only by exporting an organized sample of its roles and interactions, not of its members alone (Proposed)

From On the court agenda {P, Q, R, R iff (P and Q)} under universal domain, proposition-wise majority voting is not robustly rational, while the premise-based procedure is, and it makes the group’s judgments a function of the members’ whole judgment sets that the members’ judgments on the conclusion neither fix nor need to enter

THEORY-725 — On the court agenda {P, Q, R, R iff (P and Q)} under universal domain, proposition-wise majority voting is not robustly rational, while the premise-based procedure is, and it makes the group’s judgments a function of the members’ whole judgment sets that the members’ judgments on the conclusion neither fix nor need to enter (Active)
- THEORY-127 — On the court agenda, and on any agenda in which every proposition reaches every other by conditional entailment and some minimally inconsistent triple stays consistent with two of its members negated, a group whose judgments are robustly rational under universal domain and respect unanimity cannot form its judgment on each proposition from its members’ judgments on that proposition alone unless one member dictates; its judgments must supervene on the members’ whole sets of judgments, and so can depart from what the members judge on the same proposition while being wholly fixed by them (Active)

From A structured group is not identical to its members: two groups of one kind can have the same members at every time and still be two, which theories that identify a group with a plurality, fusion, aggregate or set of its members cannot allow, and a view of groups as realizations of structures with functionally defined nodes can

THEORY-549 — A structured group is not identical to its members: two groups of one kind can have the same members at every time and still be two, which theories that identify a group with a plurality, fusion, aggregate or set of its members cannot allow, and a view of groups as realizations of structures with functionally defined nodes can (Proposed)
- THEORY-575 — ‘Social group’ names two kinds of thing: organized social groups, such as teams, courts and families, are structured wholes whose people occupy the nodes of an internal social structure, while feature social groups, such as women, racial groups and classes, are social kinds, nodes in external social structures, and are not concrete wholes (Proposed)

THEORY-167 — The Russian wondertale is built from a small fixed set of character functions in a fixed order, so that all wondertales share one composition and differ in who performs the functions and how (Proposed)
- THEORY-1164 — Propp proposes, from his analysis of Afanasyev’s tales 50–151, that the Russian wondertale is built from 31 character functions, each defined by its consequence for the action, occurring in one order and falling to seven roles, so that wondertales share one composition and differ in who performs the functions and how (Active)

From As Raviv, Meyer and Lev-Ari report it, in a laboratory game where 12 fully connected groups of eight and 12 of four built a new communication system with total interaction held fixed, the larger groups’ languages grew more systematic faster, ended more systematic and differed less from one another, while communicative success and convergence did not differ by size; input variability, higher in the larger groups, predicted the next round’s gain in structure, an association measured and not manipulated

THEORY-164 — As Raviv, Meyer and Lev-Ari report it, in a laboratory game where 12 fully connected groups of eight and 12 of four built a new communication system with total interaction held fixed, the larger groups’ languages grew more systematic faster, ended more systematic and differed less from one another, while communicative success and convergence did not differ by size; input variability, higher in the larger groups, predicted the next round’s gain in structure, an association measured and not manipulated (Active)
- THEORY-1171 — In groups that build a new communication system, larger groups make it more systematic, and the greater variety of input they face drives the effect (Proposed)

From In one case study of the 2017 r/place Mona Lisa, what was added to or removed from the image was, once its forum opened, decided off the canvas, in a forum of 161 accounts negotiating through linked design images, where a few central members admitted or refused additions, and the forum’s changes then appeared in a region where about 77,000 accounts placed tiles, most of whom never posted; how many placers followed the designs is not measured, and the authors argue that the placement record alone cannot show who directed the image or who cooperated

THEORY-218 — In one case study of the 2017 r/place Mona Lisa, what was added to or removed from the image was, once its forum opened, decided off the canvas, in a forum of 161 accounts negotiating through linked design images, where a few central members admitted or refused additions, and the forum’s changes then appeared in a region where about 77,000 accounts placed tiles, most of whom never posted; how many placers followed the designs is not measured, and the authors argue that the placement record alone cannot show who directed the image or who cooperated (Active)
- THEORY-1181 — The content of a crowd-built r/place artwork is decided off the canvas, in negotiation through shared design images, where a few central members admit or refuse additions, and carried out by placers who mostly never join the discussion, so the placement record shows the outcome of coordination but not who directed it or who cooperated (Proposed)

From On the 2022 r/place canvas, the abrupt overthrow of a crowd-built image from a stable period is predicted up to hours ahead, on held-out images and on the 2023 edition, by a combination of many image-level series far better than by any generic slowing-down indicator; the series the predictor relies on include low past restoring of attacked pixels as well as attack outrunning restoration, and about a quarter of overthrows give no warning at the 20-minute range

THEORY-214 — On the 2022 r/place canvas, the abrupt overthrow of a crowd-built image from a stable period is predicted up to hours ahead, on held-out images and on the 2023 edition, by a combination of many image-level series far better than by any generic slowing-down indicator; the series the predictor relies on include low past restoring of attacked pixels as well as attack outrunning restoration, and about a quarter of overthrows give no warning at the 20-minute range (Active)
- THEORY-1183 — Crowd-maintained r/place images persist as an equilibrium of attack and repair, and their abrupt overthrow is preceded by several distinct precursors, including the defenders’ own conduct, rather than by one generic warning signal, while many overthrows give no warning (Proposed)

