Topics — What a work is about. Nucleation holds whatever reading the owner files here, whether or not an Anthology of the SOTA topic could also hold it; instruction for ML practice belongs to the anthology (ADR-048). The words are seeded from the clusters the reading-time triage of 2026-09-25 found, and grow by decision.
Complex systems — emergence, information decomposition, individuality, self-organization — how collective behaviour arises from parts.
150 of 1352 THEORY documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| THEORY-036 v2 | Dennett’s real-pattern criterion, compressibility against the bit map, admits almost every non-random pattern; the projectibility, perspective and scale-relativity that rainforest realism needs are later additions that do the ontological work | Dennett (1991), LIT-220 — a pattern is real “if there is a description of the data that is more efficient than the bit map, whether or not anyone can concoct it”. Ladyman (LIT-219) replaces this with indispensability to projectible generalisations, perspective- and scale-relative and lossy. Separating the two, and the claim that the added clauses carry the exclusions, is the record’s reading. The definition that would confirm it (ETMG) is unread. | Proposed |
| THEORY-058 v2 | Mental disorders are maintained by self-reinforcing causal loops rather than expressing a latent common cause, which is how they can outlast their triggers | Three traditions that do not cite one another for it put self-sustaining loops where a latent cause would be. Borsboom (LIT-547) makes disorder an alternative stable state of a strongly connected symptom network, kept up by hysteresis after its trigger has gone. Salkovskis, Sighvatsson and Sigurdsson (LIT-538) make it a threat meaning protected by the behaviour it drives. Kendler (LIT-539) finds causal loops across levels, from genes to culture. The convergence is real; the evidence is indirect, simulated or clinical, and none of it is an individual time series showing hysteresis, so the account is Proposed. It says nothing about what starts a disorder. | Proposed |
| THEORY-069 v4 | 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 | Kelso’s 2021 retrospective (LIT-601) states the Haken–Kelso–Bunz model: relative phase φ between the hands obeys dφ/dt = −a sin φ − 2b sin 2φ, so in-phase and anti-phase are both stable for b/a > 1/4 and anti-phase loses stability below it as movement speeds up. With noise the switch is a nonequilibrium phase transition, and the reported data show enhanced fluctuations and critical slowing before it. Many effectors act as one through a collective variable, with no controller choosing the pattern. Proposed, because the relaxation-time and hysteresis data are not read first-hand. | Proposed |
| THEORY-072 v4 | Organisms differ from dissipative structures such as a candle flame not by far-from-equilibrium self-maintenance, nor by circular self-maintenance alone, but by organisational differentiation: several distinct constraints that produce one another at different time scales | The record’s first answer to the demarcation question that NOTE-094 reports. Prigogine (LIT-516) supplies the dissipative structures, and nothing in his account separates them from organisms. Mossio, Saborido & Moreno (2009, LIT-570) grant a candle flame organisational closure. Its one macroscopic pattern keeps up the conditions of the combustion that makes it. They draw the line at organisational differentiation: in an organism several distinct structures make distinct contributions, where a flame makes “a unique contribution”. Montévil & Mossio (2015, LIT-582) narrow closure to mutual dependence among constraints. Each constraint must be both produced by and productive of another, with at least one slow and one fast dependence. On the record’s reading, that builds differentiation into closure, as Mossio and Bich (2014) already say of closure of constraints. No paper analyses a flame under the 2015 definitions. Neither paper says closure defines an organism. | Proposed |
| THEORY-073 v2 | 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 | Bruineberg et al. (LIT-603) and Raja et al. (LIT-598) argue, by different routes, that the Markov blanket of the free-energy literature is placed and not found. The formal core is THEORY-066: the partition takes the internal set as input. This account adds what the critiques show about practice. In Friston’s soup (LIT-526) the graph was built from one kind of coupling and steady state was assumed when the run stopped. In published applications the blanket sits wherever the model needs it: in the receptors of a brain, the membrane of a bacillus, the beam joining two pendulums. The formalism also gives the same description to a membrane the cell makes and a beam nobody’s states make. So the blanket presupposes the boundary it is offered as the criterion of. Active since 2026-10-27, narrowed to the 2013 formulation and the applications held. The later steady-state definitions have not been read, and the unrestricted claim is THEORY-1097, Proposed. | Active |
| THEORY-088 v2 | Slow large-scale activity in human cortex is organised as in a pattern-forming medium: its flow is set by interacting spiral defects centred on phase singularities, which sit on boundaries between functional regions, annihilate with spirals of opposite rotation and repel those of the same | Xu, Long, Feng & Gong (2023), LIT-638: in Human Connectome Project fMRI band-passed to 0.01–0.1 Hz, the cortical phase field holds about 16–19 spirals at a time around phase singularities, concentrated on boundaries between functional regions; their centres drift superdiffusively, opposite spirals annihilate (51% fully, 46% partly) and like ones repel (2.6%), and their positions and senses decode task condition above a raw-amplitude decoder. One source, fMRI only, at a spatial scale chosen because it maximised that decoding; the routing of activity flow is read from phase gradients and a phase-oscillator model, never perturbed. The kinship with Belousov–Zhabotinsky spirals and pattern-formation defects is the record’s analogy, not the paper’s. | Proposed |
| THEORY-099 | Tryptophan networks in microtubule architectures support collective superradiant excitonic states, and their thermal quantum yield rises with network size and survives static disorder of room-temperature size | Babcock et al. (2024), LIT-645: a radiative Hamiltonian of up to more than 10⁵ tryptophan transition dipoles, on tubulin geometry, predicts superradiant states up to about 4000 times the single-tryptophan rate in a centriole, and a thermal quantum yield that rises with network size and is almost unaffected by static disorder of about 200 cm⁻¹, though that disorder cuts the brightest state’s enhancement from about 3600 to about 20. Steady-state spectroscopy finds a tryptophan quantum yield of 17.6 ± 2.1% in microtubules against 10.6 ± 0.6% in tubulin dimers. No lifetimes were measured. This is not evidence for Orch OR or for any claim about consciousness. | Proposed |
| THEORY-128 | After mass bleaching, coral recovery on a reef crest depends on large herbivorous fishes keeping macroalgae down, because recovery runs through recruitment and survival that macroalgae suppress, not through the persistence of surviving colonies alone | Hughes et al. (2007), LIT-711. On the reef crest at Orpheus Island after the 1998 bleaching, four roofless cages kept large and medium fishes out of 25 m² plots for 30 months. Macroalgae averaged 56% cover inside against 4.1% and 1.7% in controls. Coral cover reached 7.7% inside against 19.2% and 20.2%, with recruitment about two-thirds lower and mortality of established colonies more than double. Control recovery came mainly from recruits of taxa bleaching had removed. Proposed: one site, four replicates, plot scale, and fish exclusion standing in for herbivore exclusion. The algal state vanished within 30 days of reopening. | Proposed |
| THEORY-129 | In Acropora millepora the natural night-light regime is necessary for spawning on the expected night, and displacing it delays or abolishes both release and a spawning-night programme of G-protein-coupled signalling genes | Kaniewska et al. (2015), LIT-710, at Heron Island. In 2011, colonies under natural night light spawned with the reef, while colonies lit after sunset or kept dark for the preceding week did not spawn that night (N = 6, 5, 5). 184 transcripts changed around release in ambient colonies, early in lit ones and not at all in dark ones, and the induced genes were enriched for G-protein-coupled signalling. In 2006, blue, green or white nocturnal light delayed spawning 6–8 h or two nights and red did not. Proposed: one species and site, small groups, and a cascade inferred from co-expression. It shows synchrony by a shared cue, not coupling between colonies. | Proposed |
| THEORY-130 | Coral bleaching is the breakdown of a regulated host–symbiont exchange, reached by more than one route, not a single mechanism of symbiont loss | Two reviews, with no data of their own. Boilard et al. (2020), LIT-712, collect the regulators of the coral–dinoflagellate exchange: the host controls symbiont density, bacteria shift the nitrogen supply, and heat leads either to host rejection of symbionts (apoptosis, autophagy, exocytosis, detachment, necrosis) or to pigment loss under oxidative stress. Goulet et al. (2020), LIT-709, report two octocorals whose similar symbiont losses at +3 °C came by different routes, one by fewer symbionts per host cell and one by loss of host cells. Proposed. The claim that the wider microbiome drives bleaching is declined: its own source says it cannot yet be tested. | Proposed |
| THEORY-132 | Stability, a system’s speed of return to equilibrium after disturbance, and resilience, its capacity to absorb change and persist within its domain of attraction, are distinct properties that can trade off, so a system can be highly unstable and highly resilient, and managing it for constancy can erode its resilience | Holling (1973), LIT-741. Stability is return to equilibrium. Resilience is persistence under change, set by the extent of the domain of attraction and the system’s distance from its edge. Budworm forests and insect populations in harsh climates are unstable and resilient; homogeneous lakes are stable and fragile. Active for the distinction, which has become standard. The claim that management for constancy erodes resilience rests on suggestive cases. | Active |
| THEORY-134 v2 | Whether a system can persist is a question about a constraint set fixed independently of its dynamics: a set is viable when from each of its states some evolution can stay in it, and when it is not, its viability kernel is the largest part that is, from the rest of which every evolution leaves in finite time | Aubin’s viability theory, read in his 1999 mini-course notes LIT-714. Given dynamics and a set K of required states, K is viable if some evolution from each state stays in K and invariant if every one does. Nagumo’s theorem reduces viability to a tangency condition. The viability kernel is the largest viable subset of K; from outside it every evolution exits K. The notes do not compare viability with stability; that contrast is a further step. | Active |
| THEORY-136 v3 | A persistent or abrupt change is not evidence that a system has alternative stable states: field patterns such as jumps, multimodality and dual relationships also arise from a stepwise driver or a simple threshold, and only hysteresis, initial-state dependence or a lasting shift after a temporary disturbance, with slow return excluded, comes close to showing one | Scheffer & Carpenter (2003), LIT-726. Alternative attractors predict sudden shifts, multimodal states and dual responses to a driver, and so do discontinuous drivers and threshold responses. Discriminating evidence is experimental: different starting states reaching different end states, temporary disturbances causing lasting shifts, or hysteresis. Slow recovery can mimic all three. Positive feedback must be strong to produce alternative states. Active as a methodological standard. | Active |
| THEORY-144 | Under steady cost and effort pressure, an organization’s activity migrates toward the boundary of acceptable performance, and redundant defences erode unobserved because a breach of one shows no effect, so that major accidents arise from many locally reasonable decisions rather than from coincident errors | Rasmussen (1997), LIT-743. Within a work space bounded by functional, administrative and safety constraints, a cost gradient from management and an effort gradient from the work drive activity toward the boundary. Defence in depth hides the drift, and accidents such as Zeebrügge are prepared by decision makers each optimising locally. Proposed: the account is built from accident reconstructions, which see only the cases that crossed. | Proposed |
| THEORY-153 | A system can keep its identity independently within a shared environment only if it has five interacting subsystems, the first made of elements that are themselves viable systems, so that the structure recurs at every level of recursion; when one is missing or fails, the system degrades in characteristic ways, and without the subsystem for foresight the subsystems for operational control and for identity collapse into each other, leaving a system that can only react | Beer’s Viable System Model, as he states it in his 1984 retrospective LIT-752. Five subsystems are necessary and sufficient for any organism or organization to maintain its identity independently; every viable system contains and is contained in one; breakdowns trace to a failing subsystem. Proposed: the paper declines to validate the model, the evidence is the author’s consultancy, and no observation the model forbids is named. | Proposed |
| THEORY-178 v3 | 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 | Zhao, Wytock, Reynolds and Motter (2024), LIT-854. In an ensemble of sign-consistent Boolean rules on the 87-gene core of the E. coli regulatory network, 51 genes admit a transient knockout or overexpression that moves the network to a different attractor, and every such gene reaches a strongly connected component with a positive circuit. So one genotype may hold several heritable expression states without epigenetic marks, in the model. The evidence is a model ensemble, read in full in NOTE-655, and the title claims no more. No transient perturbation has been shown to have a lasting effect in cells; that experiment is QUESTION-172. | Active |
| THEORY-179 v2 | 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 | Bhattacharyya et al. (2023), LIT-853. Single-cell swarm assays show sibling homogeneity in swarming at four generations and its loss by seven, a correlation (Spearman r = 0.71) between a Fur-repressed reporter and swarming, and longer persistence when iron is clamped. The iron link is read through a promoter reporter and is correlational, the inheritance rests on few founders and three time points, and the paper’s “memory”, “conditioning” and “decision-making” are functional labels, not evidence of learning. | Active |
| THEORY-188 v2 | In a random graph whose nodes lie in a hyperbolic disk and connect by distance, a power-law degree distribution and strong clustering follow from the geometry: the degree exponent is set by node density relative to curvature, clustering by a temperature, and removing the angular, metric part of the distance leaves the configuration model | Krioukov, Papadopoulos, Kitsak, Vahdat and Boguñá (2010), LIT-865: derived, in a large-distance approximation, for a model in the hyperbolic plane, with simulations. γ = 2α/ζ + 1; clustering falls from its maximum at T = 0 to zero at a transition at T = 1; a circle model with power-law hidden degrees is the same ensemble once degree is radial depth; dropping the angular term gives the configuration model. It does not show that any real network is hyperbolic, or that hierarchy causes heterogeneity. | Proposed |
