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.
Contextuality — whether outcomes can be explained without reference to the context of measurement — Kochen–Specker and Bell contextuality and their sheaf-theoretic form, generalized (Spekkens) contextuality, noncontextual models and their limits, and contextuality outside physics.
45 of 2004 LIT documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| LIT-003 v2 | A structure theorem for generalized-noncontextual ontological models | Schmid et al. (2020), ARXIV-2005.07161. Take a tomographically local GPT and any convex-linear, empirically adequate, diagram-preserving map M from it into FVect_R. Then M(T) = χ_B ∘ T ∘ χ_A⁻¹ for unique invertible linear maps χ_A (Thm 4.1). Consequences: - Every diagram-preserving quasiprobabilistic model is an exact (non-overcomplete) frame representation (Prop. 4.3). - Every diagram-preserving noncontextual ontological model has |Λ_A| = dim(A), the GPT dimension (Cor. 4.2, §4.3): exactly d² ontic states for a qudit and 4 for a qubit. - The paper also proves (Cor. 3.5) that three things are equivalent in arbitrary compositional scenarios, without needing tomographic locality: a noncontextual ontological model of the operational theory; an ontological model (simplex embedding) of its GPT; and a positive quasiprobabilistic model of its GPT. | Active — read in full 2026-09-25 (NOTE-017); It is the formal backbone of the “classicality = generalized noncontextuality = positive quasiprobability” position that LIT-007 and LIT-019 appeal to. The main theorem is fully proved and short. Read §3–4 and App. B. §5.2 illustrates why tomographic locality is necessary, and its numbers should be checked (see corrections). |
| LIT-007 v2 | Quasi-quantization: classical statistical theories with an epistemic restriction | Spekkens (2014), ARXIV-1409.5041. Take a classical statistical theory over a phase space Ω = R^{2n} or (Z_d)^{2n}, and allow only uniform distributions over V^⊥ + v, where V is an isotropic subspace (a Poisson-commuting set of linear “quadrature” variables) and v ∈ V gives their values. The result is operationally identical to the quadrature quantum subtheory (the stabilizer formalism for odd prime d) for continuous variables and for odd prime d; the proof is that the Wigner functions of that subtheory are exactly these distributions (§IV.A). For d = 2 the two theories are isomorphic element by element but make different predictions, because the qubit stabilizer subtheory is contextual (§IV.B). | Active — read in full 2026-09-25 (NOTE-013); the clearest single statement of the epistricted-theory programme, and the source of the weak/strong nonclassicality split that 2111.13727 and 2005.07161 build on; read it for §II–IV and App. A, not for Table II, which it asserts rather than derives. |
| LIT-016 v2 | The sheaf-theoretic structure of non-locality and contextuality | Abramsky & Brandenburger (2011), ARXIV-1102.0264. For finite measurement scenarios, an empirical model has a factorizable (local / non-contextual) hidden-variable realization iff its compatible family of distributions extends to a global section (Thm 8.1). Over the reals every no-signalling model has such a section (Thm 5.9: “negative-probability LHV” ⇔ no-signalling, because both span the same subspace, of dimension D = Σ_{U∈𝒰}(l−1)^{|U|}, Thm 5.4). Contextuality comes in three strictly nested strengths (probabilistic ⊋ possibilistic ⊋ strong), with the Bell, Hardy and GHZ(n≥3) models as witnesses of each level. | Active — read in full 2026-09-25 (NOTE-016); the founding paper of the sheaf-theoretic approach to contextuality. It is short, fully proved and still the standard reference for its definitions (empirical model, strong contextuality, the global-section criterion). Later work (cohomology, the contextual fraction, the CbD comparison) extends it but does not replace it. |