THEORY-918 — Across 642 experimental tests to September 2019, forming implementation intentions improved outcomes, with uncorrected effects on behaviour (d = .27), cognitive performance (.47), affect regulation (.66) and social judgement (.52), but publication bias was substantial: the pooled d = .36 falls to a model-averaged d = .15 under robust Bayesian meta-analysis, which still found extreme evidence of an effect (BF10 = 319), with large heterogeneity (Active)
- THEORY-1192 — Across 642 experimental tests to September 2019, forming implementation intentions reliably improved behaviour, cognitive performance, affect regulation and social judgement, but publication bias was substantial: the pooled d = .36 falls to a model-averaged d = .15 under robust Bayesian meta-analysis, with large heterogeneity (Proposed)

From In Friedkin, Jia and Bullo’s study of 30 four-person student groups discussing 15 choice dilemmas, a member’s self-reported closure to influence on the next issue rose with their influence centrality on the previous issue, an effect constant along the sequence, and with their average centrality over all prior issues, an effect that grew along it; influence networks thinned and discussions shortened while the rate of consensus did not change. Closure and centrality both come from the members’ own chip allocations, and prior closure is not controlled

THEORY-902 — In Friedkin, Jia and Bullo’s study of 30 four-person student groups discussing 15 choice dilemmas, a member’s self-reported closure to influence on the next issue rose with their influence centrality on the previous issue, an effect constant along the sequence, and with their average centrality over all prior issues, an effect that grew along it; influence networks thinned and discussions shortened while the rate of consensus did not change. Closure and centrality both come from the members’ own chip allocations, and prior closure is not controlled (Active)
- THEORY-1212 — In 30 four-person student groups discussing 15 choice dilemmas, a member’s self-reported closure to influence on the next issue rose with their influence over prior outcomes, increasingly so along the sequence, and the groups’ influence networks thinned and their discussions shortened while the rate of consensus did not change: evidence for a reflected-appraisal step by which influence concentrates in deliberating groups (Proposed)

From In Szeidl and Szucs’s model of two outlets competing for readers and advertisers, where state-owned advertisers reward only the connected outlet’s slant, with weight δ, every equilibrium has the slant gap between the connected and the unconnected outlet equal to their owners’ ideal-point gap plus a favor-exchange term (βδ/NT)·Σ over government advertisers of masⱼWⱼ, up to measurement noise; readers’ preference for slant cancels from the gap, so equal observed gaps can come from different mixes of ideology and exchange

THEORY-860 — In Szeidl and Szucs’s model of two outlets competing for readers and advertisers, where state-owned advertisers reward only the connected outlet’s slant, with weight δ, every equilibrium has the slant gap between the connected and the unconnected outlet equal to their owners’ ideal-point gap plus a favor-exchange term (βδ/NT)·Σ over government advertisers of masⱼWⱼ, up to measurement noise; readers’ preference for slant cancels from the gap, so equal observed gaps can come from different mixes of ideology and exchange (Active)
- THEORY-852 — Under Szeidl and Szucs’s structural model and its identifying assumptions (an owner’s ideology fixed across episodes; the government’s online advertising budget not driven by the outlets’ slant), both owner ideology and favor exchange drove the Hungarian portal Origo’s reduced coverage of corruption in 2013–2016: favor exchange alone cut it by 29–44%, the new owner’s ideology alone by 63% after 2016, and the two together by 74%, so they are substitutes; and a ban on targeted government advertising would have raised it by 41–79% (Active)
  - THEORY-1217 — Both owner ideology and favor exchange drove the Hungarian portal Origo’s reduced coverage of corruption in 2013–2016: favor exchange alone cut it by 29–44%, the new owner’s ideology alone by 63% after 2016, and the two together by 74%, so they are substitutes; and a ban on targeted government advertising would have raised it by 41–79% (Proposed)

THEORY-329 — In 32,479 Weibo posts of 2011–2016 scored with an emotion lexicon, posts tagged fake by the platform’s fact-checkers carry more anger and less joy than true posts from verified accounts, and a post’s share of anger words goes with more retweets after controls; whether the gap is due to falsity, and whether anger causes the spread, is not shown (Active)
- THEORY-1221 — Anger makes fake news viral online: on Weibo fake news carries more anger and less joy than true news, and the anger in a post causes it to be shared more (Proposed)