| THEORY-190 v3 | 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 | García-Pérez, Boguñá and Serrano (2017), LIT-869: derived to first order, checked on one synthetic network over eight layers and on six real ones. Supernode hidden degree is the ℓ^β sum of its members’; β is invariant. The record proves the closure at large separation, with relative error of order β times the block width over the separation, and computes that between neighbouring supernodes the union of member links exceeds the S¹ form (by 32–86% in a worked example with two-node blocks). The source’s flow ⟨k⟩′ = r^ν⟨k⟩, with ν = 2/(γ−1) − 1 or 2/β − 1, rests on the closure at every separation and is QUESTION-203. It does not show that real networks are generated by a hidden metric space, or that the nested layers are a hierarchy found in the data. | Active |
| THEORY-193 v2 | A simplicial complex grown by gluing each new d-simplex to one (d − 1)-face, by a rule that uses no geometry, has a 1-skeleton whose geodesic triangles are 2-thin at every size, so hyperbolic network geometry in Gromov’s sense is an outcome of the combinatorial growth; the equal-link drawing as ideal simplices is a convention, a tiling of the hyperbolic plane for d = 2 in the manifold flavor but not a tiling of hyperbolic d-space for d ≥ 3 | Bianconi and Rahmede (2017), LIT-867, read in NOTE-678, for the model, the degree law and the ideal-simplex drawing. The coarse hyperbolicity of the graph is the record’s: the source asserts the hyperbolic geometry through its drawing and measures nothing on the graph. Every such complex has a 1-skeleton with 2-thin geodesic triangles, whatever the flavor, so its hyperbolicity is a property of the network. The constant-curvature picture, ideal simplices in the Poincaré ball, is a choice of drawing. It tiles H² for d = 2 in the manifold flavor, and for d ≥ 3 it is not a tiling. This says nothing about whether real networks grow this way. | Active |
| THEORY-214 v2 | 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 | Falmagne, Stephenson & Levin (2026), LIT-1025, read in NOTE-731. On about 10,900 compositions of 2022, a predictor of time-to-overthrow built from 19 composition-level series with a 7-hour memory catches half the overthrows within 20 minutes at 3.6% false positives, beats variance and autocorrelation trends, and transfers to 2023. Its precursors are many: attack outrunning defence, a turbulent history, little accepted change, uncoordinated activity, simple images, low past defence; slowing down and speeding up both occur. About a quarter of overthrows are unpredictable at the 20-minute range. The social readings of the precursors are the authors’ conjecture from SHAP curves; nothing here measures plans. | Active |
| THEORY-216 v3 | 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 | Wu & Silva (2024; v3 2026), LIT-1032, read in NOTE-724. On the 2017 r/place log, coalitions are recovered, failed ones included, from co-placement and image segmentation. Under 7% of updates survive to the end. Recovered coalition sizes shift toward large over the event. A decision tree on four state features predicts survival at F1 0.09 in the first 48 hours and 0.67 later. Illustrated successful coalitions keep working under overwriting throughout, and some failures look like successes until a late collapse. The recovery is validated only against final artworks, the lifecycles are ten examples, and nothing here observes plans. | Active |
| THEORY-217 | A trait malfunctions when its operation does not correlate positively with the normative field, the minimum change its organization’s viability requires at each precarious state; malfunction needs no second order of regulatory norms, and regulatory traits can malfunction by the same standard | Barandiaran (2025), LIT-1026, read in NOTE-730. Against the dual-order account (Saborido, Moreno and Mossio), on which malfunction needs a regulated switch between regimes that a first-order trait fails to follow. The viability space has viable, precarious and terminal regions, and the normative field is the minimum constant parametric change at each precarious state that avoids disintegration. A trait functions when it correlates positively with the field, subfunctions when it does so too slowly, malfunctions when it does not, and is nonfunctional when it does nothing where it must. The standard is the organization’s own viability, not a designer’s goal, a selection history or a statistical norm. Proposed: a conceptual proposal with a one-variable model, and the field has not been computed for any real organism. | Proposed |
| THEORY-218 v2 | 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 | Litherland & Mørch (2021), LIT-1037, read in NOTE-733. For the 2017 Mona Lisa, 76,774 accounts placed tiles in the region while a forum of 161 negotiated the design through linked blueprints. Central members admitted a Swiss flag and a rainbow frame and erased r/trees objects, and a tag on the image recruited newcomers. Whether an addition was vandalism or part of the image was settled by negotiation. The authors argue placement data cannot reveal cooperation, conflict or affiliation. One artwork; the link from forum power to placements is “unclear” by the authors’ account; their “instruction” is Reddit’s rules, not planners’ orders. | Active |
| THEORY-222 v5 | 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 | Hoel, Albantakis & Tononi (2013), LIT-1038, read in NOTE-726. Effective information is effectiveness (determinism minus degeneracy) times the log of the state-space size, measured under uniform interventions. Coarse-graining always loses size and can gain effectiveness; when the gain wins, causal emergence CE = EI(macro) − EI(micro) is positive. Four toy systems give 0.40 to 2.92 bits. The arithmetic is exact. The reading that the EI peak marks where causation happens rests on imposing a uniform distribution at each level, which the paper adopts and does not test; the title does not claim it. Promoted on 2026-10-20 on a construction. Whether the gain survives one intervention distribution at both levels is QUESTION-204. | Active |
| THEORY-223 | When the rules that govern a system lie within its own action space, the governor’s authority rests on the system leaving them alone, and further controls in the same space inherit that dependence | Lahiri (2026), LIT-1039, read in NOTE-732. In agent frameworks that keep configuration as files the agent can read and write, the coordination architecture sits inside the technical subsystem’s action space, and the social subsystem’s authority “depends on the agent’s continued acceptance of-or inability to circumvent-the constraints”. Permission lists and sandboxes kept in the same space inherit the exposure; only constraints outside it escape. Argued from one platform’s architecture. The paper’s vignettes show agents adding channels and artifacts, not altering a control, so the recursion itself is argued and not observed. | Proposed |
| THEORY-224 v2 | Objects on a crowd-built canvas are of two kinds, those that spread by a local recolouring rule and are predictable from the canvas itself, and those copied from designs coordinated off the canvas, which local canvas dynamics do not predict | Adams, Fernandez & Witkowski (2022), LIT-1034, read in NOTE-725. A CNN predicting each pixel of the 2017 r/place canvas from its 3×3 neighbourhood recovers objects that grow by recolouring neighbours (Blue Corner, German flags) and misses designs whose plan lived off the canvas (EU dove, US flag) at a six-hour horizon. Short-horizon predictability dips mid-event. The evidence is one illustrated output and SSIM curves with no persistence baseline. The paper models no participants and does not measure anyone’s reach. | Proposed |
| THEORY-232 | A whole can be compared in agency with its parts, relative to a chosen decomposition and the parts’ goals: it is more agential when its organization lets the parts resolve more of their mutual constraints than they do at the equilibrium of their former organization, which shows as delayed gratification | Watson (2023), LIT-990, read in full in NOTE-747. A perspective essay’s definition, argued and illustrated, not proved: agency of a whole with respect to its parts is the reduction in the parts’ total frustration at equilibrium that its organization achieves over the old one, shown by directed uphill excursions to lower minima. It is comparative and sets no threshold for an additional agent, and it holds only relative to a stipulated decomposition and stipulated goals for the parts. That bottom-up learning produces such organization is reported from earlier models, not shown here. | Proposed |
| THEORY-257 v3 | 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 | Barnett and Seth (2023), LIT-979, read in NOTE-791: definition (2.6), transitivity (App. A), the causal-graph form (2.12), the state-space closed form (3.18) and the genericity count (3.25)–(3.26). The target is the coarse variable’s own future, so no outside query is chosen. But the system/environment boundary, the scale and the class of maps are inputs, and the worked theory’s candidates are linear subspaces, not partitions. The cost is multimodal, with every local optimum kept. The criterion is closure, not locality or informativeness about a far environment. | Proposed |
| THEORY-262 | Neural complexity is low when a system’s parts are independent or uniformly dependent and high when small subsets are nearly independent while large ones are integrated; in simulated cortex, patchy grouped connectivity raises it above random connectivity of equal density | Tononi, Sporns & Edelman (1994), LIT-998, read in NOTE-767. C_N, the summed shortfall of average subset integration below a linear rise with subset size, is shown low at both extremes for one family of Gaussian matrices, and higher for patchy than for uniform wiring in one simulated visual area and in small linear systems. All of it is simulation. It is a whole-system number over fixed components: it does not find units or boundaries. | Proposed |
| THEORY-265 v2 | In Tissot et al.’s minimal model of bistable deciders in a modular coordination game (thirty modules of ten, couplings 1 within modules and 0.01 across), phase synchronisation within modules, driven only by each unit’s local payoff, makes each module respond to the others as a single unit, and synchrony matched to the modules is needed: global synchrony does not | Tissot, Levin, Buckley & Watson (2024), LIT-989, read in full (v1 and v2) in NOTE-773. In a simulated game of 30 modules of 10 units, each unit shifts the phase of its decision cycle toward partners it currently agrees with. Phases synchronise within modules, each module then switches as a block, and 50 of 50 runs with the learned phases reach a global optimum against 0 of 50 without; imposed global synchrony does not help. It is one clean game, one parameter setting, and a postulated phase rule; the modules are built in, and timing finds them rather than drawing them. Re-implemented by the sixteenth status review: learned phases, and module-matched phases imposed by hand, reach the optimum; random, shuffled and global phases do not make modules respond as units. The effect matters only where modules are isolated enough to lock in. | Active |
| THEORY-268 | An emergence measure averaged over a system’s transitions can be positive while its local value is negative at particular transitions, so emergence so measured belongs to the ensemble, not to each moment | Varley (2022), LIT-953, read in full in NOTE-756. Both the coarse-graining and the ΦID accounts of emergence score an expected past–future mutual information, whose pointwise values can be negative. In one constructed toy, about 53% of informative micro transitions map to misinformative macro ones, and the synergy atom is negative for 31.25% of transitions, while both averages are positive. The paper argues the possibility is general and shows it in instances; it does not show that every system flickers, or say what a negative moment means. | Proposed |
| THEORY-272 v3 | 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 | Hopfield (1982), LIT-918, read in full in NOTE-819. The energy descent is derived for symmetric couplings; everything quantitative, including the 0.15N figure, the basin width and the merging distance, is simulation at N = 30 and 100. It does not say that spurious states come only from overload: they appear at a low load too. Nor does it say anything about addressed memory. | Active |
| THEORY-276 v2 | In a network of identical units with identical couplings, the orbits of the graph’s symmetry group are synchronization clusters, and a cluster moved only by its own factor of the group can lose synchrony while the others keep it | Pecora et al. (2014), LIT-960, read in NOTE-775: eq. (1), the IRR block-diagonalization (eqs. 2–4, 7), the derivation of isolated desynchronization (eq. 6) and an 11-node electro-optic experiment. The pattern of clusters that can synchronize is read from the wiring; whether each does is set by the dynamics through its own transverse block. Symmetry is sufficient, not necessary, for clusters. The paper says nothing about external drive, so the use of these clusters as a sign of coupling rather than a shared cue is the record’s inference. | Active |
| THEORY-285 v2 | 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 | Linson, Anderson, Markham, Friston & Moutoussis (2025), LIT-983, read in version 2 in NOTE-766. A mechanism shown in illustrative simulation: policy selection by expected information gain, plus a confident prior, withholds the only disambiguating observations, so the prior keeps itself up. Exposure that restores sampling rebalances it; a precision-lowering drug only suppresses symptoms. No data are fitted, parameters are unreported, and the two intervention arms differ in environment as well as mechanism. | Active |
| THEORY-304 v3 | In plane Couette and Poiseuille flow, measured in the energy norm, the linearized operator is so far from normal that perturbations can grow transiently by O(R) and resonate by O(R²), and operator perturbations of size O(R⁻²) make it eigenvalue-unstable, at Reynolds numbers where eigenvalue analysis predicts stability | Trefethen, Trefethen, Reddy & Driscoll (1993), LIT-932, read in NOTE-839: computed scalings for the linearized shear flows, mostly credited to earlier work and unified through pseudospectra. That transition to turbulence happens by this linear growth plus nonlinear mixing is the paper’s conjecture from a 2 × 2 model, and is not part of this account. | Active |
| THEORY-307 | Adaptation by trial and error in a large system finishes in a practicable time only if partial adaptations are retained, and retention requires that the system not be fully joined: parts must be separated, at least temporarily, by constancies, and where later learning shares parts with earlier learning it destroys it | Ashby (1952; 2nd ed. 1960), LIT-787, read in NOTE-851, Chs. 10–16. If all successes must occur at once, trials grow as (1/p)ᴺ; if each is kept once found, roughly as N/p. Keeping one needs no channel from what is still changing to what is already right, so a fully joined system, the Homeostat included, cannot accumulate; constancies cut a system into independent parts. Exact for the type-cases; the fit to brains is qualitative. | Proposed |
| THEORY-323 v2 | 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 | Farrell & Ioannou (1996), LIT-931, read in NOTE-831, with Trefethen et al. (1993), LIT-932, read in NOTE-839. Linear algebra, worked in examples: growth over time t is the top singular value of the propagator, which the eigenvalues bound from below, not above. It holds in a chosen norm, and a norm can always be found that makes the operator normal. It says nothing about whether the growth survives nonlinearity. | Proposed |