| LIT-019 v2 | Why interference phenomena do not capture the essence of quantum theory | Catani et al. (2021), ARXIV-2111.13727. The paper models each mode of the interferometer as a classical bit pair (occupation N ∈ Z_2, phase Φ ∈ Z_2). The agent may know only one of N, Φ or N⊕Φ and must be maximally ignorant of the rest. Under that restriction, local and deterministic dynamics (beamsplitter: swap N_L with Φ_L⊕Φ_R) reproduce exactly the 50-50, ϕ ∈ {0, π} Mach–Zehnder statistics (output port L with probability 1 or 0; ½/½ with a which-way detector). The same holds for the Elitzur–Vaidman bomb tester (success probability ¼), delayed choice and the quantum eraser (interference within each P_A outcome class, none within each Q_A class). So this “TRAP” slice of interference phenomenology cannot, on its own, force wave–particle complementarity, observer-dependence or nonlocal/retrocausal influence. The paper leaves aside the phenomenology it concedes is nonclassical (unbalanced beamsplitters, the functional form of the duality relation, Hardy’s interferometer). | Active — read in full 2026-09-25 (NOTE-018); It is the worked, experiment-by-experiment defence of placing interference in the weakly nonclassical column of LIT-007’s Table II, and it has the clearest statement of the Spekkens-school methodology (define classicality, formalize it, prove a no-go). Read §3, §4.3, §5.1 and App. A. Its noncontextuality claim for the toy field theory is asserted, not proved. |
| LIT-077 v2 | What is quantum in quantum randomness? | Grangier & Auffèves (2018), ARXIV-1804.04807. The paper argues informally, with no equations anywhere in it, that an “elementary” system with a fixed finite number N of exclusive modalities, placed in contexts that can change, must behave unpredictably when the context changes: a return to C1 after C2 cannot be certain, or there would be N² > N joint modalities (p. 4). It asserts that adding one condition, that the context can change continuously, makes the N×N modality-transition matrix unistochastic, i.e. Born-rule, but defers that proof to refs [14, 15] (p. 5). The anti-reductionist conclusion follows from Postulate 0, which declares contexts non-contextual by definition, not from an argument against decoherence. | Active — read in full 2026-09-25 (NOTE-044); a short, equation-free position paper whose one argument of its own is an informal N = 2 reductio; read it for the position, and read the derivation it cites — Auffèves & Grangier, Sci. Rep. 2017 (arXiv 1610.06164) and arXiv 1801.01398, neither held here — for the result. |
| LIT-263 v2 | Kochen-Specker contextuality | Budroni et al. (2021), ARXIV-2102.13036. A review that fixes a minimal definition of Kochen–Specker contextuality: no global joint distribution has the context distributions as marginals (Fine’s theorem). The sheaf-theoretic framework is one of several formulations it calls substantially equivalent. It argues that contexts defined by joint measurability alone make every nondisturbing correlation classically realizable, so contexts must be repeatable, nondisturbing measurements. It treats Spekkens’s generalized contextuality as a different notion, covered only to mark the difference. | Active — read 2026-10-14 (NOTE-844) in full; worth reading as the record’s reference for Kochen–Specker contextuality: the marginal-problem definition (Fine’s theorem) with the sheaf form among its equivalents, why contexts need more than joint measurability, how experiments model disturbance, and Vorob’ev’s acyclicity theorem. Read it knowing that it is a review whose results are mostly cited, that it describes Spekkens’s notion only to set it apart and never derives the relation between the two, and that its account of maximally noncontextual models is one noise model among several, with no behavioural data. |
| LIT-264 v2 | Is there contextuality in behavioral and social systems? | Dzhafarov, Zhang & Kujala (2016; arXiv 2015), ARXIV-1504.07422. Under Contextuality-by-Default a binary cyclic system is noncontextual iff ΔC ≤ 0, a criterion that separates context-dependent marginals from contextuality and reduces to Leggett–Garg, CHSH and KCBS when marginals agree. Every behavioural and social data set re-analysed meets it: 73 question-order pairs, Schröder staircases, conjoint choices, 23 primed word combinations and millions of dichotomisations of psychophysical matching. The QQ equality of the quantum question-order model implies it. | Active — read 2026-10-09 (NOTE-235); worth reading as the paper that separates context-dependent response distributions (“inconsistent connectedness”) from contextuality proper in behavioural data, and finds no contextuality in any data set it re-analyses: 73 question-order pairs, Schröder staircases, conjoint animal–sound choices, 23 primed word combinations and psychophysical matching. Its sharpest result is that the quantum question model’s QQ equality implies noncontextuality, so the best-supported quantum-cognition model of order effects predicts its absence. Read with the caveats that the criterion it applies is proved elsewhere, that the absence of contextuality is offered as a working hypothesis, and that its worked Rose–Jackson example contains a labelling slip (the conclusion survives). |