THEORY-889 — In Acharya, Blackwell and Sen’s analysis, white Southerners surveyed in 2006 to 2011 who live in counties where a larger share of the 1860 population was enslaved are, with state fixed effects and 1860 county controls, somewhat less Democratic, less supportive of affirmative action and more racially resentful, by about 0.09 to 0.13 standard deviations per standard deviation (about 20 points) of slave share, and in 1984 to 1998 survey data colder toward black Americans, by about 0.32; a cotton-suitability instrument gives larger estimates; the direct effect estimated with present black share held fixed is about as large; and in county presidential vote, which counts black voters until disenfranchisement, the difference appeared after Reconstruction rather than before the war (Active)
- THEORY-1225 — Among white Southerners today, those living in counties where a larger share of the 1860 population was enslaved are somewhat less Democratic, less supportive of affirmative action and more racially resentful, by about a tenth of a standard deviation per standard deviation of slave share, and colder toward black Americans, by about a third; the difference is not accounted for by the present black share of the population, and in voting it appeared after Reconstruction rather than before the war, which supports its origin in the incentives that emancipation gave planters to repress black labor and votes, and its local transmission thereafter (Proposed)

THEORY-123 — In ad hoc groups of students negotiating a bonus, a confederate’s randomly assigned pleasant or unpleasant mood spread to the other members, by their self-reports and by blind coders, and the more pleasant mood a group caught, the more cooperative and the less conflicted coders rated it, a correlation with measured contagion rather than a manipulated effect; members’ moods also converged without a confederate, and members did not credit their mood for how effective they judged themselves; the numbers are from the 2001 manuscript (Active)
- THEORY-126 — A collective emotion is the synchronous convergence of individuals’ affective responses to one event or object, produced bottom-up by shared appraisal, contagion and group membership, and it needs no group subject; it takes a We-mode form when the appraisals rest on collectively intentional states (Proposed)
- THEORY-826 — In work groups, a member’s displayed mood spreads to the others, and pleasant emotional contagion improves the group’s cooperation and reduces its conflict (Proposed)

From In the 66 Pew surveys of 2001–2011 that varied the order of two back-to-back binary questions, question order changed the answers but not the probability of giving the same answer to both, the equality a Lüders projection model predicts for any projectors; a classical model with order events can meet the same equality, so the regularity does not by itself favour quantum over classical probability

THEORY-166 — In the 66 Pew surveys of 2001–2011 that varied the order of two back-to-back binary questions, question order changed the answers but not the probability of giving the same answer to both, the equality a Lüders projection model predicts for any projectors; a classical model with order events can meet the same equality, so the regularity does not by itself favour quantum over classical probability (Active)
- THEORY-796 — Question-order effects in surveys generally leave the probability of giving the same answer to both questions unchanged, the QQ equality that a projection model of judgment predicts, as Wang, Solloway, Shiffrin and Busemeyer propose (Proposed)

From On the 2017 r/place log, with coalitions recovered from co-placement, which coalitions survive is barely predictable early from their size, area, start time and colour entropy, the size distribution of recovered coalitions shifts toward large ones as the event proceeds, and some lasting artworks were held by coalitions that kept working under sustained overwriting

THEORY-216 — On the 2017 r/place log, with coalitions recovered from co-placement, which coalitions survive is barely predictable early from their size, area, start time and colour entropy, the size distribution of recovered coalitions shifts toward large ones as the event proceeds, and some lasting artworks were held by coalitions that kept working under sustained overwriting (Active)
- THEORY-795 — On a contested shared canvas, the artworks that last are those whose coalitions keep repairing them under continual overwriting, not those left unattacked, as Wu and Silva propose from the 2017 r/place lifecycles (Proposed)

From In Rasooly and Rozzi’s pre-registered randomised experiment on 817 Manifold play-money markets, about 59% of a ±5-point price shock that no trader’s information supported was still present after 60 days in unresolved markets, with about 25% undone in the first week and reversion slowing after it; their model of finitely many risk-averse traders gives such partial, slowing reversion even with no learning, and the data cannot tell that from learning

THEORY-267 — In Rasooly and Rozzi’s pre-registered randomised experiment on 817 Manifold play-money markets, about 59% of a ±5-point price shock that no trader’s information supported was still present after 60 days in unresolved markets, with about 25% undone in the first week and reversion slowing after it; their model of finitely many risk-averse traders gives such partial, slowing reversion even with no learning, and the data cannot tell that from learning (Active)
- THEORY-810 — Prediction-market prices keep much of an arbitrary price shock that no trader’s information supports for weeks, rather than correcting it within a short transition phase (Proposed)

From In a community whose consensus model is replaced whenever a single sound pairwise model comparison favours the challenger, which models get proposed can hold the consensus on a false model while results reproduce almost perfectly, so reproducibility need not track convergence to truth

THEORY-295 — In a community whose consensus model is replaced whenever a single sound pairwise model comparison favours the challenger, which models get proposed can hold the consensus on a false model while results reproduce almost perfectly, so reproducibility need not track convergence to truth (Active)
- THEORY-814 — In scientific communities that settle on a consensus model by sound model comparisons, high reproducibility of results is not evidence that the consensus is true, because which alternatives get proposed can hold a false consensus whose results reproduce almost perfectly (Proposed)