| THEORY-324 v2 | A system adapts by trial and error to an environment whose effect on its essential variables it does not know only if a second feedback, through those variables, changes the parameters of its primary feedback; and if that feedback changes them whenever, and for as long as, an essential variable is outside its limits, every equilibrium the whole can reach is an adapted one | Ashby (1952; 2nd ed. 1960), LIT-787, read in NOTE-851. Necessity is deduced from the trial-and-error rule against a Black Box environment; sufficiency from the theorem that a whole is at equilibrium only where every part is. The Homeostat shows it in a four-unit linear machine. The parameter changes are blind, and the account says nothing about how long the search takes, which is the subject of THEORY-307. | Active |
| THEORY-325 v2 | Random variation of a linear operator destabilizes it generically, needing less variation the more non-normal and less damped the mean operator | Farrell & Ioannou (1996), Part II, LIT-929, read in NOTE-842. Necessity is a one-line bound and the destabilization is shown in explicit examples. Genericity under noise is cited to stochastic stability theory, and the role of non-normality and damping is shown numerically. In the deterministic examples, variation much faster than the transient growth destabilizes nothing. Since 2026-10-19 the deterministic half is THEORY-724, and this THEORY holds the stochastic half. | Proposed |
| THEORY-402 | Collective intelligence is resource-bounded computation with a larger but modular budget: a group has more sensing, state, processing and action than any member, and pays coordination and cooperation costs because those resources sit in autonomous members, so the same trade-off yields groups that do better than their members on some tasks, groups that run qualitatively different algorithms, and groups that do worse | Pilgrim et al. (2025), LIT-046, read in NOTE-055, with Dreyer et al. (2024), LIT-075 (NOTE-040), as its clearest instance. The paper offers a four-dimensional resource inventory, a catalogue of constraints and a mapping to mechanisms, and calls the known forms of collective intelligence “aspects of a single unifying principle”. NOTE-055 judges it a vocabulary, not a derivation: the resources have no metric, so the framework cannot say in advance when costs outweigh gains, and its “predictions” are research questions. Filed Proposed as the account the vocabulary states. | Proposed |
| THEORY-412 v2 | 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 | Perunov, Marsland & England (2014; Phys. Rev. X 2016), LIT-518, read in NOTE-416, on England’s macrostate relation (LIT-515). The log-odds of two driven outcomes decompose exactly into an internal-entropy term, a reversal term and Ψ − Φ = −ln⟨e^{−βΔQ}⟩. The “tendency towards adaptation” needs the first two held equal. The paper does that only in two single-particle hopping models, argues the many-body case, and reports a driven-chemistry simulation without showing it. The reading judged the claim too weak to source a THEORY; it is filed as Proposed for that reason. “Adapted” here means absorbing the drive’s work, nothing more. | Proposed |
| THEORY-424 v2 | Mutual-information measures of individuality that read a system’s next state off its present one measure association, not propagation: I(S_{t+1}; S_t) can be positive when the environment alone drives the system, and conditioning on the environment does not recover the interventional quantity, which can be zero where the conditional measure is one bit | Bourrat (2024), LIT-005, read in NOTE-027, against Krakauer et al.’s (2020) measures. A* = I(S_{t+1}; S_t) is confounded when the environment is a common cause (his Fig. 2), and his interventional  = I(S_{t+1}; Ŝ_t) is zero there. His claim that  equals Krakauer’s conditional A = I(S_{t+1}; S_t | E_t) is false in general: NOTE-027’s XOR system has A = 1 bit and  = 0.  itself depends on an intervention distribution the paper leaves open, and neither it nor the paper’s two ETI criteria is computed on any system. | Active |
| THEORY-459 | Exact duality between two models is incompatible with ontological emergence of one from the other, reductive or not, because a duality is an isomorphism and ontological emergence needs a non-injective linkage whose interpretation does not commute; the two combine only when emergence runs beside the duality, as along the radial direction in AdS/CFT, through an approximate duality, or as merely epistemic emergence | De Haro & Butterfield (2025), LIT-199, §14.3, read in NOTE-178. A formal argument from equinumerosity and surjectivity, strictly proven for finite sets, with an argument on physical grounds for the infinite case (fn. 57). Emergence here is novel reference in the domain of application, a semantic condition, not an information measure. AdS/CFT is the exception only to incompatibility with non-epistemic emergence, not to “not indicative”: on a theoretically equivalent reading, its “emergent gravity” “cannot be understood as ontological emergence”. | Proposed |
| THEORY-497 v3 | A macro feature can supervene on a system’s present state and still carry information about its future that no small set of parts carries, because supervenience is instantaneous and prediction runs across time; on the integrated-information-decomposition account a system can host such a feature of order k exactly when its dynamics carry k-th-order synergy about their own future, relative to a fixed partition into parts and a chosen redundancy function | Mediano et al. (2022), LIT-025, read in NOTE-010, a review of Rosas et al. (2020), which the record does not hold. A feature V_t is causally emergent of order k iff Un⁽ᵏ⁾(V_t; X_t′ | X_t) > 0, and a system has one iff Syn⁽ᵏ⁾(X_t; X_t′) > 0, split as downward causation plus causal decoupling. The review proves none of this and names no redundancy function. The quantities change with the partition, which the review calls intended, and they are predictive, not interventional, unless p(X′ | X) is a do()-distribution. | Proposed |
| THEORY-508 | The Belousov–Zhabotinsky oscillation is a relaxation cycle in which autocatalytic production of HBrO₂ oxidizes the catalyst until bromide, released as the oxidized catalyst is reduced by bromomalonic acid, switches the autocatalysis off; a closed batch oscillates only as a transient of up to several thousand cycles on its way to equilibrium, and only a continuous-flow reactor sustains it indefinitely | Zhabotinsky (2007), LIT-531, the Scholarpedia review by one of the reaction’s eponyms, read in full in NOTE-414. The mechanism is supported by the primary kinetics and by phase-resetting with injected Br⁻, Ag⁺ and Ce⁴⁺. The closed/open distinction matters for any use of BZ as a dissipative structure: a closed batch is a long transient, and only a fed reactor holds it away from equilibrium. The Oregonator models oscillation and excitability qualitatively, not quantitatively. The “several thousand cycles” figure is uncited, and the primary papers were not read. | Active |
| THEORY-532 | Systems pushed from equilibrium develop structures that degrade the applied gradient faster, and living systems and maturing ecosystems are such structures, continuous with convection cells and vortices, so the second law is a necessary but not a sufficient condition for life | Schneider & Kay (1994), LIT-533, read in NOTE-415. A “restated second law” (systems moved from equilibrium “utilize all avenues available to counter the applied gradients”) is offered as a corollary of Kestin’s equilibrium stability result and not derived from it. Bénard convection and a draining vortex show organized structures raising dissipation. One stressed marsh, airborne thermal scans and a satellite comparison are read as ecosystems developing to degrade more sunlight; all are correlational and confounded with water. The reading judged the claims too weak to source a THEORY; filed as Proposed. The paper itself says thermodynamics cannot tell the living from the non-living. | Proposed |
| THEORY-543 | Diffusion can destabilise a homogeneous steady state of reacting chemicals that is stable without it: a disturbance of one finite wave-length grows while uniform and very short ones decay, so a stationary pattern appears whose spacing is set by the reaction and diffusion constants and only its phase by the disturbance; with two substances this needs them to diffuse at different rates, and the pattern lasts only while free energy is supplied | Turing (1952), LIT-535, read in full in NOTE-427, with Cross & Hohenberg (1993), LIT-527, skimmed in NOTE-418. The linear analysis on a ring gives exact conditions (9.4a, b) for stationary waves of finite wave-length; that equal diffusibilities cannot satisfy them is NOTE-427’s substitution, not stated in the paper. Turing calls the linearity assumption “a serious one” and the biological examples “very limited”. Stationary chemical Turing patterns were first made in 1989–1991, and real morphogens implementing the mechanism had not been identified by 1993. | Active |
| THEORY-548 v2 | In generalised Lotka–Volterra dynamics with every interaction delayed by τ, a feasible equilibrium that is stable without delay loses stability at τ_c = min over the community eigenvalues j of arctan(|Re j|/|Im j|)/|j|, where a pair of roots crosses the imaginary axis transversally (a Hopf point), so the critical delay is set by the community spectrum; in Saeedian et al.’s strongly self-regulated setting the oscillating regime’s community-level variability falls with diversity (S = 10–70) while species-level variability rises, and the fall matches what summing independently oscillating species gives | Saeedian et al. (2021), LIT-057, a preprint, read in NOTE-061. The critical delay is the linear delay result on the community matrix, re-derived on promotion (2026-10-20), and matched simulation at one strongly self-regulated setting. The variability result was re-run on promotion against an averaging null and matches it: the fall of community-level variability with diversity is what summing independent oscillators gives, not an ecological stabilisation. | Active |
| THEORY-571 | A brain state is better characterised by how its activity moves among a repertoire of weakly coupled metastable substates on a landscape constrained by the structural connectome, their occupancy, dwell times and transitions, than by static connectivity; health is a metastable balance, depression a landscape of over-deep attractors and psychedelic states a flattened one, at opposite ends of one spectrum | Vohryzek, Cabral, Vuust, Deco & Kringelbach (2022), LIT-080, read in full in NOTE-054. A perspective that reports no data: the landscape (Fig. 6) is illustrative, adapted from earlier work, and no study measures depressed and psychedelic states on a common scale. The psychedelic evidence is small cited studies of broader substate repertoires and higher signal diversity; the depression evidence is three studies, one in remitted patients, though the abstract calls depression “excessively rigid”. The modelling section is heavily self-citing. Proposed: a framing, with the record’s rival description of the same dynamics as continuous in THEORY-088. | Proposed |
| THEORY-581 v2 | Two exactly consistent descriptions of one system, at different scales and with equal past–future mutual information, can differ in how that information divides between redundancy and synergy; in small generated Boolean networks under the Williams–Beer redundancy measure the coarser scale was more synergy-biased in every random stochastic system, by at most about 0.07, and the direction mostly reversed for redundancy-biased deterministic ones | Varley & Hoel (2022), LIT-027, read in NOTE-020. Expanding a 3-node macroscale TPM into 4- and 5-node equivalence-class microscales holds mutual information fixed with zero inconsistency, so any change in the PID’s synergy bias B_syn is a change of type, not of amount. In 200 |Gaussian| TPMs the macroscale gained B_syn by about 0.005–0.07; in 185 near-deterministic ones most with macro B_syn < 0.5 lost it. One redundancy function, no robustness check, tiny constructed systems. The “unifying theory” of the title is a framing: the EI section re-describes a known decomposition and adds no result. | Active |
| THEORY-598 | Group interactions that enter a dynamical model linearly, as in diffusion on simplices or random walks on hypergraphs with edge-independent vertex weights, reduce to pairwise dynamics on a reweighted graph; the abrupt transitions, bistability and stabilised coexistence reported for higher-order systems come from group terms that enter nonlinearly, and whether the representation is a hypergraph or a simplicial complex is itself a modelling claim that changes the outcome | Battiston et al. (2020), LIT-091, read in full in NOTE-057. A review: the reduction results are its own §V algebra (three-body linear diffusion becomes a weighted graph Laplacian, Eq. 35) and reported results it cites (Chitra and Raphael on hypergraph walks); the nonlinear effects are one-paper summaries, mostly mean-field or all-to-all. The review never states the condition as a result, and its introduction and conclusion claim more (“no doubts”, “profoundly change”), citing linear cases among the examples. The condition is the record’s reading of what its body shows; edge-dependent vertex weights also escape the reduction. | Active |
| THEORY-607 | The individuals that appear in an evolutionary transition in individuality are a summary description of lower-level processes, not a new level of fact: a partition of particles defines collectives when predicting at the coarse grain matches predicting at the fine grain across many traits, and collectives become indispensable, and so quasi-ontological, only because finite measurement and computation make the fine-grained prediction unaffordable | Bourrat (2023), LIT-049, read in full in NOTE-064. The formal criterion is accuracy only: δC = C_T − C_P < θ across many traits, conditioned on the environment where ecology could fake coherence. As written it is degenerate, since the identity partition gives δC = 0. The cost that is meant to stop that is argued separately, by a 100-operation toy, and never put into the criterion. Nothing is computed on any model. The abstract’s “cannot be dismissed as merely epistemic” is stronger than the body, which says one “pretend[s] as if” the level of selection changed. | Proposed |
| THEORY-608 | On one physical piano-movers puzzle scaled to the solver, ant groups solve better as they grow, while human groups barred from speech and gesture solve worse than individuals and talking groups only about as well as one; large ant groups keep speed and heading through wall contact, and the human mechanisms offered are fitted or observed informally, not tested | Dreyer et al. (2024), LIT-075, read in NOTE-040. Large ant groups (n = 28) reach about 90–100% success by 400 corridor widths, small groups (n = 48) about 50%. Restricted human groups do 32% worse than individuals’ median, talking groups 7% better. There are no inferential statistics. The ant persistence mechanism is supported by collision data and a model whose carrier numbers fall short of calibration; the human majority vote (greediness P = 0.2) is tuned by eye, and “follow a random leader” rests on unrecorded observation. One ant species, one puzzle, one human population. | 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 | Sharpee (2019), LIT-875, read in NOTE-670: a four-page programmatic review with no new result, whose reading filed no THEORY because its claims are conjectures. Filed here as Proposed because the record holds two tests of the conjecture, odour statistics (LIT-872, THEORY-191) and CA1 place-cell correlations (LIT-876, THEORY-192), and no general statement. The Zipf step shows a shared exponential count of states, which any system with extensive entropy has, not a shared metric, and the rigidity argument for three dimensions cites Mostow’s theorem, which concerns finite-volume manifolds and says nothing about noisy point clouds. | Proposed |