| LIT-265 v2 | The contextual fraction as a measure of contextuality | Abramsky, Barbosa & Mansfield (2017), Phys. Rev. Lett. 119, 050504. For any no-signalling empirical model in the sheaf-theoretic framework, the contextual fraction CF (one minus the largest weight of a noncontextual sub-model) is a linear programme. Its dual gives a generalised Bell inequality whose normalised violation equals CF, the largest attainable (Theorem 1). CF does not increase under relabelling, translation of measurements, coarse-graining or mixing, and is multiplicative in NCF under products (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC computing a non-linear function (Theorem 3). | Active — read 2026-10-09 (NOTE-236); worth reading as the reference definition of the contextual fraction CF. CF is the weight left over when the largest noncontextual sub-model is taken out of a no-signalling empirical model. It is a linear programme over the incidence matrix (Eq. 3). Its dual reads off a Bell inequality whose normalised violation is exactly CF, the maximum over all Bell inequalities (Theorem 1). CF is invariant under relabelling, cannot increase under translation of measurements, coarse-graining of outcomes or mixing, and composes as max under choice and as 1 − NCF·NCF′ under product (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC and of k-consistent games (Theorems 3–4). All proofs are complete in the supplement. It is defined only for models with compatible marginals. |
| LIT-266 v2 | Contextuality for preparations, transformations, and unsharp measurements | Spekkens (2004), arXiv:quant-ph/0406166. Defines a noncontextual ontological model as one in which operationally equivalent procedures (preparations, measurements, transformations) get identical representations; shows the traditional Kochen–Specker notion is the special case of measurement noncontextuality for sharp measurements plus outcome determinism, proves that preparation noncontextuality implies that outcome determinism, and proves preparation contextuality already for a qubit. | Active — read 2026-10-14 (NOTE-748) in full; worth reading as the source of generalized noncontextuality and as the bridge between it and the Kochen–Specker notion. It proves preparation, transformation and unsharp-measurement contextuality for a qubit, shows the traditional notion is measurement noncontextuality plus outcome determinism for sharp measurements, and proves that preparation noncontextuality implies that determinism. Read it knowing that every result assumes exact operational equivalence and gives no inequality; that the bridge (§VIII.A) uses the full set of quantum preparations; and that “any classical theory is necessarily noncontextual” is a footnote assertion. It argues from ontological economy and does not mention Leibniz. |
| LIT-270 v2 | Identifying Quantum Structure in AI Language: Evidence for Evolutionary Convergence of Human and Artificial Cognition | Aerts et al. (2025), DOI-10.3390/e28060622. Two findings are reported. First, prompted as test subjects on Aerts & Sozzo’s “The Animal Acts” concept-combination test, ChatGPT and Gemini give CHSH values above 2: 4 in a guided conversation, and 2.25 and 3.09 (ChatGPT) and 2.12 and 4 (Gemini) over 81 prompted repetitions, against 2.42 for the 81 humans in 2011. Second, the rank–frequency curve of one 2,861-word Gemini story fits a Bose–Einstein form with energy E_i=(i−1)^0.8 far better than a Maxwell–Boltzmann form. The authors take these as showing that LLM “code is structurally quantum” (§5). The CHSH analysis never checks the strongly context-dependent marginals in its own tables. One reported expectation value disagrees with its table. The Bose–Einstein fit is numerically a Zipf–Mandelbrot law. | Rejected — read in full 2026-09-27 (NOTE-243); not worth a reader’s time for its claims, because the data cannot carry them. The CHSH values come from tables whose marginals shift by up to 0.96 between contexts, the confound the record already holds against this very test design (LIT-264). The “Bose–Einstein versus Maxwell–Boltzmann” contest compares a Zipf-like curve with an exponential one on a single primed story. No goodness-of-fit statistic is reported, and there is no significance test for any LLM result. |