THEORY-674 — In Garg and Fetzer’s GPT-4o-mini extraction from NBER and CEPR working papers, edges kept in at least 4 of 9 passes and never checked against human labels, the mean per-paper share of edges labelled as backed by difference-in-differences, instrumental variables, an RCT, regression discontinuity or synthetic control rose from 7.7% in 1990 to 31.7% in 2020, at every overlap threshold; the paper’s per-decade slope fell from +0.0187 to +0.0011 once method mix and graph volume were controlled (Active)
- THEORY-824 — Economics working papers increasingly back their claims with design-based identification: in NBER and CEPR working papers the share of claims supported by difference-in-differences, instrumental variables, an RCT, regression discontinuity or synthetic control rose about fourfold from 1990 to 2020 (Proposed)

From Among 100 Chapel Hill voters not yet committed to a candidate in September–October 1968, the aggregate emphasis they gave the campaign’s main issues correlated +.967 with the emphasis of the major news items, and +.979 with that of the minor items, in the nine media serving them; and voters leaning to a candidate agreed more with all the news than with news about their own side in 18 of 24 comparisons

THEORY-842 — Among 100 Chapel Hill voters not yet committed to a candidate in September–October 1968, the aggregate emphasis they gave the campaign’s main issues correlated +.967 with the emphasis of the major news items, and +.979 with that of the minor items, in the nine media serving them; and voters leaning to a candidate agreed more with all the news than with news about their own side in 18 of 24 comparisons (Active)
- THEORY-838 — The emphasis the news media give to issues raises the importance their audiences assign those issues, independently of any change in the audiences’ attitudes toward the issues: the media influence what people think about more than what they think (Proposed)

From In one laboratory study of a nine-period game between an infrastructure operator, two elected policymakers and three voters (59 groups), letting the first elected policymaker fix the price for all periods moved prices toward the efficient level when policymakers’ payoffs were set equal to consumers’, and moved them further above it when their payoffs were set equal to the operator’s; under period-by-period regulation voters’ pressure lowered the operator-aligned policymakers’ prices without removing the gap

THEORY-856 — In one laboratory study of a nine-period game between an infrastructure operator, two elected policymakers and three voters (59 groups), letting the first elected policymaker fix the price for all periods moved prices toward the efficient level when policymakers’ payoffs were set equal to consumers’, and moved them further above it when their payoffs were set equal to the operator’s; under period-by-period regulation voters’ pressure lowered the operator-aligned policymakers’ prices without removing the gap (Active)
- THEORY-858 — Insulating a regulator from future political revision fixes in place the interests of whoever designs the insulated regime: it relieves the underinvestment that electoral pressure on prices causes when the designers serve consumers, and entrenches capture, removing the electoral pressure that would correct it, when they serve the regulated industry (Proposed)

society-and-governance

From Agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, and planning, autonomy, memory and coordination do not supply one

THEORY-219 — Agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, and planning, autonomy, memory and coordination do not supply one (Proposed)
- THEORY-1165 — Kahl proposes that agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, which is absent where revision is imposed from outside, escaped by rollback or chosen by a fixed higher-order rule; that allocating authority among a system’s own components coordinates execution without authoring its governing frame; and that planning, autonomy, memory and coordination do not supply that standpoint (Active)

THEORY-327 — Fricker, in her 2003 paper and her 2017 retrospective, holds that testimonial injustice is an ingenuous misjudgement of a speaker’s credibility caused by prejudice, and that deliberate manipulation of credibility is a related but distinct wrong; and she accepts Anderson’s structural form, which is pre-emptive: a speaker’s word is never sought because of a structural prejudice, such as an un-refreshed list, that is no one’s fault (Active)
- THEORY-1173 — Testimonial injustice is an ingenuous misjudgement of a speaker’s credibility caused by prejudice, not a deliberate manipulation of it, and it can be structural and pre-emptive, when a speaker’s word is never sought because of a structural prejudice that is no one’s fault (Proposed)

From In one case study of the 2017 r/place Mona Lisa, what was added to or removed from the image was, once its forum opened, decided off the canvas, in a forum of 161 accounts negotiating through linked design images, where a few central members admitted or refused additions, and the forum’s changes then appeared in a region where about 77,000 accounts placed tiles, most of whom never posted; how many placers followed the designs is not measured, and the authors argue that the placement record alone cannot show who directed the image or who cooperated

THEORY-218 — In one case study of the 2017 r/place Mona Lisa, what was added to or removed from the image was, once its forum opened, decided off the canvas, in a forum of 161 accounts negotiating through linked design images, where a few central members admitted or refused additions, and the forum’s changes then appeared in a region where about 77,000 accounts placed tiles, most of whom never posted; how many placers followed the designs is not measured, and the authors argue that the placement record alone cannot show who directed the image or who cooperated (Active)
- THEORY-1181 — The content of a crowd-built r/place artwork is decided off the canvas, in negotiation through shared design images, where a few central members admit or refuse additions, and carried out by placers who mostly never join the discussion, so the placement record shows the outcome of coordination but not who directed it or who cooperated (Proposed)

From Of 853 national states of emergency declared from 1949 to 2017, 115 were reported as unlawful in declaration, conduct or duration; 97 of those followed political turmoil and 14 natural disasters, democracies were less likely than autocracies to run one, and those so coded lasted 321 days on average against 115, a gap the coding of overlong emergencies as unlawful builds in part