| THEORY-634 | In a ring of M chemotactic catalyst species, each making the next one’s substrate, the non-reciprocal nearest-neighbour coupling makes the linear stability matrix circulant: the uniform state goes unstable for any M exactly when the species attract themselves, and the parity of M then decides the leading mode, a real alternating mode for even M that sorts species by parity into two static clusters, and a complex pair for odd M that can give persistent oscillation | Ouazan-Reboul, Agudo-Canalejo & Golestanian (2023), LIT-036, read in NOTE-079. The spectrum is exact for equal parameters (Eq. 5): instability iff μ(p) − μ(s) < 0, real leading eigenvalue for M = 2K, complex pair for M = 2K + 1. Brownian dynamics at M = 5 and 6 bear out the linear predictions; the odd-cycle oscillation is one trajectory at one parameter set. Direct metabolic partners repel; same-parity second neighbours co-aggregate. The origin-of-life reading is untested, and the authors concede in the review file that real catalysts’ mobilities are unknown. | Active |
| THEORY-655 v3 | 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 | Hamilton (1964), Part II, LIT-506, read in full in NOTE-410, on Part I’s model (LIT-497). Two general principles, offered as an “unrigorous” generalization, are applied to alarm calls, kin recognition in gulls and terns, clones and colonies, Hymenoptera, and fights. The evidence is anecdotal. The author calls his positive indications for distant relatives “rather few and feeble”, reports Australian halictines and multiple mating as running against the hymenopteran hypothesis, and explains anomalies case by case. | Proposed |
| THEORY-657 | If an emergent property is one that the parts lack, then the other marks commonly given for emergence are separate questions about each system and not consequences of it: established condensed-matter cases show novelty without a discontinuity, a discontinuity without novelty, novelty that was predicted first, and novelty with a rigorous reduction | McKenzie (2025), LIT-141, read in full in NOTE-124. The book stipulates novelty as the definition and runs twelve further characteristics over about thirty cases. The Kondo effect is novel and continuous; the liquid–gas transition is discontinuous and adiabatically connected; BKT and the hexatic phase were predicted before they were found; Butterfield’s cases are novel and reduced. NOTE-124 says no THEORY follows from the book as a whole, since its definition is stipulated and the logical relations among the characteristics are left open by its own account (§24). This is the part of it that the counterexamples carry. | Proposed |
| THEORY-661 v4 | In a nearly decomposable system, where interactions within subsystems are much stronger than interactions between them, each subsystem’s short-run behaviour is approximately independent of the others’, and in the long run each depends on the others only through their aggregate behaviour; so a hierarchy defined by intensity of interaction separates fast dynamics inside its parts from slow, aggregate dynamics between them | Simon (1962), LIT-905, read in full in NOTE-704. The two propositions are a theorem of Simon & Ando (1961), cited and not reproduced, and established since. Simon widens hierarchy to any nesting of interrelated subsystems, with or without authority, identified by who interacts with whom. His claims that social systems are prominently nearly decomposable, and that higher organisational levels run slower, are assertion and “probably true”. The watchmaker argument is a model whose numbers he says not to take seriously. | Active |
| THEORY-671 | An open system with enough internal degrees of freedom settles, among its possible steady states, into the one of maximum entropy production; this selects among steady states, where Prigogine’s minimum principle only describes relaxation toward one | Kleidon (2009), LIT-519, read in NOTE-423, states the principle and its Earth-system applications; Dewar (2003), LIT-521, read in NOTE-426, is the derivation it cites. The applications are model fits, mostly reported, many by the author. The derivation needs a mean-field step and an unargued assumption about path counts, and does not reach far from equilibrium (THEORY-676). “Sufficient degrees of freedom” is never made operational, so a failure can always be read as too few. The advocate’s own verdict: “no one would argue that MEP is well established”. | Proposed |
| THEORY-692 | Near equilibrium, steady states minimise entropy production and every fluctuation decays; far from it, the excess entropy production has no fixed sign, so a fluctuation can grow past an instability of the thermodynamic branch into a structure held up only by flows across the boundary, and thermodynamics supplies only a sufficient condition for such a state’s stability, not a principle selecting which structure appears | Prigogine (1977), LIT-516, read in full in NOTE-424, with Cross & Hohenberg (1993), LIT-527, skimmed in NOTE-418. Minimum entropy production holds only for steady states in the strictly linear regime with constant coefficients. Beyond it, non-negative excess entropy production suffices for stability; its failure is not shown to give instability. Which structure appears is left to kinetics, boundaries and fluctuations. Cross and Hohenberg find no global minimisation principle beyond a perturbative one near threshold, and quote Glansdorff and Prigogine conceding that “the search for a universal kinetic potential has proved to be unsuccessful”. | Active |
| THEORY-696 | Under multiplicative noise, members that pool their resources and split them equally every step each grow faster in the long run than they would alone, with no complementarity, relatedness or reciprocity: N independent geometric Brownian motions that share keep their drift μ and cut their volatility to σ/√N, raising each one’s time-average growth rate from μ − σ²/2 to μ − σ²/(2N); the gain vanishes as their noise becomes perfectly correlated, and a sufficiently superior member does better alone | Peters & Adamou (2022), LIT-035, read in NOTE-059. The mathematics is correct and self-contained, and the paper’s own citations show the pooled-variance result was known (McCloskey, Winterhalder, Uitdehaag, Kennedy et al.). Sharing leaves the ensemble rate μ unchanged, so an analysis built on expectations sees no reason to share, and the time average does. It holds with zero cost, perfect pooling, unbreakable pacts and unconstrained growth. It is a sufficient mechanism in that model, not a solution of the cooperation puzzle, and not a showing that cooperation is the default. | Active |
| THEORY-701 | Transfer entropy from J to I is the Kullback entropy of assuming that I’s next state, given its last k states, does not depend on J’s last l states, so it is zero exactly when J’s past leaves I’s transition law unchanged; unlike time-delayed mutual information it is directional and zero for static correlation, and it removes a common input only when that input is conditioned on | Schreiber (2000), LIT-1177, read in full in NOTE-940. The definition and its zero condition are exact. The comparison with delayed mutual information rests on two coupled-map lattices built with a known direction. The continuum-limit claim is asserted. The Letter’s tent-map coefficient appears to be twice what Eq. 4 gives. | Active |
| THEORY-702 | Transfer entropy is the average of a local transfer entropy, the log ratio at each observation of the destination’s next-state probability with and without the source given the destination’s past, which can be negative where the source misleads; the destination history must be long for it to exclude self-influence routed through the source; and conditioning also on the destination’s other causal parents (complete) can differ sharply from conditioning on its past alone (apparent), as in elementary cellular automata, where long-history local transfer entropy picks out gliders and domain walls | Lizier, Prokopenko & Zomaya (2008), LIT-1175, read in full in NOTE-943. The local decomposition is an identity; the claims about k and about particles rest on elementary CAs judged by eye. Three worked local values and rule 18’s XOR structure were checked. | Active |
| THEORY-703 | Weak emergence of a macro variable from micro variables can be measured as its Granger autonomy from them times their mean Granger causation of it, which is zero if it is independent of them or fully predicted by them; in a boids model this measure, and Granger causation from the flock’s centre of mass to each boid, are higher for flocking than for random or rigid motion; the measure is relative to the regression model and the variables included | Seth (2010), LIT-1170, read in full in NOTE-935. The definitions are exact; the evidence is two hand-built agent-based models. Seth concedes that the measure is epistemic: positive with missing micro factors or a limited predictor, zero with the system as predictor. Proposed until it is tested against an independent criterion. | Proposed |
| THEORY-706 | Any redundancy function defined on the lattice of antichains of sources, ordered so that a collection precedes another when every source of the latter contains one of the former, has a Möbius inverse that decomposes the information a set of sources gives about a target into partial-information atoms summing to every subset’s mutual information; for the I_min redundancy, the expected minimum specific information over sources, every atom is nonnegative, and interaction information is a signed mixture of atoms, synergy minus redundancy for three variables | Williams & Beer (2010), LIT-1172, read in full in NOTE-936. The lattice, the inversion and the nonnegativity proofs are exact and were followed; the lattice sizes and the 3-parity and three-copies cases were checked. What is not shown is that I_min measures shared information: it gives two independent copied bits 1 bit of redundancy (Harder, Salge and Polani). | Active |
| THEORY-707 | Transfer entropy is not a measure of information flow, if flow means information in the target attributable solely to the source’s past: as a conditional mutual information it equals the source’s unique information plus its synergy with the target’s past, so it can credit a full bit to a source that carries none alone (Y_t = X_{t−1} ⊕ Y_{t−1}) and report zero from each of two sources that jointly fix the target (Z_t = X_{t−1} ⊕ Y_{t−1}); conditioning on further series, as causation entropy does, has the same fault | James, Barnett & Crutchfield (2016), LIT-1165, read in full in NOTE-945. Both examples are exact and were recomputed. The decomposition holds for any PID satisfying the consistency equations. The conclusion is relative to the authors’ definition of flow; transfer entropy as predictive improvement is untouched. The claims about directed information, interventional flow and Gaussian systems are cited or conjectured. | Active |
| THEORY-708 v2 | The I_min redundancy credits two independent sources with shared information whenever each reduces surprise about the same target outcomes, so it gives two independent bits copied into the target 1 bit of redundancy; a bivariate redundancy built from reverse information projections, each source’s information less what is lost when its posteriors are replaced by mixtures of the other’s, satisfies symmetry, self-redundancy, monotonicity and the identity property, gives a nonnegative decomposition, and agrees with I_min on XOR, AND and their compositions but can reverse the state-independent/state-dependent split of a transfer entropy | Harder, Salge & Polani (2013), LIT-1168, read in full in NOTE-944. The counterexample, the axioms and the nonnegativity proof are exact; every Table I value and both transfer-entropy examples were recomputed. The identity axiom and “mechanistic redundancy” are positions, not results, and the measure is bivariate only. | Active |
| THEORY-712 v2 | Under any partial information decomposition satisfying the consistency equations, a transfer entropy T_{Y→X} = I(X_{t+1}; Y_t | X_t) is the unique information of the source’s past about the target’s next state plus the synergy of source and target pasts, which Williams and Beer call state-independent and state-dependent transfer; time-delayed mutual information is the unique part plus redundancy; with the I_min redundancy the state-independent part alone suffices for perfect open-loop control, and a two-source transfer entropy splits into unique, redundant and synergistic transfer | Williams & Beer (2011), LIT-1176, an arXiv preprint, read in full in NOTE-942. The two-term split is an identity from the consistency equations; the sizes of the terms, the control theorem and the multivariate split use I_min, which the record has not read argued for. The one-parameter family and the three corners of the two-source family were recomputed. The text mislabels the synergistic corner. | Active |
| THEORY-714 v3 | 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 | Ince (2017), LIT-1201, read in NOTE-968. The definition is exact, and three of the four properties the paper claims hold; subset equality does not (re-derived 2026-10-20), and every discrete example reproduces. That the dropped properties ought to be dropped, and that unique information depends on the source–source marginal, are argued, not proved. Whether it is the right redundancy measure is open (QUESTION-223). It meets Rauh et al.’s −1-bit bound exactly on XORCOPY. | Active |
| THEORY-715 | The information about a target carried by the Gács–Körner common random variable of the sources is a redundancy defined for any number of sources that is nonnegative, target-monotone and no larger than I_min, and is a lower bound on any reasonable redundancy; but it counts only deterministic shared structure, so it is zero whenever the sources’ joint distribution has full support, fails the identity property and continuity, and gives negative synergy already for two sources | Griffith, Chong, James, Ellison & Crutchfield (2014), LIT-1203, read in full in NOTE-967. The lemmas are short and exact, and every example was recomputed, with Harder, Salge and Polani’s copy family and a three-source copy added. It is the record’s one redundancy defined for n > 2, and it does not have the identity property there or for n = 2. | Active |
| THEORY-716 | Synergy can be measured as the whole minus the union of its parts, the union information being the least information the sources could jointly carry about the target with every source–target marginal held fixed; this synergy lies between WholeMinusSum and Williams and Beer’s S_max, which respectively subtract redundancy and count unique information as synergy, and for two sources it is Bertschinger et al.’s synergy, so independent sources can share information about a target, as the authors’ last version concedes against their first | Griffith & Koch (2014), LIT-1204, read in full in NOTE-966, v6 with v1–v2 checked. The definitions, the criticisms of S_max, WMS and ΔI, and the properties proved in Appendix C are exact, and every Table 1 value for two predictors was recomputed. The identification with Bertschinger et al.’s CI for two sources is the record’s derivation. Monotonicity is claimed without proof, and the S_max bound’s proof has a step the reading could not reconstruct. | Active |
| 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 | Farrell & Ioannou (1996), Part II, LIT-929, eq. 12 and §4, read in NOTE-842. Because d‖u‖²/dt = u†(A + A†)u, the first Lyapunov exponent lies between the long-run averages of the extreme eigenvalues of the Hermitian part. For a normal operator the largest is the spectral abscissa, so an operator that is normal and uniformly stable at every instant has a negative exponent. A switched undamped oscillator, neutral at every instant, has a positive exponent. The stochastic genericity results are THEORY-325’s, still Proposed. | Active |