| LIT-276 v2 | Measurement as Sheafification: Context, Logic, and Truth after Quantum Mechanics | Ghose (2025), ARXIV-2512.12249. A programmatic essay, with no theorem, model or computation of its own. It restates Abramsky & Brandenburger’s criterion (contextuality and non-locality as the absence of a global section of a presheaf over measurement contexts) and adds three things. It declares measurement to be “sheafification” of that presheaf, without specifying a site, a topology or any sheafified object. It asserts that the Ghose–Patra seven-valued logic is a finite Heyting algebra, without giving an order or operations. It proposes that a quantum potential scaled by 0 ≤ λ ≤ 1 (λ = 1 quantum, λ = 0 classical Hamilton–Jacobi) carries the presheaf continuously to a sheaf, which is asserted, not modelled. Its one worked formula, v2’s Čech “obstruction class” [δs], is a coboundary and so is zero by construction. | Rejected — read in full 2026-09-27 (NOTE-249); not worth a reader’s time for its formal claims. Every correct formal ingredient is Abramsky & Brandenburger’s (LIT-016) or Abramsky, Mansfield & Barbosa’s, and both are better read in the original. What the essay adds is either unconstructed (the sheafification, the site, the seven-element Heyting algebra, the link from λ to cohomology) or wrong as stated: v2’s obstruction class is zero by construction, a sheaf topos is not in general Boolean, and classical distributions do not form a sheaf. Its only value is as a record of a position, the Bohrian reading of contextuality plus the Jaina analogy. |
| LIT-277 v2 | The Cohomology of Non-Locality and Contextuality | Abramsky et al. (2011), ARXIV-1111.3620. For an empirical model e on a measurement cover 𝒰, a section s in the support of context C₁ gets a class γ(s) ∈ Ȟ¹(𝒰, F_{C̄₁}), relative Čech cohomology of the presheaf F = F_ℤ S_e of formal ℤ-linear combinations of support sections. γ(s) = 0 iff s belongs to a compatible family of such ℤ-combinations (Prop 4.2), so a non-vanishing γ(s) is a sufficient but not necessary witness of contextuality (Prop 4.3). γ is non-zero for every section of the PR box, GHZ, the triangle, the 18-vector Kochen–Specker set and the Peres–Mermin square, but it vanishes on the Hardy model and on one strongly contextual Kochen–Specker cover (§8). | Active — read in full 2026-09-27 (NOTE-250); worth reading as the short original source of the cohomological obstruction to contextuality. It gives the exact definition (a relative Čech class, with coefficients the free abelian group on support sections), the vanishing criterion and its two known failures. Its results are small propositions with complete proofs, plus machine-checked mod-2 computations that I reproduced. It calls itself “work in progress” (§1), and the conceptual development is in Abramsky et al.’s CSL 2015 “Contextuality, Cohomology and Paradox” (not read). A reading of that paper would show whether it supersedes this one. |
| LIT-278 v2 | Contextuality, Cohomology and Paradox | Abramsky et al. (2015), ARXIV-1502.03097. For a model whose outcomes live in a commutative ring R, “All-vs-Nothing” (AvN_R) means the R-linear equations satisfied by every support section of every context have no global solution. Theorem 21 proves AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S), so every AvN model is strongly contextual with a non-vanishing Čech obstruction on every section. Theorem 4 shows that every state stabilised by an “AvN triple” of n-qubit Pauli operators yields such a model. The “paradox” half proves much less: an informal identification of Liar cycles with strongly contextual support tables, one exact case (the length-4 Liar cycle is the PR box), and the cohomology of paradox deferred to future work. | Active — read in full 2026-09-27 (NOTE-251); worth reading as the place where the cohomological witness is shown complete for a named, large class (every All-vs-Nothing model, over any ring). It is also where the obstruction is defined as a connecting homomorphism. It does not supersede LIT-277. It re-derives none of LIT-277’s computations, and it drops its negative results: the Hardy false positive and the strongly contextual cover with γ = 0. Its “paradox” title-word is supported by about one page of analogy and one worked identification. |