THEORY-911 — Of 853 national states of emergency declared from 1949 to 2017, 115 were reported as unlawful in declaration, conduct or duration; 97 of those followed political turmoil and 14 natural disasters, democracies were less likely than autocracies to run one, and those so coded lasted 321 days on average against 115, a gap the coding of overlong emergencies as unlawful builds in part (Active)
- THEORY-1198 — Unlawful states of emergency, in declaration, conduct or duration, follow political turmoil far more often than natural disasters, are rarer in democracies than in autocracies, and last much longer than lawful ones (Proposed)

From In Gangneung, South Korea, in 2022, people the authors observed by chance in beach-front cafés outwardly ignored routine retaliatory missile launches and military sites on the beach, while a missile malfunction at night under a news embargo was followed by tweets voicing fear of war with North Korea and blaming the government and military

THEORY-898 — In Gangneung, South Korea, in 2022, people the authors observed by chance in beach-front cafés outwardly ignored routine retaliatory missile launches and military sites on the beach, while a missile malfunction at night under a news embargo was followed by tweets voicing fear of war with North Korea and blaming the government and military (Active)
- THEORY-1202 — A securitised public keeps its routines by not attending to routine security practices, which keeps an anxiety about the threat latent rather than absent; when the security apparatus itself disrupts those routines the anxiety surfaces, and with it blame of the apparatus (Proposed)

From Clark (2020), synthesising open writing by Russian military authors of 2015 to 2020, chiefly Chekinov and Bogdanov, Bartosh and Kiselev, with no stated rule for choosing sources, reports that they use “hybrid war” for a category of war marked off by its objective, reshaping a target state’s governance, geostrategic orientation and worldview, in which every action up to conventional military force is subordinate to an information campaign, and not, as he says Western usage does, for a set of means below the threshold of armed conflict

THEORY-488 — Clark (2020), synthesising open writing by Russian military authors of 2015 to 2020, chiefly Chekinov and Bogdanov, Bartosh and Kiselev, with no stated rule for choosing sources, reports that they use “hybrid war” for a category of war marked off by its objective, reshaping a target state’s governance, geostrategic orientation and worldview, in which every action up to conventional military force is subordinate to an information campaign, and not, as he says Western usage does, for a set of means below the threshold of armed conflict (Active)
- THEORY-1207 — Russian military writers of 2015–2020 use ‘hybrid war’ for a category of war marked off by its objective, reshaping a target state’s governance, geostrategic orientation and worldview, in which every action up to conventional military force is subordinate to an information campaign, and not, as in Western usage, for a set of means below the threshold of armed conflict (Proposed)

From In Szeidl and Szucs’s model of two outlets competing for readers and advertisers, where state-owned advertisers reward only the connected outlet’s slant, with weight δ, every equilibrium has the slant gap between the connected and the unconnected outlet equal to their owners’ ideal-point gap plus a favor-exchange term (βδ/NT)·Σ over government advertisers of masⱼWⱼ, up to measurement noise; readers’ preference for slant cancels from the gap, so equal observed gaps can come from different mixes of ideology and exchange

THEORY-860 — In Szeidl and Szucs’s model of two outlets competing for readers and advertisers, where state-owned advertisers reward only the connected outlet’s slant, with weight δ, every equilibrium has the slant gap between the connected and the unconnected outlet equal to their owners’ ideal-point gap plus a favor-exchange term (βδ/NT)·Σ over government advertisers of masⱼWⱼ, up to measurement noise; readers’ preference for slant cancels from the gap, so equal observed gaps can come from different mixes of ideology and exchange (Active)
- THEORY-852 — Under Szeidl and Szucs’s structural model and its identifying assumptions (an owner’s ideology fixed across episodes; the government’s online advertising budget not driven by the outlets’ slant), both owner ideology and favor exchange drove the Hungarian portal Origo’s reduced coverage of corruption in 2013–2016: favor exchange alone cut it by 29–44%, the new owner’s ideology alone by 63% after 2016, and the two together by 74%, so they are substitutes; and a ban on targeted government advertising would have raised it by 41–79% (Active)
  - THEORY-1217 — Both owner ideology and favor exchange drove the Hungarian portal Origo’s reduced coverage of corruption in 2013–2016: favor exchange alone cut it by 29–44%, the new owner’s ideology alone by 63% after 2016, and the two together by 74%, so they are substitutes; and a ban on targeted government advertising would have raised it by 41–79% (Proposed)

THEORY-329 — In 32,479 Weibo posts of 2011–2016 scored with an emotion lexicon, posts tagged fake by the platform’s fact-checkers carry more anger and less joy than true posts from verified accounts, and a post’s share of anger words goes with more retweets after controls; whether the gap is due to falsity, and whether anger causes the spread, is not shown (Active)
- THEORY-1221 — Anger makes fake news viral online: on Weibo fake news carries more anger and less joy than true news, and the anger in a post causes it to be shared more (Proposed)