| 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 | Farrell & Ioannou (1996), LIT-931, read in NOTE-831, with Trefethen et al. (1993), LIT-932, read in NOTE-839. For du/dt = Au with A fixed: a normal A grows as its spectral abscissa; in general the growth over t is σ_max(e^(At)), with the eigenvalues as a floor; the initial growth rate is the top eigenvalue of (A + A†)/2; and harmonic forcing is bounded by κ(E)/dist(iω, Λ(A)), with the ε-pseudospectrum marking where eigenvalues of A + Δ, ‖Δ‖ < ε, can go. All in a chosen norm. The white-noise variance result is THEORY-323’s, still Proposed. | Active |
| 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 | Barnett and Seth (2023), LIT-979, read in NOTE-791: the VAR form (3.26) of exact dynamical independence, with the genericity count of §3.3 made a proof for VAR order p ≥ 2 and corrected for p = 1. Linear coarse-grainings Y = LX of a stable VAR with non-degenerate Gaussian innovations only. It says nothing about near-independence, about the multimodality of the dynamical-dependence cost, or about nonlinear maps. | Active |
| THEORY-752 | In optimal prediction of a Hamiltonian system with separable energy and canonical initial law, projected by conditional expectation, the t-model’s leading-order memory term tPLQLx makes the effective Hamiltonian of the resolved variables non-increasing, so the eliminated variables’ memory enters as a damping that grows linearly in time | Chorin, Hald and Kupferman (2002), LIT-1253, read in NOTE-975: Theorem 8.1. Truncating the Mori–Zwanzig memory integral at its lag-zero value gives dΦ/dt = R̄(Φ) + tS(Φ), and under the stated conditions this dissipates the free energy of the resolved variables. It says nothing about how well the t-model predicts, gives no error bound, and holds only for the conditional-expectation projection; the true conditional mean’s effective energy need not decrease. | Active |
| THEORY-755 v3 | For a linear system x(t+1) = Ax and a partition of its variables, the vector of block sums evolves autonomously from every initial state iff the partition is column-equitable, every column of each block of A summing to the same value; the aggregate operator then keeps a subset of A’s eigenvalues, and the blocks may interact as strongly as they like, so exact aggregation is not decomposition | Shpak, Stadler, Wagner and Hermisson (2004), LIT-1242, read in NOTE-979. Exact aggregation of a linear system by the sums over a partition holds for every state iff the partition is column-equitable, a symmetry condition, not a weak-coupling one. Linear dynamics and partition-based aggregation only. It says nothing about approximate aggregation, the Simon–Ando case, or nonlinear systems, where the paper itself shows aggregations with no linear counterpart. | Active |
| THEORY-756 | The growth rate of thermodynamic depth is the entropy rate of whichever macrostate process it is computed over, so the choice of macrostates sets it anywhere from zero up; among partitions of a stationary process’s past that predict every finite stretch of the future as well as the whole past does, the causal states are the coarsest, so they have the least entropy and the least entropy rate | Crutchfield and Shalizi (1999; arXiv 1998), LIT-1260, read in NOTE-978. Depth is not a structure measure until its states are fixed. Predictive sufficiency, then minimality, fixes them as the causal states. The choice is canonical only given the measuring instrument and the target, the observed future. The redefined dive is still an entropy rate. | Active |
| THEORY-758 | For a self-adjoint diffusion operator T with spectrum in [0, 1], the diffusion-wavelet levels are spectral subspaces spanned by the eigenvectors with λ^s ≥ ε for an increasing sequence of powers s, so they form an exactly nested chain of T-invariant subspaces whose differences are bands of the spectrum, about one octave each, not spatial blocks; Coifman and Maggioni’s compressed algorithm, which squares the operator on each orthonormal basis, thresholds T^{2^{j−1}} at level j and not T^{2^j−1} as the paper states; and since each level is T-invariant, each is an exactly closed linear coarse-graining whose localized basis is a choice the level does not fix | Coifman and Maggioni (2006), LIT-1256, read in full in NOTE-984. The levels of a diffusion multiresolution are spectral bands. Nesting is exact and needs no decay of the spectrum, and decay buys only compression. The compressed algorithm as written keeps ⟨ξ : λ^{2^{j−1}} > ε⟩ at level j, and the paper’s own Fig. 9 follows this, not its stated count. Each level closes exactly under T, but a single scaling function does not. It says nothing about non-self-adjoint operators, about error bounds for the compressed scheme with finite precision, or about when a localized basis exists. | Active |
| THEORY-762 | For a reversible diffusion dX = −∇U dt + √2 dW in equilibrium, an eigenfunction ψ of its generator with eigenvalue −λ is closed in conditional mean with no memory: E[ψ(X_{t+s}) | the whole past] = e^{−λs}ψ(X_t), and so the same given ψ’s own past; but ψ(X_t) is a Markov process in its own right only when |∇ψ|² is a function of ψ, since otherwise its own path reveals a noise intensity that its present value does not fix | The record’s answer to what Nadler et al. (LIT-1250, read in NOTE-986) leave implicit when they call slow eigenfunctions reaction coordinates. Such a coordinate is exactly closed in its mean: its best prediction from the full state and from its own history coincide, with no memory term, and the Mori–Zwanzig memory and noise for it vanish under Mori’s projection. It is closed in law only under a further condition. In the paper’s examples the condition holds; the mixed Ornstein–Uhlenbeck eigenfunction x₁x₂ shows it can fail. This does not say that diffusion maps estimate the eigenfunctions well from finite data, or that a slow eigenfunction is the best coarse variable. | Active |
| THEORY-764 | For a deterministic supervenient feature V of a Markov system, Rosas et al.’s practical emergence criterion Ψ equals the parts’ whole-minus-sum information about V’s next value minus V’s one-step dynamical dependence I(V_t′; X_t | V_t), so dynamical independence raises Ψ but neither implies emergence nor is implied by it, and the paper’s sufficient test for causal decoupling can pass a feature that is not dynamically independent | Rosas et al. (2020), LIT-1244, read in NOTE-987, with Barnett and Seth’s dynamical independence (LIT-979, THEORY-257). By the chain rule, Ψ(V) = [I(X_t;V_t′) − Σ_j I(Xʲ_t;V_t′)] − I(V_t′;X_t|V_t). The subtracted term vanishing implies Barnett–Seth independence for Markov X. Examples show each condition without the other. The independent sum of AR(1) channels has both, with no interaction. The identity concerns Ψ, the paper’s practical criterion; it says nothing about the PID quantities themselves. | Active |
| THEORY-771 v4 | In a singularly perturbed system, fast states x̃ with dx̃/dt = f̃(x̃, y) + ε·δf and slow states y with dy/dt = ε·ḡ(y) + ε·δg, if f̃ is contracting in x̃ at rate β_f with x̃ = 0 an equilibrium for every y, ḡ is contracting at rate β_g, and |δf| ≤ d_f + α_f,x|x̃| + α_f,y|y|, |δg| ≤ d_g + α_g,x|x̃|, then for every ε below ε_c = β_f / [χ_f(α_f,x + χ_g α_f,y α_g,x / β_g)] the fast states stay within O(ε) of zero and, when d_g = 0, the slow states within O(ε) of the reduced trajectory, uniformly in time after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε) | Bousquet and Slotine (arXiv 2015), LIT-1292, read in NOTE-1025. It gives Simon’s near-decomposability the rate in ε that THEORY-661’s open statement lacks: first order, for trajectory error, uniform in time, under contraction of the modules and of the reduced slow system, with an explicit, sufficient ceiling on ε. Deterministic. It bounds how far the states are from the reduced description, not how much information the rest of the system sends into the slow variables, and it takes the slow coordinates as given. | Active |
| THEORY-773 v2 | In a singularly perturbed system ẋ = f(x, z, t), εż = g(x, z, ε), if g is contracting in z at rate λ_z for every x and ε, the reduced system ẋ = f(x, γ(x), t) on the slow manifold z = γ(x) is contracting at rate λ_x, f is α-Lipschitz in z, |∂γ/∂x · f| ≤ d and g is K-Lipschitz in ε, then for all t ≥ 0 the fast state is within χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z of the manifold and the slow state within χ_x|x(0) − x_γ(0)|e^{−λ_x t} + ε[C₁(e^{−λ_x t} − e^{−λ_z t/ε}) + C₂(1 − e^{−λ_x t})] of the reduced trajectory; partial contraction of f in x alone is not enough | Del Vecchio and Slotine (IEEE TAC 2013), LIT-1295, read in NOTE-1020, with their Theorem 3’s hypothesis corrected. A global, uniform-in-time, first-order bound on the error of the quasi-steady-state reduction, with every constant explicit and no requirement that ε be small. The reduction is Markovian: eliminating a fast, contracting variable costs O(ε) in trajectory error for all time, so no memory term is needed to that order. Constant metrics, deterministic, and the bound d on how fast the manifold moves is needed along the trajectory. | 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 | Zhao, Wytock, Reynolds and Motter (2024), LIT-854, read in full in NOTE-655. Their abstract and discussion carry the model result to cells: one genotype may hold several heritable expression states without epigenetic marks, and history picks among them. They predict that a transient crp knockdown leaves zraR, melR or rhaRS changed (NOTE-655, C8, untested). THEORY-178 is the model result this rests on. No such perturbation has been done in cells (QUESTION-172). | Proposed |
| THEORY-787 v2 | 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 | Hoel, Albantakis and Tononi (2013), LIT-1038, argue that macro can beat micro: the scale at which effective information peaks is where a system’s causal structure is, so causation can emerge at a macro level. The record holds as shown, in THEORY-222, only the arithmetic: under uniform interventions imposed separately at each level, a macro description can have more effective information. This is the causal thesis, filed under ADR-047 v4 when the narrowing dropped it. | Rejected — Rejected on 2026-10-30, at the owner’s direction, by its own refutation condition: the macro advantage disappears whenever the same intervention distribution over micro states is used at both levels. THEORY-1167 proves this for a coarse-graining of one transition, by data processing, and Comolatti and Hoel concede it (LIT-1843). The independent interventionist test the promote_when asks for, causal feature learning (LIT-1833), can confirm that a macro is proportionate but never that it beats the micro (THEORY-1169). What the record holds instead is CLAIM-497, by ARG-041. Hoel’s temporal and spatio-temporal macros, which the bound does not reach, are QUESTION-335. The arithmetic it rested on, THEORY-222, stands. |
| 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 | Wu & Silva (2024; v3 2026), LIT-1032, read in NOTE-724, claim C6 (weak). The persistence thesis drawn from ten illustrated lifecycles, filed apart from what the 2017 log measured (THEORY-216) under ADR-047 v4. It is the population form of what CLAIM-280 needs. | Proposed |
| 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 | The degree flow that THEORY-190’s title carried before the promotion audit (batch 048) narrowed it to the asymptotic closure of the S¹ connection law. García-Pérez, Boguñá and Serrano (2017), LIT-869, derive it to first order by treating the coarse-grained graph as an S¹ model at every separation. The record’s exact computation shows that treatment fails between nearby supernodes, so the flow is unproved. Filed under ADR-047 v4. | Proposed |
| 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 | The large-N and forgetting clauses that THEORY-272’s title carried before the promotion audit (batch 051) narrowed it to Hopfield’s simulations at N = 30 and 100. Hopfield (1982), LIT-918, expects the 0.15N figure “for all large N” from a crosstalk estimate, and states in words that overload makes every memory irretrievable unless old ones are forgotten, with saturating couplings as the mechanism. Filed under ADR-047 v4. | 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 | The empirical claim THEORY-190 set aside when the promotion audit (batch 048) narrowed it to the model. García-Pérez, Boguñá and Serrano (2017), LIT-869, find six embedded real networks self-similar under the renormalization and call this new evidence of hidden metric spaces. The collapse was judged by eye over two to five layers with no null ordering of the circle. Filed under ADR-047 v4. | Proposed |
| 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 | The thesis THEORY-714’s title carried before the promotion audit (batch 056) corrected and narrowed it to what I_ccs is and which properties it has. Ince (2017), LIT-1201, proposes I_ccs as the redundancy measure and reads its negative atoms as misinformation and mechanistic redundancy. Filed under ADR-047 v4. The re-derivation found that I_ccs fails subset equality, the property Ince argues licenses the lattice, which counts against the thesis without settling it. | Proposed |
| THEORY-818 v2 | In two-scale Lorenz 96 without time-scale separation (ε = 0.5, K = 18, J = 20, F = 10), a discrete-time closure for each resolved variable that fits the one-step discrepancy of a fourth-order Runge–Kutta step as a NARMAX series, with one autoregressive lag, at most two lags of the resolved variable and at most one moving-average term, reproduces the mean, spread, distribution, autocorrelation and neighbour cross-correlation of x₁ (Kolmogorov–Smirnov distance 0.005) and, with observations every 0.05, extends the 20-member ensemble forecast lead time at anomaly correlation 0.6 from about 2 to about 4 time units over a polynomial-plus-AR(1) closure; the model structure was chosen by trial and error, differs between the two observation spacings, and carries no error bound | Chorin and Lu (2015), LIT-1304, read in NOTE-1028. A data-driven closure with one or two steps of discrete memory, plus one moving-average innovation, is enough on this test to reproduce the resolved statistics and to double forecast skill against a memoryless-in-x closure. It is the record’s first evidence that a short discrete memory suffices in a chaotic system with no scale gap. It is one system, the structure was tuned, and there is no bound, so it does not give the controlled prediction error QUESTION-222 asks for. | Active |
| THEORY-819 | For an invertible map with an invariant measure, projecting onto the closed span of the whole past of chosen functions of the resolved variables (a Wiener projection) turns the exact discrete Mori–Zwanzig equation into one with no memory term: the next resolved value is a causal linear filter of those past functions plus a residual orthogonal to all of them, so the memory is carried by the coarse variable’s own history; a finite NARMAX form follows only by a rational approximation of the filter, uncontrolled in general but exact for a scalar coarse variable of a stable linear Gaussian VAR(1) | Lin and Lu (2021), LIT-1307, read in NOTE-1029. Mori–Zwanzig’s memory is what a projection onto the present leaves out; a projection onto the whole past absorbs it, leaving the causal Wiener filter on the coarse variable’s own history and a residual orthogonal to all of that history. Second-order (linear-span) projections, invertible dynamics and stationarity only. It gives no finite augmentation with a bounded error in general: the NARMAX step is the paper’s uncontrolled approximation, exact only in the linear Gaussian class shown here. | Active |
| THEORY-820 v2 | For a stationary discrete-valued process X and any coarse-graining Z of its past, informational closure at every horizon is the same condition as zero one-step transfer entropy from the micro past, Barnett and Seth’s dynamical independence, and as causal closure, Z’s ε-machine coinciding with its υ-machine; for a coarse-graining of the present state it also makes Z’s causal states a coarse-graining of X’s, which does not imply closure, and the closed coarse-grainings are not closed under common refinement | Rosas et al. (arXiv 2024), LIT-1306, read in NOTE-1030, with Barnett and Seth’s definition (LIT-979). For discrete stationary processes, the information-theoretic criterion of emergent closure (dynamical independence), the interventionist one (the macro’s own causal states are all the micro distinctions that matter for it) and, for maps of the present state, a computational one are nested as stated. The first two are one condition. It says nothing about approximate closure. It does not make closure a mark of unity. “Causal” is only in the Granger sense unless X is fully observed. | Active |