| LIT-279 v2 | On the Cohomology of Contextuality | Carù (2017), ARXIV-1701.00656. Carù gives a possibilistic model on the two-party, two-setting, four-outcome Bell cover that is strongly contextual, yet whose Čech obstruction vanishes on all 22 support sections (Fig 4, Table 1). That refutes Abramsky–Mansfield–Barbosa’s Conjecture 8.1 for any symmetry or connectedness condition the Bell cover meets. I confirmed the refutation over ℤ and ℤ/2. The rest is weaker. Odd-degree “higher obstructions” form a hierarchy (Prop 5.5), but every level above the first is empty for every no-signalling model (Prop 5.6). Prop 6.3 restates the known torsor description of Ȟ¹. Prop 6.1 is false in the “only if” direction; the PR box refutes it. | Active — read in full 2026-09-27 (NOTE-252); worth reading for one result, the first complete false negative of the cohomological witness (§4 and Appendix A, about two pages), which is correct as I verified it. §5 proves its own programme empty for no-signalling models, and §6 is a standard correspondence plus one false proposition. The follow-up, arXiv 1807.04203, may supersede it; that is unverified, since it was not read. |
| LIT-280 v2 | Towards a complete cohomology invariant for non-locality and contextuality | Carù (2018), ARXIV-1807.04203. For cyclic scenarios, where the contexts’ intersection graph is a single chordless N-cycle, Carù proves that ℤ/2 Čech cohomology of the (N−1)-th iterated “joint model” is a complete invariant for logical and strong contextuality (Thm 7.7). I confirmed this on the Hardy model, on his own 2017 counterexample (detected on all 22 sections at level 3) and on 659 random contextual cycle models. The extension beyond cycles does not hold as stated. Prop 8.2 is false on the paper’s own Table 5 model. Worse, the one non-cyclic case the paper claims to repair, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa, is never detected at any level under the paper’s definitions (my proof and computation). That refutes the paper’s closing Conjecture 9.1. | Active — read in full 2026-09-27 (NOTE-253); worth reading for §§4–7, a correct and genuinely new result: a complete cohomological test for contextuality on cyclic scenarios, which closes the Hardy and Carù-2017 gaps. Read it knowing that §8’s generalisation, its §8.1 Kochen–Specker example and Conjecture 9.1 are wrong as stated. |
| LIT-281 v2 | Logical and Topological Contextuality in Quantum Mechanics and Beyond | Carù (2019), DOI-10.5287/ora-kznbp4k4y. The thesis’s cohomology chapters republish the 2017 and 2018 papers with the “joint model” renamed the “line model”. The definitions, Thm IV.32 (complete on cyclic scenarios at level |M|−1), Prop IV.34, Thm IV.36 and Conjecture IV.41 carry over unchanged. So does every defect found in the preprint. Under the thesis’s own Def IV.26, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa is still detected on only 6 of 15 sections at levels 0–3 (my recomputation), and a lifted ℤ/2 family shows it is never fully detected at any level. The context-wise reading the worked examples use detects all 15 sections at level 2. The genuinely new material lies outside the cohomology programme: a probabilistic line-model construction (§IV.7), a valuation-algebra “theory of disagreement” with inference algorithms (Ch V, bound O(k^n·l^{k(n−1)+1}) on (n,k,l) scenarios), and the AvN-triple characterisation of stabiliser states (Ch VI). | Active — read in full 2026-09-27 (NOTE-254); worth reading as the consolidated source for Carù’s cohomology programme (Chs III–IV), for the valuation-algebra reformulation of contextuality (Ch V) and for the AvN triple theorem (Ch VI). Chs III–IV repeat, uncorrected, the errors recorded for LIT-279 and LIT-280, so read them with NOTE-252 and NOTE-253 at hand. |
| LIT-298 v2 | The problem of hidden variables in quantum mechanics | Kochen & Specker (1967), DOI-10.1512/iumj.1968.17.17004. Theorem 1 exhibits a finite partial Boolean algebra D, generated by 117 directions in E³, that admits no homomorphism onto Z₂. §4 embeds D in the projections of the spin-1 space of orthohelium’s 2³S₁ state, via α ↦ P J_α². So no assignment of 0/1 values to those quantum propositions respects all the Boolean relations within each commeasurable set. Theorem 0 makes the target precise: a partial Boolean algebra embeds in a Boolean algebra iff its Z₂-homomorphisms separate points. That is, hidden variables in the paper’s sense need enough “global valuations”, and quantum mechanics in dimension 3 has none. The same paper builds a classical hidden-variable model for spin ½ (§6). It argues that von Neumann’s no-go needed additivity on non-commuting observables (Theorem 3). | Active — read in full 2026-09-29 (NOTE-279); worth reading. It is the primary source of the Kochen–Specker theorem and of the partial-algebra formulation that the sheaf and topos programmes generalise. The 117-vector construction is superseded for economy by later small KS sets (e.g. Peres’s 33, Cabello’s 18 in dimension 4); the review LIT-263 covers those. |