THEORY-889 — In Acharya, Blackwell and Sen’s analysis, white Southerners surveyed in 2006 to 2011 who live in counties where a larger share of the 1860 population was enslaved are, with state fixed effects and 1860 county controls, somewhat less Democratic, less supportive of affirmative action and more racially resentful, by about 0.09 to 0.13 standard deviations per standard deviation (about 20 points) of slave share, and in 1984 to 1998 survey data colder toward black Americans, by about 0.32; a cotton-suitability instrument gives larger estimates; the direct effect estimated with present black share held fixed is about as large; and in county presidential vote, which counts black voters until disenfranchisement, the difference appeared after Reconstruction rather than before the war (Active)
- THEORY-1225 — Among white Southerners today, those living in counties where a larger share of the 1860 population was enslaved are somewhat less Democratic, less supportive of affirmative action and more racially resentful, by about a tenth of a standard deviation per standard deviation of slave share, and colder toward black Americans, by about a third; the difference is not accounted for by the present black share of the population, and in voting it appeared after Reconstruction rather than before the war, which supports its origin in the incentives that emancipation gave planters to repress black labor and votes, and its local transmission thereafter (Proposed)

From In five formerly stable democracies selected because they later autocratized (South Korea, India, Hungary, Venezuela and Turkey), formal emergency powers, extraordinary powers functionally equivalent to them, and emergency rhetoric each contributed, on the authors’ unsourced country accounts, to weakening democratic institutions

THEORY-906 — In five formerly stable democracies selected because they later autocratized (South Korea, India, Hungary, Venezuela and Turkey), formal emergency powers, extraordinary powers functionally equivalent to them, and emergency rhetoric each contributed, on the authors’ unsourced country accounts, to weakening democratic institutions (Active)
- THEORY-1229 — Formal emergency powers, extraordinary powers functionally equivalent to them and emergency rhetoric are linked to the accelerated erosion or dismantling of democracy and to the consolidation of autocratic rule, though the convergence is not inevitable (Proposed)

From Disinformation incidents labelled with DISARM techniques can be carried in standard STIX 2.1 without extending it, as an intrusion set that uses attack patterns, is attributed to a threat actor and targets locations; the DISINFOX mapping that moved 118 incidents into OpenCTI this way keeps only a name, a free-text description, a first-seen date and labels beside those three relationships, with no object or property for an incident’s narratives, channels, evidence or countermeasures or for the confidence of its attribution

THEORY-651 — Disinformation incidents labelled with DISARM techniques can be carried in standard STIX 2.1 without extending it, as an intrusion set that uses attack patterns, is attributed to a threat actor and targets locations; the DISINFOX mapping that moved 118 incidents into OpenCTI this way keeps only a name, a free-text description, a first-seen date and labels beside those three relationships, with no object or property for an incident’s narratives, channels, evidence or countermeasures or for the confidence of its attribution (Active)
- THEORY-1238 — Disinformation incidents carried in STIX 2.1 as the DISINFOX mapping carries them are an interoperable representation that survives exchange between organisations, and sharing them in this form improves analysts’ correlation and exposure of incidents and their reuse of labels (Proposed)

From On the court agenda {P, Q, R, R iff (P and Q)} under universal domain, proposition-wise majority voting is not robustly rational, while the premise-based procedure is, and it makes the group’s judgments a function of the members’ whole judgment sets that the members’ judgments on the conclusion neither fix nor need to enter

THEORY-725 — On the court agenda {P, Q, R, R iff (P and Q)} under universal domain, proposition-wise majority voting is not robustly rational, while the premise-based procedure is, and it makes the group’s judgments a function of the members’ whole judgment sets that the members’ judgments on the conclusion neither fix nor need to enter (Active)
- THEORY-127 — On the court agenda, and on any agenda in which every proposition reaches every other by conditional entailment and some minimally inconsistent triple stays consistent with two of its members negated, a group whose judgments are robustly rational under universal domain and respect unanimity cannot form its judgment on each proposition from its members’ judgments on that proposition alone unless one member dictates; its judgments must supervene on the members’ whole sets of judgments, and so can depart from what the members judge on the same proposition while being wholly fixed by them (Active)

From A platform that sits between the two sides of a market and can cheaply change how it allocates value tends to decay in three stages, first good to its users, then abusing them to favour its business customers, then abusing both to claw the value back for itself, each stage made possible by the lock-in the one before it built; how far it can go is set by the cost of leaving, so lowering that cost is the remedy

THEORY-150 — A platform that sits between the two sides of a market and can cheaply change how it allocates value tends to decay in three stages, first good to its users, then abusing them to favour its business customers, then abusing both to claw the value back for itself, each stage made possible by the lock-in the one before it built; how far it can go is set by the cost of leaving, so lowering that cost is the remedy (Proposed)
- THEORY-470 — Platform decay of the enshittification kind is caused not by a change in the character of those who run platforms but by the weakening of four constraints that used to discipline them, competition, regulation, interoperability and the bargaining power of tech workers, so it is reversible if those constraints are durably restored (Proposed)

From Among 100 Chapel Hill voters not yet committed to a candidate in September–October 1968, the aggregate emphasis they gave the campaign’s main issues correlated +.967 with the emphasis of the major news items, and +.979 with that of the minor items, in the nine media serving them; and voters leaning to a candidate agreed more with all the news than with news about their own side in 18 of 24 comparisons