| THEORY-822 | For a Markov chain and a partition of its states, strong lumpability, every state of a block sending the same probability into each block, is the column-equitability of the master-equation operator and makes the block label informationally closed; when the chain is irreducible, an informationally closed label that is itself Markov is strongly lumpable, with no aperiodicity needed; weak lumpability gives a Markov label that need not be closed; and the same holds on the causal-state chain of any stationary process | Rosas et al. (arXiv 2024), LIT-1306, read in NOTE-1030, on Kemeny–Snell lumpability, joined to Shpak et al.’s column-equitable partitions (LIT-1242, THEORY-755). Exact closure of a block label in a Markov chain is a condition on transition rows, a symmetry, and not weak coupling. The converse needs the label to be Markov: a closed label can have memory. Exact results only, with no bound for nearly lumpable partitions. | Active |
| 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 | Perunov, Marsland and England (2016), LIT-518, Eq. (8), read in full in NOTE-416 (which checked Eqs. 3–8), building on England’s macrostate relation (2013), LIT-515; re-derived by the promotion audit. An exact identity from microscopic reversibility, under a fixed protocol and uniform starting densities; the paper itself uses near-uniformity. It says nothing about which macrostates a real many-body system can be divided into with the other two terms equal. | Active |
| THEORY-863 | In Arthur’s model of two agent types arriving in random order, each adopting whichever of two technologies pays it more, with returns linear in previous adoptions, increasing returns make the adoption difference a random walk with absorbing barriers: one technology takes the whole market with probability one, A with probability Δ_R/(Δ_R + Δ_S) where Δ_R and Δ_S are the gaps at which each type switches, the winner need not be the better technology, and once it has locked in no small change in returns alters later choices; with diminishing returns the barriers reflect and the shares go predictably to one half | Arthur (1983 working paper; published 1989), LIT-1314, read in NOTE-1052. A result about one model: two types, linear returns, no expectations, no pricing, no sponsors. It does not show that any historical technology won by small events, and the “efficiency” it says can fail is a weak search criterion. Its lock-in is absorption in accumulated adoptions, which a small reversal of returns cannot undo. | Active |
| THEORY-885 | Politics is more prone than markets to increasing returns, and so to path dependence, because it is dense with compulsory institutions and policies, rests on collective action and is opaque, so that institutions, collective action and shared interpretations of politics all feed back on themselves; and once on a path political actors leave it less readily than market actors, because competition and learning correct weakly, elected officials discount the future at the electoral cycle, and institutions are built to be hard to change | Pierson (1997 working paper; published 2000), LIT-1372, read in full in NOTE-1076 through its precursor only, the EUI Jean Monnet Chair Paper 44. An argument with no data and one illustration (two failed US health-care reforms, 1993–96). It extends North’s claim for institutions in general (THEORY-429) to a claim about politics as a domain. It does not show that any political outcome was locked in by an early contingent event. | Proposed |
| THEORY-891 v2 | In dyads of untrained dancers facing each other, shared music and visual contact each raised interpersonal movement synchrony, with no interaction detected, in different movement components: lateral and hand movements synchronized through seeing the partner, anteroposterior movements such as head bobs through the shared music, and vertical bounce through both; which factor drove a component’s synchrony was classified from its spatial direction and not from the periodicity of the synchrony | Bigand, Bianco, Abalde & Novembre (2024), LIT-1424, read in full in NOTE-1093 from PubMed Central. A 2 × 2 within-dyad experiment, 35 dyads, that removes the shared external drive and the coupling channel independently and finds synchrony from each in different components, no interaction, and a timing signature that cannot tell them apart. It answers QUESTION-063 in part: separate interventions on drive and channel do discriminate, and a timing-only measure does not. Direction of coupling, and a perturbation of one member, are not in it. | Active |
| THEORY-899 v2 | In an infinite, well-mixed population whose members each sample the quality of their own type, imitation of success has the expected update of a single Cross Learning bandit agent (the Taylor replicator dynamic), and the weighted voter model, in which members broadcast their type in proportion to their sample and adopt the first type they perceive, has the expected update of Maynard-Cross Learning, π_a ← π_a + (r_k / v^π)(1[a = k] − π_a), the Maynard-Smith replicator dynamic; the population vector is the policy and no member learns | Soma et al.’s Propositions 1 and 2, read in full from arXiv v6 in NOTE-1132, with both proofs and the aggregation in Eqs. 8–14 re-derived. An exact identity of expected updates under idealizing assumptions: infinite N, large well-mixed neighbourhoods, and imitation decoupled from evaluation. Each member is one parallel action sample, α = 1/N plays the learning rate, and the macro-policy is a stateless bandit policy over the members’ types. | Active |
| THEORY-905 | In simulations of the two imitation protocols, populations small enough to be finite often settle on a suboptimal option because pure imitation can remove a type entirely, and once settled the protocols give no way to re-explore when the environment changes; variants in which members compare quality estimates converge faster than blind weighted voting, at the cost of more communication and comparison | Soma et al.’s simulations and limits section, read in NOTE-1132. With ten options and arbitrary parameters that the authors call qualitative, swarms of about 500 track the replicator dynamic and smaller ones often lock onto worse options. The authors state that pure imitation can delete a type and has no mechanism for escape in non-stationary environments. Hybrids that compare qualities are faster. Simulation-scoped, not a theorem. | Proposed |
| THEORY-927 v2 | 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 | Shazeer et al. (2017), section 4 and Appendix A, read in full in NOTE-1145 from arXiv v1. The imbalance is reported in one paragraph, with Table 6’s no-loss row as its evidence: one architecture, ten epochs, no variance. The mechanism is the authors’ explanation, not a tested result. The fix penalizes unequal use regardless of competence. | Active |
| THEORY-928 v2 | Two softmax learners that update every action’s reward memory each step with its average reward against the other’s current play, the memories decaying at rate α, follow, in the limit of learning slow relative to play, coupled replicator equations plus a term α Σ x_j log(x_j / x_i) toward uniform play; in a two-player matrix game, without decay the dynamics preserve phase volume in log-ratio coordinates and are Hamiltonian when the game is zero-sum with an interior Nash equilibrium, and with decay the volume contracts at rate (N − 1)(α_X + α_Y) | Sato & Crutchfield (arXiv 2002; PRE 2003), LIT-1520, read in full from arXiv v2 in NOTE-1144, Eqs. 1–13. The entropy-like term is what memory decay becomes under softmax choice. The limit, the dissipation rate and the Hamiltonian were re-derived, second-checked at the status review of 2026-10-30. The model updates every action’s memory at the same rate, so it is the full-information case, not Q-learning as usually run. | Active |
| THEORY-936 | In two-agent rock–scissors–paper played by softmax reinforcement learners, nonzero tie rewards give quasiperiodic tori and Hamiltonian chaos without memory decay, and limit cycles and chaotic attractors with it, over a large share of the tie-reward range simulated | Sato & Crutchfield (arXiv 2002; PRE 2003), LIT-1520, read in full in NOTE-1144, pp. 2–3 and Figs. 1–3. Numerical results for one game, two agents and the parameters shown; the zero-decay, zero-sum case was first reported in Sato, Akiyama and Farmer (PNAS 2002), which is not held. | Proposed |
| THEORY-939 | In a multi-agent system run against a world utility, a private utility is factored, ranking each agent’s own alternatives with the others fixed as the world utility ranks them, exactly when it is an increasing function of the world utility given the components the agent cannot affect; the world utility minus its value with the agent’s effect set clamped is factored, and in a sixty-agent bar problem clamping to the average action learned better than clamping to absence or using the world utility itself | Wolpert & Tumer’s NASA report (LIT-1506), read in full in NOTE-1153 because their Advances in Complex Systems paper (LIT-1515) could not be reached. The characterisation is stated with its proof deferred; the “if” half and the WLU’s factoredness were checked when filed. Factoredness is ordinal, member by member, and needs no scale shared among members. The WLU is not a causal counterfactual: it re- scores the actual history with one member’s part replaced. Removal was not the best clamp. | Proposed |
| THEORY-942 v2 | 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 | Omidshafiei et al. (2019), LIT-1516, Theorems 2.1.2, 2.4.24 and 2.5.1, read from arXiv v4 in NOTE-1156, proofs followed. Irreducibility is immediate, and the α → ∞ limit is of transition matrices. That each MCC lies in one chain component follows from chain transitivity, re-derived and second-checked on 2026-10-30. The converse, that attracting components contain MCCs, rests on a conjugation step that is asserted, not carried out, and is THEORY-1233. Not shown: what the stationary distribution tends to, how it splits among several sinks, or invariance to duplicated strategies, which the reader found it lacks. | Active |
| THEORY-945 v2 | 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 | Yang et al. (2018), read in full in NOTE-1148 from arXiv v5. The expansion and the remainder bound were re-derived when filed and second-checked on 2026-10-30; the appendix states an equality where only an inequality holds, which leaves the bound intact. The convergence theorem is reported, not held. Its contraction step needs a high temperature while it assumes a greedy limit, and the cancellation of remainders is argued from homogeneity, not proved. The approximation requires agents interchangeable within a neighbourhood. Whether the method works as claimed is THEORY-1231. | Active |
| THEORY-1052 | A quasi-static attractor describes an open system only while the drift of that attractor in state space is slow against the system’s slowest relaxation, scaled by the distance to the tipping threshold; rate-induced tipping is the failure of that time-scale separation, and it can occur while the parameter itself moves slowly, if the attractor is sensitive to it, and with no slowing of any mode | Ashwin, Wieczorek, Vitolo and Cox (2012), LIT-1654, read in NOTE-1215. Shown for the linear tipping-radius model, where the sufficient condition for tracking is a rate comparison, and stated for a fast-slow example (“rate dependent tipping occurs when the slow and very slow timescales are no longer separable”). Proposed because for nonlinear systems the paper says “no exact analogy possible” and gives no general condition. It bears on CLAIM-478, whose constraints hold over a window between time scales. | Proposed |
| THEORY-1054 | One arrangement of harmonic particles on a finite graph, with the same wiring, potential and inverse-mass response, gives the discrete wave equation when the integrator stores position and momentum and the discrete heat equation when it stores position alone; both are governed by the symmetrized graph Laplacian, and the coupling data do not decide between them | Spivak (2026), LIT-1649, read in NOTE-1225 from arXiv v2. In the operad ArrSm, harmonic particles wired along a finite directed graph form one closed arrangement with potential Σ(κ_e/2)(q_tgt − q_src)². The phase semantics gives m_v q̈_v = −(L_G q)_v (Proposition 7.4.1) and a symplectic step; the configuration semantics gives m_v(q_v(t+1) − q_v(t)) = −(L_G q(t))_v (Remark 7.4.2). Active: both are short exact computations. It says nothing about time scales, and what differs between the two is a modelling choice, not the coupling. | Active |
| THEORY-1061 | Flack makes downward causation operational as components tuning their behaviour to their own estimates of aggregate variables they collectively compute: weak (apparent) when at least one component tunes to a variable slower than the behaviour producing it, strong (effective) when the variable is predictive, robust, used nearly universally, agreed on and gaining mutual information with the microscopic behaviour; she speaks of a new level only at the strong form, and regeneration of the variable by its use is not among her criteria | Flack (2017), LIT-1630, read in NOTE-1216, §1, §3 and §7. Active as a reading of what the paper proposes, which its text states as lists. It does not say the proposal is right, that it meets any definition of causation (the paper sets that aside), or that the macaque system meets the strong criteria; the evidence for that is in earlier papers not reproduced here. The sharp edge for the record: what reinforces itself in Flack is the convergence of members’ estimates, not the variable’s persistence. | Active |
| THEORY-1073 | An aggregate variable much slower than the behaviour producing it becomes an environment the components tune to whether or not it summarizes the underlying regularities well, which helps coordination; if it summarizes them poorly the system likely fails in the end, later the larger the separation | Flack (2017), LIT-1630, read in NOTE-1216, §6. Argued informally, with social institutions and the macaque power distribution as illustrations, and no model or data. Proposed. It does not say slowness makes an aggregate accurate, or that separation alone makes the components use it: in Flack the aggregate constrains only through the components’ tuning to it. | Proposed |
| THEORY-1079 | An open system can tip away from an attractor that exists, keeps its stability and keeps its basin at every frozen value of its parameter, when the attractor moves faster than the system can track it; with steady drift what bifurcates is the ability to track, in a co-moving frame, not the attractor, and the critical rate depends on the relaxation rate, the distance to the tipping threshold and the starting state | Ashwin, Wieczorek, Vitolo and Cox (2012), LIT-1654, read in NOTE-1215 from the corrected arXiv v5. Active as mathematics: shown in a linear model with a fixed stable matrix (so no bifurcation and no noise), in normal-form examples worked exactly, and in a fast-slow system whose equilibrium is unique and globally stable. It does not say that a system that tips is lost for good, that the threshold is a basin boundary, or that any real system has tipped this way. | Active |