| LIT-299 v2 | A topos foundation for theories of physics: III. The representation of physical quantities with arrows δ̆°(A): Σ̲ → ℝ̲≽ | Döring & Isham (2008), DOI-10.1063/1.2883777. Each bounded self-adjoint  becomes an arrow δ̆^o(A) : Σ → ℝ^⪰, where ℝ^⪰ is the presheaf of real-valued order-reversing functions on ↓V. At λ ∈ Σ_V it sends V′ ⊆ V to λ(δ^o(Â)_{V′}), and δ^o(Â)_V is the spectral-order-least operator in V above Â. The map  ↦ δ̆^o(A) is injective, but ℝ^⪰ is only an additive monoid, not the real-number object. Its Grothendieck completion k(ℝ^⪰) is an abelian-group object whose global elements are exactly the functions of bounded variation on V(H). | Active — read in full 2026-09-29 (NOTE-290); worth reading as the source of the quantity-value object ℝ^⪰ and of the explicit statement that Gel’fand points of a context are ultrafilters of its projection lattice. Its results are small, mostly proved or cited from de Groote and Döring. The physically motivated parts (k(ℝ^⪰), “intrinsic dispersion”, unitaries) are definitions with no consequences drawn. |
| LIT-325 v3 | A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations | Isham & Butterfield (1998), DOI-10.1023/A:1026680806775. Isham & Butterfield show that, for bounded self-adjoint operators with discrete spectrum, a global section of the spectral presheaf Σ(Â) = σ(Â) on the category O_d, whose arrows are functional relations B̂ = f(Â), is literally a valuation obeying the value rule and FUNC. A global section of the dual presheaf D(W) = Hom(W, {0,1}) on the poset W of Boolean subalgebras of projectors is literally a Kochen–Specker 0/1 assignment. The Kochen–Specker theorem (cited, not reproved) is therefore equivalent to “no global sections” when dim H > 2 (§§2.2–2.3). In place of the missing global valuations, they construct contextual truth values in the Heyting algebra of sieves: ν^ρ(A ∈ Δ) = {f: B̂ →  | tr(ρ Ê[B ∈ f(Δ)]) = 1}. For every density matrix these satisfy functional composition, null, monotonicity, exclusivity and unit conditions (§4.4, §5.3, Theorem 5.1). | Active — read in full 2026-09-29 (NOTE-271); worth reading as the founding paper of the topos approach to quantum theory and the origin of “Kochen–Specker = no global section of a presheaf over contexts”, credited as such by Abramsky & Brandenburger (LIT-016). The reformulation itself is definitional and its proof is two paragraphs. The paper’s own contribution is the sieve-valued generalized valuation, whose later development (von Neumann algebras as base, daseinisation) lives in Parts III–IV and in Döring–Isham (LIT-343, LIT-335, LIT-299, LIT-310). |
| LIT-335 v2 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory | Döring & Isham (2008), DOI-10.1063/1.2883742. A projector P̂ is mapped to the clopen sub-object δ(P̂) of the spectral presheaf Σ over the poset V(H) of abelian von Neumann subalgebras, by “daseinisation” (de Groote’s V-support): δ(P̂)V = the least projector in V above P̂. The map is injective and preserves ∨, but only δ(P̂∧Q̂) ⪯ δ(P̂)∧δ(Q̂). This represents the propositional language PL(S) in the Heyting algebra Sub_cl(Σ). A pure state gives the truth object T^ψ, and “A ε Δ” gets the sieve-valued truth value {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}. | Active — read in full 2026-09-29 (NOTE-292); worth reading as the core reference for “quantum propositions as clopen sub-objects of the spectral presheaf” (daseinisation, Sub_cl(Σ), truth objects). It is a cleaned-up restatement of Isham–Butterfield (LIT-325) plus de Groote’s operation, not a new no-go result. Several proofs are sketched or omitted, and the “no global elements” statement it relies on needs dim H > 2, which it never says. |