THEORY-842 — Among 100 Chapel Hill voters not yet committed to a candidate in September–October 1968, the aggregate emphasis they gave the campaign’s main issues correlated +.967 with the emphasis of the major news items, and +.979 with that of the minor items, in the nine media serving them; and voters leaning to a candidate agreed more with all the news than with news about their own side in 18 of 24 comparisons (Active)
- THEORY-838 — The emphasis the news media give to issues raises the importance their audiences assign those issues, independently of any change in the audiences’ attitudes toward the issues: the media influence what people think about more than what they think (Proposed)

From In one laboratory study of a nine-period game between an infrastructure operator, two elected policymakers and three voters (59 groups), letting the first elected policymaker fix the price for all periods moved prices toward the efficient level when policymakers’ payoffs were set equal to consumers’, and moved them further above it when their payoffs were set equal to the operator’s; under period-by-period regulation voters’ pressure lowered the operator-aligned policymakers’ prices without removing the gap

THEORY-856 — In one laboratory study of a nine-period game between an infrastructure operator, two elected policymakers and three voters (59 groups), letting the first elected policymaker fix the price for all periods moved prices toward the efficient level when policymakers’ payoffs were set equal to consumers’, and moved them further above it when their payoffs were set equal to the operator’s; under period-by-period regulation voters’ pressure lowered the operator-aligned policymakers’ prices without removing the gap (Active)
- THEORY-858 — Insulating a regulator from future political revision fixes in place the interests of whoever designs the insulated regime: it relieves the underinvestment that electoral pressure on prices causes when the designers serve consumers, and entrenches capture, removing the electoral pressure that would correct it, when they serve the regulated industry (Proposed)

thermodynamics

From For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient

THEORY-026 — For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol’s dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient (Active)
- THEORY-817 — Once an agent’s actions affect the percepts it receives, maximal prediction of those percepts is generally not what thermodynamic efficiency requires: in percept-action loops with feedback, maximally predictive agents generally extract less work than the most efficient agent (Proposed)

From In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²))

THEORY-779 — In Kunin et al.’s underdamped Langevin model of SGD with momentum on linear regression, with gradient-noise covariance σ²H independent of the parameters, the stationary state is Gaussian with parameter covariance ησ²H(H + λI)⁻¹/(2S(1 − β)) and velocity covariance σ²H/(S(1 − β²)): for small weight decay it spreads equally along every nonzero Hessian direction and not along the null space, each Hessian eigenmode is a damped oscillator with damping ratio (1 − β)/√(2η(1 + β)(ρ + λ)), and the expected squared step is η²σ² tr(H)/(S(1 − β²)) (Active)
- THEORY-209 — A converged network trained with SGD is not at a point and does not sample its training loss: under a quadratic approximation it samples a Gibbs distribution on a modified loss, isotropic in the Hessian’s column space, with probability currents that keep it oscillating, so its speed and spread are set by the learning rate, momentum and batch size (Proposed)

From In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise

THEORY-211 — In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise (Active)
- THEORY-790 — In deep networks trained by SGD on real data, minibatch noise drives parameter groups related by a rescaling symmetry toward the point where their gradient noise balances, at a rate set by learning rate over batch size, so that at stationarity they sit neither where gradient flow would leave them nor spread out as under isotropic noise, as Ziyin, Li and Ueda propose (Proposed)

From For a system driven by a protocol fixed in advance, with microscopic reversibility and with forward and reverse processes started uniformly on their macrostates, the log-odds of two outcomes II and III reached from I equal the difference in their log state counts, plus the log-ratio of their reversal probabilities, plus the difference in minus the log mean of e^(−βQ) over the paths reaching each, where Q is the heat released; so when the first two terms are equal the likelier outcome is the one whose paths release more heat on average with less variation

THEORY-831 — For a system driven by a protocol fixed in advance, with microscopic reversibility and with forward and reverse processes started uniformly on their macrostates, the log-odds of two outcomes II and III reached from I equal the difference in their log state counts, plus the log-ratio of their reversal probabilities, plus the difference in minus the log mean of e^(−βQ) over the paths reaching each, where Q is the heat released; so when the first two terms are equal the likelier outcome is the one whose paths release more heat on average with less variation (Active)
- THEORY-412 — Driven matter tends toward states that look adapted to the drive, with no replication needed: of two outcomes of a system driven by a fixed external field, equal in internal entropy and in the probability of reversal, the likelier is the one whose formation absorbed work from the drive and dissipated it more reliably (Proposed)

From For classical systems with Markovian, microscopically reversible dynamics, the entropy production of a trajectory obeys an exact finite-time symmetry against the time-reversed process, P_F(+ω)/P_R(−ω) = e^ω, wherever ω is odd under reversal; Jarzynski’s work equality and the mean second law follow as corollaries, and the symmetry constrains fluctuations, not which steady state a system selects