| THEORY-1085 | In Toner and Tu’s hydrodynamic theory, a flock’s ordered state is a broken continuous symmetry of a coarse-grained velocity field: speed fluctuations relax in a finite time of order 1/α and are dropped, leaving as slow degrees of freedom the transverse velocity and the conserved density, which propagate as sound at ω = ±cq⊥; no individual member appears in the description, which is offered for a universality class of microscopic rules | Toner and Tu (1995), LIT-1670, read in NOTE-1228 from the arXiv preprint. Active as a statement of what the theory is: its variables, the degree of freedom it eliminates and why, and the modes that remain, all in the text (pp. 3–8). It does not say that the theory’s exponents are right (THEORY-1095), that the universality claim holds, or that the order so described is an agent or an individual. | Active |
| THEORY-1087 | In the symmetric Sakaguchi–Kuramoto model on a triangle and a pentagon sharing one vertex, the lag α gives each ℓ-cycle a phase-independent path mismatch (ℓ − 2)α, and the remote-synchronized locked branch passes through an exact two-group state at α = π/3, where the pentagon’s mismatch reaches π; but that branch has already lost transverse stability, in the pentagonal symmetry sector, at αc ≃ 1.0164, so the point of maximal cycle frustration marks the geometry of the phase relations, not the end of the synchronized pattern | Torres-Hugas, Duch, Gómez and Arenas (2026), LIT-1668, read in NOTE-1218 from the arXiv preprint v2. The mismatch (ℓ − 2)α is a three-line derivation (eqs. 15–18); the two-group state at π/3 is exact algebra (eqs. 38–40); the threshold αc is a numerical continuation with det M⊥ = 0 (Supplementary Material). All three were reproduced. Active for this motif and branch; the paper’s wider reading, that remote synchronization is a manifestation of cycle-level constraints, is not what this says. | Active |
| THEORY-1095 | In Toner and Tu’s hydrodynamic model of a flock, true long-range orientational order survives in two dimensions, where the Mermin–Wagner theorem forbids it at equilibrium and the model’s own linearized theory loses it, because the convective coupling of the velocity field to its own transport, relevant below four dimensions, keeps the transverse velocity fluctuations bounded; the authors’ heuristic is that motion keeps changing each member’s neighbours, so originally distant members come to interact | Toner and Tu (1995), LIT-1670, read in NOTE-1228 from the arXiv preprint. The linear theory’s loss of order in d ≤ 2 and the relevance of the convective nonlinearity below d = 4 are calculations. The survival of order in d = 2 follows from χ = −1/5 < 0, which is exact only if a cubic vertex stays irrelevant (“we strongly suspect”), or from χ = (z − d)/3 with z < 2, which is expected “on physical grounds”. The change-of- neighbours mechanism is a heuristic. Proposed for those two steps. | Proposed |
| 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 | The wider claim dropped when THEORY-073 was narrowed to Friston’s 2013 formulation and promoted on 2026-10-27 (ADR-047 v4: a narrowing’s dropped claims go to Proposed entries). Both critiques argue the general form, and the record holds neither of the later formulations it would have to reach. Its promote_when is THEORY-073’s own, moved here unchanged. | Proposed |
| THEORY-1122 | Weiss’s mark of a system, that the whole varies far less than the summed variances of its parts, is met by any aggregate of independent parts when the whole is measured as their average; when it is measured as their sum, it holds only if the parts’ fluctuations compensate one another, with ∑ᵢ<ⱼ Cov(xᵢ, xⱼ) close to −½ ∑ᵢ Var(xᵢ) | Nicholson (2026), LIT-1821, read in NOTE-1340, reports Weiss’s one formula, VS ≪ ∑ (va + … + vn), as a property “we would invariably find, if we studied any system”, and as a way of identifying systems empirically. The variance of a sum settles what it can show. Active: an identity, checked numerically. It does not say Weiss’s systems lack compensating covariance, only that the inequality as stated does not test for it. | Active |
| THEORY-1126 | In animal development the organ-level outcome is reproducible while the trajectories of individual cells differ between embryos raised alike, so developmental regularity is imposed on variable cell behaviour rather than read out from a fixed sequence of cellular events (Weiss, as Nicholson reconstructs him), with invariant lineages such as that of C. elegans the conceded exception | Nicholson (2026), LIT-1821, read in NOTE-1340. Weiss’s “determinacy in the gross despite demonstrable indeterminacy in the small”, shown as a redrawn schematic (Fig. 5) and supported by citations to work on stochastic gene expression and cell-fate decisions. This is the only one of the paper’s four critiques that rests on evidence, and the paper presents none of the evidence itself. Proposed. | Proposed |
| THEORY-1161 v2 | 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 | Friedkin (2011/2012), LIT-1404, read in NOTE-1374. Under reflected appraisal, the next self-weight set to the member’s share of control over the last issue, with each issue settled by Friedkin–Johnsen anchoring, his simulations give a single leader in 99.0–100% of 100,000 random networks. Re-runs on sparser networks gave 93% and 96%. The contrast with the DeGroot step, and which step real groups follow, are not this reading’s: the first is THEORY-1232. | Active |
| THEORY-1162 | In a consumer-resource model whose species-resource benefits and impacts have correlation rho, the replica-symmetric cavity fixed point loses stability where rho squared equals surviving species over non-depleted resources | Blumenthal, Rocks and Mehta (2024), LIT-1236: in MacArthur’s consumer–resource model with benefits c and impacts e correlated by ρ, the replica-symmetric steady state is unstable when ρ < ρ*, ρ² = S/M* at the boundary. With ρ = 1 competitive exclusion keeps S*/M* ≤ 1, so the symmetric model never crosses it. What this does not say: that non-reciprocity by itself breaks the optimization picture; it does so only past a boundary set by species packing, and that the other side is chaotic is shown numerically, not derived. | Active |
| THEORY-1167 v2 | 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 | The test QUESTION-204 asks for has an answer that needs no system: with one distribution over micro states at both levels, EI(macro) = I(f(X); f(Y)) ≤ I(X; Y) = EI(micro) by data processing. Hoel’s causal emergence uses the uniform distribution at each level, which is a non-uniform micro distribution whenever blocks differ in size, so its positive values come from the micro chain scored under that other distribution. Comolatti and Hoel (LIT-1843) concede the stationary case; Eberhardt and Lee (LIT-1839) name the distribution as the source of the artefact. It says nothing about measures that data processing does not bound, nor about whether EI’s level agrees with an interventionist test. | 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 | The wide form THEORY-285 held before the eighth status review narrowed it, kept as a Proposed reading by the ADR-047 kind-1 sweep of 2026-10-30: the source asserts it, and THEORY-285, Active, is its evidence. Linson et al. (2025), LIT-983, recast post-traumatic stress as lost context disambiguation and offer the loop as its mechanism; their simulations show it once per phenotype, with unreported parameters, and fit no patient data. | Proposed |
| 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 | The wide form THEORY-218 held before the tenth status review narrowed it, kept as a Proposed reading by the ADR-047 kind-1 sweep of 2026-10-30: the source asserts it, and THEORY-218, Active, is its evidence. Litherland & Mørch (2021), LIT-1037, give a framework in which the image on the canvas carries out a design decided in the forum, and argue that placement data cannot show cooperation, conflict or affiliation; they studied one artwork, and call the link from forum power to placements unclear. | Proposed |
| 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 | The wide form THEORY-214 held before the twelfth status review narrowed it, kept as a Proposed reading by the ADR-047 kind-1 sweep of 2026-10-30: the source asserts it, and THEORY-214, Active, is its evidence. Falmagne, Stephenson and Levin (2026), LIT-1025, frame a defended image as an equilibrium of attack and repair and read several of the precursors of its overthrow as the defending community’s conduct. The framing is presupposed by their state variable, and the social reading is their own conjecture from SHAP curves. | Proposed |
| 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 | The wide form of Bhattacharyya et al. (2023), LIT-853, kept here when the status review of 2026-10-30 narrowed THEORY-179 to what the assays measured and promoted it (ADR-065; ADR-047, kind 1). The paper asserts a heritable iron memory. Iron was read only through a plasmid promoter reporter, inheritance rests on few founders and four sampled generations, and the delay model is fitted to two time points. | Proposed |
| 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 | The authors’ explanation in Shazeer et al. (2017), LIT-1507, p. 5, stated and not tested. Kept here when the twenty-second status review promoted THEORY-927 as the report (ADR-065; ADR-047, kind 1). | 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 | What the Haken–Kelso–Bunz paper proposes and what the first report of critical fluctuations measured, split from THEORY-069 by the twenty-third status review on THEORY-1110’s pattern (ADR-058, ADR-065), so that a claim needing only the model and that report can stand on it. Both papers are read in full (NOTE-1424, NOTE-1433). Whether the switch is a nonequilibrium phase transition with critical slowing and hysteresis as its warning signs, which needs relaxation-time and hysteresis data neither paper shows, stays THEORY-069’s question, Proposed. | 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 | The wide form of Yang et al. (2018), LIT-1512, kept here when the twenty-fourth status review narrowed THEORY-945 to the re-derived expansion and to what the paper states and ran, and promoted it (ADR-047, kind 1). The paper asserts that mean field Q-learning converges to the Nash Q-value under its assumptions, and that it makes many-agent learning tractable. Its proof does not reconcile a high-temperature contraction with a greedy limit, and its many-agent evidence is one task at two sizes. | Proposed |
| 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 | The general contrast, filed beside THEORY-1161 by the twenty-fourth status review (ADR-065 §3; ADR-047 kind 1). THEORY-1161, Active, holds what Friedkin’s simulations showed; whether the anchored step’s autocratic fixed points attract almost every start, and the DeGroot step’s do not, is this entry’s question. Re-runs on sparse networks of density 0.4 gave 93% and 96% single leaders; the DeGroot side rests on THEORY-916, Proposed. | Proposed |
| 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 | The half of Omidshafiei et al.’s Theorem 2.4.24 that THEORY-942 does not assert, kept here when the twenty-fourth status review promoted THEORY-942 on the other half (ADR-047, kind 1). The paper’s proof conjugates the interior flow to a volume-preserving one, and then argues inductively that an attractor reaches a vertex. NOTE-1156 found that step asserted, not carried out. | Proposed |
| THEORY-1248 | Catalysts cancel out of the stoichiometric matrix, so a cycle of mutual catalytic production among species is not a stoichiometric cycle: the cycles that carry a reaction network’s steady currents and affinities are fuel-to-waste pathways through it, and who catalyses whom enters only the rates, never the thermodynamic driving | From the definitions of Dal Cengio, Lecomte and Polettini (2023), LIT-1897 (NOTE-1476), and Rao and Esposito (2016), LIT-1889 (NOTE-1464), neither of which says it in these words. It makes exact A13’s “a productive cycle need not be a stoichiometric cycle” from the exchange “Analyze nucleation concepts”. Checked on a reversible rewriting of CASE-189’s first construction. Active: definitional and checked. It does not say that catalytic cycles are thermodynamically idle. They decide which of the states the driving allows is reached, and bistability needs them; but the budget they work within is set elsewhere. | Active |
| THEORY-1250 | A chemostat is an emergent structure of a closed reaction network: when exchange reactions are fast and the reservoir species are abundant by a factor of order Omega, the closed network behaves to leading order as the open one, with its local detailed balance and dissipation, over a window that opens once exchange equilibrates and closes on a time of order Omega, after which a driven network relaxes to equilibrium; neither separation alone suffices | Remlein, Esposito and Avanzini (2026), LIT-1887, read in NOTE-1468: a multiscale expansion in the rate ratio ε and a WKB argument in the abundance ratio Ω, for arbitrary stoichiometry. The lifetime scaling was checked by integrating the paper’s §IX network, T_δ = 0.00483 Ω against its fit 0.0048 Ω + 0.33. Active. It does not say that every driven closed network departs from open behaviour at T ~ Ω, only that it can from then on, nor anything about systems that are not chemical. | Active |
| THEORY-1251 | Mossio, Montévil and Longo take organization, as closure of constraints, to be a theoretical principle of biology, posited a priori as physics posits its phase space and defining the domain together with variation, so that every system meeting both is biological by definition and organisms are one class of such systems; on this reading “organisms are organized” is constitutive, not an empirical finding, and the principle is to be judged by what it makes intelligible | Mossio, Montévil and Longo (2016), LIT-1888, §§1 and 4, read in NOTE-1461; the definitions are Montévil and Mossio’s (LIT-582, TERM-070). The paper states the principle’s status and scope and supports it by history and analogy; it derives nothing from it here. Proposed: what the principle buys is untested in this paper. It does not say that closure is observed in every organism, nor that only organisms realize it. | Proposed |
| THEORY-1266 | In an open reaction network of reversible elementary mass-action reactions, sustained currents need an emergent cycle with nonzero affinity: each chemostatted species either breaks a conservation law or opens an emergent cycle, so one chemostat never suffices; with every emergent affinity zero the network relaxes to a unique equilibrium fixed by its unbroken conserved quantities; and a nonequilibrium steady state dissipates exactly the sum over emergent cycles of current times affinity | Rao and Esposito (2016), LIT-1889, read in NOTE-1464, with the one-chemostat proof of Dal Cengio, Lecomte and Polettini (2023), LIT-1897 (NOTE-1476). Active: rank–nullity, Wegscheider’s condition and two lines of algebra, checked numerically on a reversible rewriting of CASE-189’s first construction, which has a single equilibrium at zero affinity and is bistable only above 3.52 RT. It does not say that a driven network is far from equilibrium, multistable or organized; only that without an emergent affinity it cannot be, and what it costs when it is. | Active |
| THEORY-1267 | A safe set that a nominal barrier-function controller keeps invariant can be left under an arbitrarily small input disturbance, but if the barrier inequality holds with a disturbance penalty, ḣ ≥ −α(h) − ι(‖d‖), the state stays in a larger set {h + γ(‖d‖∞) ≥ 0} with γ class-K, and adding a damping term along the barrier’s input direction to any barrier controller gives this with an enlargement of ‖d‖²/(4ε) in units of h | Kolathaya and Ames (2019), LIT-1893, Theorems 1–2 and Examples 1–2, read in NOTE-1477. Short proofs, checked by hand, and both examples checked by simulation. Active. Matched input disturbances only, local, and the enlargement is in barrier units and quadratic in the universal construction, not proportional to the disturbance. | Active |
| THEORY-1271 | Mossio, Montévil and Longo argue that closure of constraints both stabilizes and propagates variation: it regenerates each constraint beyond the time scale at which it is conserved, and removes a variation only if the variation does not reach the constraints that regenerate the varied one; when it does reach them, the variation propagates and the system either loses closure and disintegrates or comes to realize a new closure, which they say a rock, a flame or a table cannot do | Mossio, Montévil and Longo (2016), LIT-1888, §5, read in NOTE-1461. The conditional is close to definitional. That regeneration actually recovers is a dynamical claim the paper does not model, and the contrast with non-organized systems rests on three examples. Proposed. It does not say how large a variation closure can absorb, nor that recovery follows whenever the structural condition holds. | Proposed |
| THEORY-1273 | In a driven chemical system with two stable states and a finite number of molecules, each maintained state is metastable: it is left by fluctuation after a time exponential in system size, set by a barrier in a nonequilibrium quasi-potential that closes at the saddle-node, and the long-run share of time in each state is fixed by the parameters and system size, not by the initial condition; the deterministic rate equation, where the initial condition decides, holds only on times short of that escape time | Vellela & Qian (2009), LIT-1890, read in NOTE-1474, on the Schlögl model, with Falasco & Esposito’s general frame (LIT-1892, §§III.C, IV.H, VI; NOTE-1472). Active: the slow eigenvalue, the barriers, the initial-condition independence and the closing of the barrier at the fold were checked numerically on the paper’s model. It does not say which state is the more stable without computing the quasi-potential, and the paper’s own large-volume rate formula carries a spurious 1/V factor. | Active |
| THEORY-1292 | Whether a coupled oscillator sustains its own rhythm or is a damped oscillator driven from outside is settled directly by removing the drive, and, because the two arrangements are not dynamically equivalent, may also show in how they respond to a shift of phase: in Leise and Siegelmann’s model, damped peripheral clocks regain their phase relation to the central clock quickly, while self-sustained ones take longer, need strong coupling, and sometimes resynchronize by a long delay instead of an advance | Bechtel and Abrahamsen (2010), LIT-1936, §3.6, p. 328, read in NOTE-1492. Peripheral clocks were once taken for “slaves” of the SCN, damping without it. Later evidence suggested they keep oscillating and lose synchrony, so the SCN is “a conductor” rather than “a ringmaster”. The paper’s reading is that models of both kinds “can reveal the empirical signatures that each may leave”. This is a biological instance of the record’s question where maintenance is realized. It is also a case outside CLAIM-522’s ideal limit: the outside drive does not reproduce the inside law, so a perturbation of state may tell them apart. | Proposed |
| THEORY-1301 | In the Haken–Kelso–Bunz potential V(φ) = −a cos φ − b cos 2φ, switching the parameters in time from no landscape (a = b = 0) to two wells (a = b = 1) to one well (a = 1, b = 0) makes the spread of the relative phase across trials rise while both wells are open and fall after one is removed, the pattern Sánchez-Cañizares compares qualitatively with post-stimulus quenching of trial-to-trial variability in decision tasks; under Gaussian white noise of the stated strength q = 1 the spread of cos φ settles near 0.40, not at 0 | Sánchez-Cañizares (2026), LIT-1992, read in full in NOTE-1506 from the publisher’s open-access PDF. Active for the model’s behaviour, which this reading checked: the bistability condition |a| < 4b analytically, and the rise and fall by re-simulation (20,000 paths). The paper’s statement that the spread decreases “to 0” holds without noise and fails at q = 1. It does not say that the model fits the cited EEG data, which the author calls “only qualitative”, or that the parameter changes are the mind’s (THEORY-1302). | Active |
| THEORY-1302 | Sánchez-Cañizares reinterprets Irruption Theory for binary decisions (IAT*): irruption is the onset of a landscape with one attractor per option and absorption the passage to a single attractor, both correlated in mind and brain, reversing the sign of Froese’s variance hypothesis; the mind’s causal role is to set the landscape’s control parameters by “selecting the relevant physical coarse-graining”, not to push the state, so on this account the mind controls which outcomes exist and not which one the state reaches | Sánchez-Cañizares (2026), LIT-1992, read in full in NOTE-1506. The interpretation is the paper’s (pp. 6–7, 18–20); the last clause is this reading’s description of the control architecture it implies, from the model and simulation of pp. 8–12, in which the choice between wells is settled by initial condition and noise. Proposed: the paper says IAT* “cannot be tested with empirical results without further assumptions” (p. 7). Bears on CLAIM-533 and QUESTION-071. | Proposed |
| THEORY-1306 | In hierarchically organized systems, levels are demarcated by sharp discontinuities in characteristic process and relaxation rates, the embedding level being slower than its components, and a level controls another by constraint only while their rates stay separated: when the cycling rates converge, the control asymmetry and the distinction between the levels are lost, whatever the signal amplitude; Juarrero’s statement of Salthe’s and Allen and Starr’s hierarchy theory | Juarrero (2023), LIT-1995, ch. 12 (pp. 183–191) with ch. 6 (pp. 68, 72, 77), read in NOTE-1532. A philosophical statement of hierarchy theory, quoting Salthe 1985 and Allen, O’Neill and Hoekstra 1984, with examples from academic administration and tumours; no model. Proposed: the rate-demarcation of levels is the hierarchy theorists’, and the loss of control when rates converge is asserted, not shown. It is the closest prior art the record holds for CLAIM-478. | Proposed |
| THEORY-1309 | For a small-noise diffusion the Freidlin–Wentzell quasi-potential equals 2 inf ∫ |b| sin²(θ/2) ds over curves, θ the angle between curve and drift, and the minimizing curve is the path a noise-driven transition follows with probability tending to one; with gradient drift and identity noise that curve runs against the drift, the time-reversed relaxation path, while a far-from-gradient drift, as in the Maier–Stein model above β = 4, makes it depart from the heteroclinic orbit | Heymann and Vanden-Eijnden (2008), LIT-1981, read in part in NOTE-1504 (§§1, 2.1–2.2, 4.1, 5). Active: the geometric formula and the gradient case were re-derived for this reading; the path concentration is Proposition 2.3, whose proof was not read but whose statement is standard Freidlin–Wentzell theory; the Maier–Stein threshold is the paper’s computation agreeing with Maier and Stein’s. It does not say anything at finite noise, or for degenerate noise, which the paper’s ellipticity assumption excludes. | Active |
| THEORY-1314 | Contextual emergence, on which an emergent phenomenon’s parts are necessary but not sufficient, needs a functional boundary between the phenomenon and its context, without which the context could be folded into the base and the account collapse into weak emergence or global supervenience; and the subject of attention is contextually emergent in this sense, its boundary being the skull, meninges and brain stem, so that its macro-level patterns constrain micro-level activity with a power that is not reducible to the parts and needs neither downward nor circular causation | Jennings (2026), LIT-1963, §§4–6, read in NOTE-1508, with Bishop and Atmanspacher 2006 (LIT-1987) and Bishop, Silberstein and Pexton 2022 (LIT-1993), both unread, for contextual emergence. The boundary condition is the paper’s own addition (fn. 15). Proposed, with a sharp edge: the paper’s reason that the boundary resists micro redescription is that its properties “are macro-level ones” (p. 20), which is the point at issue, and its neural evidence fits weak emergence too. Bears on CLAIM-334, whose supervenience premise it denies. | Proposed |
| THEORY-1315 | As a stochastically forced system approaches a fold bifurcation its dominant eigenvalue goes to zero, so it recovers ever more slowly from small perturbations, and under ongoing perturbation this critical slowing down shows as rising lag-1 autocorrelation and rising variance; skewness and flickering are further warnings not caused by slowing; but the same signals, less pronounced, also precede thresholds that are not bifurcations, so they mark proximity to a broad class of thresholds and not which kind, nor when the shift will come | Scheffer et al. (2009), LIT-1956, read in NOTE-1536 (Skimmed) from the version reprinted as chapter 2 of Dakos’s 2011 thesis. Active for the slowing-down result, which is standard (Wissel 1984), checked in the chapter’s Box 2.2 example, and re-derived and simulated here for variance σ²/(2|λ|) and autocorrelation e^{λΔ} since the chapter’s Box 2.3 was not extractable. The scope and timing caveats are the authors’ (p. 31). It does not say the signals have been shown in chaotic systems, which the review calls “less well studied”, or that any of the reviewed empirical cases was checked here. | Active |
| THEORY-1321 | Froese proposes that irruptions combined with attunement, modelled on Watson et al.’s self-optimization of a Hopfield network by occasional resets to arbitrary states plus Hebbian learning, answer the libertarian’s randomness challenge, how behaviour can be effective though underdetermined; but in the model offered the resets are arbitrary, so what makes the behaviour effective is the learned attractor landscape, and nothing in it distinguishes a motivated irruption from noise | Froese (2023), LIT-1950, read in full in NOTE-1521. The first half is the paper’s proposal (§§4, 5.2.3, 6); the second half is this reading’s observation about the model it cites, from the paper’s own description of it (“reset to an arbitrary configuration”). Proposed because the proposal is stated as potential (“holds potential to provide”) and no model with motivated irruptions is given. It bears directly on CLAIM-533 and on the luck objection (THEORY-328, THEORY-040). | Proposed |
| THEORY-1323 | Under weak local noise a many-body operator decays at a rate proportional to its size, so a Hamiltonian’s small-sized integrals of motion, which chaotic dynamics does not spread, decay slowly while generic operators, which it spreads, decay fast; in exact diagonalization of Ising and Heisenberg chains of up to nine sites at weak depolarizing noise, the slowest Lindbladian eigenoperators are linear combinations of the known exact and approximate small integrals of motion, and which approximate ones rank among the slowest moves with the noise strength | Yang and Abanin (2025), LIT-1958, read in NOTE-1534. Active as a statement of what the paper derived exactly and what its exact diagonalization showed at the sizes it reached. It does not say that the correspondence holds at large system size (THEORY-1343), that every integral of motion is found (those above size 3–4 are not), or that slowness ranks conserved quantities independently of the environment. | Active |
| THEORY-1333 | Bishop’s contextual emergence, as stated in his 2024 SEP revision, is an ontological account: stability conditions not given at smaller and faster scales (parameter values, dimensionality, large-scale flows) constrain or enable what is possible and actual at those scales, as convection rolls constrain the motions of fluid molecules, without the brute new causal powers of strong emergence; laws bound possibility and contexts fix actualization | Bishop, SEP “Chaos” (revision of 2024-10-11), LIT-1961, §§6.2–6.3, read in NOTE-1535, summarizing Bishop, Silberstein and Pexton 2022 (LIT-1993, not reached). Active as a statement of the view, checked verbatim. It corrects the record’s gloss (THEORY-1314, NOTE-1508) that Bishop’s contextual emergence concerns only the well-definedness of a higher-level description; it is not evidence that the view is true, and it sets the view against causal powers, not for them. | Active |
| THEORY-1335 | In Juarrero’s constraint account, context-independent constraints take conditions away from equiprobability and context-dependent ones away from independence (after Gatlin); enabling constraints couple separate processes into a coherent dynamic whose constitutive constraints then act top down as second-order governing constraints that confine its components to a subregion of state space; a constraint is defined as what raises or lowers barriers to energy flow without transferring kinetic energy, carries no time-scale condition, and the governing regime is held not to be other than the components | Juarrero (2023), LIT-1995, read in NOTE-1532 (Skimmed; chapters 2, 3, 6, 7, 11, 12, 15 and 16 in full). Active as a statement of what the account is: its kinds of constraint, its definition, and its top-down step, all stated in the text (pp. 25, 40, 67, 71, 79–84). It does not say that the account is true, that constraints transfer no energy, or that her constraint is the record’s (TERM-070’s). | Active |
| THEORY-1340 | Training forms a task-defined, low-dimensional subregion of prefrontal and premotor population activity (a preparatory subspace, an axis of choice organized as a line attractor) that constrains a monkey’s delayed response top down, holding behavior to what the task requires while the trial’s cue and evidence select within it; Juarrero’s reading of Churchland et al. 2012 and Mante, Sussillo et al. 2013 as governing constraints and as neural correlates of prior and proximate intentions | Juarrero (2023), LIT-1995, ch. 15 (pp. 223–232), read in NOTE-1532; the primary studies are not held. The subspaces and their predictive power are the studies’ findings as the chapter reports them. That the subspace controls behavior top down, is “the standing cause of the response” (p. 230) and correlates with prior and proximate intentions (pp. 229–231) is the book’s reading. Proposed for that step: the evidence reported is dimensionality reduction, prediction and a network simulation, with no intervention on the pattern. | Proposed |
| THEORY-1343 | The first-order picture in which a small integral of motion Q of a many-body Hamiltonian is, under weak local noise, the slowest Lindbladian eigenoperator holds at large system size whenever its leak ε out of span{Q} satisfies ε ≲ γ ≪ 1, because the gap controlling the perturbation is not the exponentially small level spacing but one of order min(γL, √(v_Bγ)), set by the ballistic growth of the operators Q leaks into | Yang and Abanin (2025), LIT-1958, §IV, read in NOTE-1534. Proposed because the argument assumes fully chaotic operator growth for the complement, is checked numerically only at nine sites, and is qualified by the authors themselves: hydrodynamic modes give a continuum of slow modes in the Mori limit, so the gap “might not exist at all” outside sectors orthogonal to them. | Proposed |
| THEORY-1348 | In Froese’s Irruption Theory an agent’s motivations make a difference to behaviour only as an increase in its material underdetermination, measurable at best indirectly as neural or behavioural entropy, and such irruptions are by the theory’s own terms undirected: scaled up by chaos, 1/f noise or criticality, they loosen the current organization but “cannot force the formation of the behavior that is to come”, so the theory places behaviour’s direction in the body’s attunement and not in the irruption | Froese (2023), LIT-1950, read in full in NOTE-1521 from PubMed Central’s copy. Active as a statement of what the theory says: its three axioms, its three theses, and the commitment that irruptions are undirected, all in the text (§§5.1–5.2.3). It does not say that the theory is true, that neural entropy measures motivation, or that attunement answers the luck objection (THEORY-1321, Proposed). | Active |