THEORY-431 — For classical systems with Markovian, microscopically reversible dynamics, the entropy production of a trajectory obeys an exact finite-time symmetry against the time-reversed process, P_F(+ω)/P_R(−ω) = e^ω, wherever ω is odd under reversal; Jarzynski’s work equality and the mean second law follow as corollaries, and the symmetry constrains fluctuations, not which steady state a system selects (Active)
- THEORY-482 — For a closed quantum system that starts thermal and is measured projectively in energy before and after a unitary drive, the two-point-measurement work obeys the Jarzynski equality and the Tasaki–Crooks relation in exactly the classical form, because those energy measurements erase the coherences in which a quantum difference could appear (Active)

translation

From In Goldwasser, Gruber, Kalai and Paradise’s common-nonsense model, where the source language and the plausibility prior are uniform over sets from which shared random nonsense is removed and translators are one-to-one, maximum likelihood with the prior alone translates without parallel data with error O(log|Θ| / (α·min(m, |T|))) with high probability, falling in the shared nonsense α, in the size of the source language and in the number of samples

THEORY-270 — In Goldwasser, Gruber, Kalai and Paradise’s common-nonsense model, where the source language and the plausibility prior are uniform over sets from which shared random nonsense is removed and translators are one-to-one, maximum likelihood with the prior alone translates without parallel data with error O(log|Θ| / (α·min(m, |T|))) with high probability, falling in the shared nonsense α, in the size of the source language and in the number of samples (Active)
- THEORY-806 — Translation without parallel data, as of an unknown animal communication system into a human language, is learnable from a plausibility prior over the target language alone, with error falling as the common ground between the languages grows and as the source language grows richer (Proposed)

nucleation

Explorer

Lines of explanation

Graph View

agency

From A condition on free agency stated only in present psychological structure is met by a manipulated agent, so the account must count that agent free or add a historical condition, and no historical condition in the record’s readings is shown to escape

From A regulation is autonomous to the degree it is integrated with one’s other values and needs: autonomy is graded, and is not set by the order of the attitude that endorses it

From Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one

From Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency

From Agency in the sense that bears on responsibility needs a standpoint that settles and carries forward revised evaluative standards when its inherited ones break down, and planning, autonomy, memory and coordination do not supply one

From Moral responsibility requires the normative competence to recognise and act on the True and the Good, which identification with one’s own motives does not supply

From An agent can be morally responsible for what she did although she could not have done otherwise, when what made the action unavoidable played no part in bringing it about, so the principle of alternate possibilities is false

anthology-candidate

From In a linear-in-parameters model with Gaussian noise and a Gaussian prior, the evidence is exactly the best-fit likelihood times an Occam factor set by the posterior curvature, and when the design grows linearly with n that factor costs (k/2) log n

From Linear connectivity from a shared, already-stable start holds only when the objectives the runs optimise share structure, and breaks as that structure is removed

From Two ε-dropout-stable solutions of a multilayer ReLU net, under a loss convex in the output, are joined by a piecewise-linear path whose loss never exceeds the worse endpoint’s by more than ε, and joined pairwise paths compose with no higher barrier than their worst

behavioral-integration

From Self-control over time needs no faculty of will: interests that dominate at different delays bargain, and resolve is the recursive staking of expected future choices on the present one

From Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning

From How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control

causality

From With uniform interventions imposed separately at each level, a coarse-grained macro description of a known discrete mechanism can have more effective information than the micro description it supervenes on, so the level at which effective information is highest can be a macro level

cognition

From How much cognitive control to exert, and on what, is decided by a cost-benefit computation (the expected value of control) in a specification system distinct from the structures that implement the control

complex-systems

From Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning

From In Friedkin’s simulations of reflected appraisal with Friedkin–Johnsen anchoring on initial positions, 99.0–100% of 100,000 random strongly connected networks of 5 to 25 members, complete or with random zero weights, evolve to a single leader and the rest to two

From For a linear system du/dt = A(t)u, non-normality is necessary for time dependence to destabilize an operator that is uniformly stable at every instant, and a non-normal operator neutral at every instant can be asymptotically unstable under switching

compositionality

From The Russian wondertale is built from a small fixed set of character functions in a fixed order, so that all wondertales share one composition and differ in who performs the functions and how

consciousness

From Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled

From Expert texts of Shamatha, Transcendental and Stillness Meditation describe their so-called contentless goal-states as lacking thoughts, perceptions, images and memories but keeping many forms of content, and as differing across the practices in attentional stability and vividness

contextuality

From An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality

From Classical Boolean logic is restored when a presheaf of contextual data acquires a global section or is sheafified

databases

ecology

embodied-cognition

From Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning

From Minimal agency requires individuality, interactional asymmetry and self-generated normativity, and individuality without the other two is not agency

From Brushing a seen rubber hand in synchrony with the hidden real hand makes touch be felt at the rubber hand, makes it be reported as one’s own, and shifts the felt position of the real hand toward it; a small asynchrony removes these effects

emotion-and-affect

From Felt affect is the experience of the body’s regulatory state, valenced by what that state means for staying alive and present throughout waking experience, not only in emotion; whether what is felt is a prediction of that state or a map of it is unsettled

endosymbiosis

epistemology

From When a group learns only about the options its members pursue and good evidence is scarce, sparser communication makes it more likely to settle on the better option, at a large cost in speed; when evidence is plentiful, sparsity only slows learning