| LIT-343 v2 | A topos foundation for theories of physics: I. Formal languages for physics | Döring & Isham (2008), DOI-10.1063/1.2883740. The programme paper. Its contention is that a theory of a physical system S is a representation of a typed “local language” L(S) (ground types Σ and R, function symbols A : Σ → R) in a topos. Physical quantities become arrows Σ_φ → R_φ, and propositions become sub-objects of Σ_φ, which form a Heyting algebra. Classical physics is the representation in Sets. The paper proves nothing. Its one argument is that the axiom ¬(A ε Δ) ⇔ A ε ℝ∖Δ is not validated by non-Boolean Heyting representations (§3.2.1). | Active — read in full 2026-09-29 (NOTE-277); worth reading as the statement and vocabulary of the Döring–Isham programme (neo-realism, PL(S) and L(S), state object and quantity-value object, representation in a topos). It is discursive and contains no results, and every piece of quantum mathematics is deferred to papers II–III (LIT-335, LIT-299). |
| LIT-378 | ’What is a Thing?’: Topos Theory in the Foundations of Physics | Döring & Isham (2008), arXiv:0803.0417. A 212-page restatement of the topos programme of papers I–IV. A theory of a system S is a representation of a typed language L(S) in a topos. In quantum theory the topos is presheaves on the poset V(H) of abelian von Neumann subalgebras, propositions are clopen sub-objects of the spectral presheaf Σ via daseinisation, a pure state gives sieve-valued truth values ν(A ε Δ; ψ)V = {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}, and each bounded self-adjoint  becomes an arrow Σ → ℝ^↔ whose values are monotone intervals over subcontexts. Classical physics fits its axioms for composite systems exactly; quantum theory fits the disjoint sum but not the tensor-product composite. | Active — read in full 2026-10-01 (NOTE-326); worth reading as the one-volume statement of the Döring–Isham topos programme. It is, by the authors’ own account (p. 14), “partly an amalgam” of papers I–IV (LIT-343, LIT-335, LIT-299, LIT-310), with corrections and some new material. The new material is the value of a quantity in a pseudo-state and the inverse-image link between L(S) and PL(S) (§§8.5–8.8, Thm 8.6), the algebra of the candidate quantity-value objects (§9.3), the Boolean-subalgebra base category (§5.5.3), Kochen–Specker as a lifting obstruction (§6.6), and a long speculative §14–15. It adds no new no-go result. Its proofs are as sketchy as the series’ at the same places: Sub_cl(Σ) is Heyting (Thm 16.1) with ⇒ left unwritten, and ι : O → P_clΣ natural and monic is left to the reader. A reader who has read I–IV gains only §§8.5–8.8, 9.3 and 14; a reader new to the programme should start here. |
| LIT-380 v2 | A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category | Hamilton et al. (1999), arXiv:quant-ph/9911020. Hamilton, Isham & Butterfield move the series’ base category from operators (O) and Boolean subalgebras (W) to the poset V of commutative von Neumann subalgebras of B(H). There they restate Kochen–Specker as “the spectral presheaf Σ(V) = Gelfand spectrum of V has no global elements” (dim H > 2, cited, §2.2). They identify the coarse-graining presheaf G(V) = L(V), G(i)(P̂) = inf{Q̂ ∈ L(V₂) | P̂ ≤ Q̂} (Def. 3.1), with the presheaf CloΣ of clopen subsets of the spectra; the isomorphism is asserted, and part IV proves it. A state ρ yields the sieve-valued valuation ν^ρ_V(P̂) = {V′ ⊆ V | ρ(G(P̂)) = 1} (Def. 3.4) and the support projectors inf{P̂ ∈ L(V) | ρ(P̂) = 1}. These form a global element of G and hence an “interval valuation”, a subobject of Σ (§§4.2–4.4). The probability-r valuations give subobjects of G but not global elements. | Superseded — by LIT-382; read in full 2026-10-01 (NOTE-327); covered better by part IV (Butterfield & Isham 2002, read alongside), which restates this paper’s V-framework definitions nearly verbatim, proves the G ≅ CloΣ isomorphism that this paper only asserts and develops its interval valuations into theorems, and by Döring–Isham II (LIT-335), whose spectral presheaf over V(H) and clopen propositions are the mature form; it remains the citation for introducing V as the base category (as LIT-335 credits it). |
| LIT-381 v2 | A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues | Butterfield & Isham (1998), arXiv:quant-ph/9808067. Butterfield & Isham argue that the sieve-valued “generalised valuations” of part I are not a quantum peculiarity but the natural valuations for any presheaf of propositions on any small category. Three supports are offered. (1) Classical physics: the value presheaf Υ(Ā) = Ā(S) on the category M of measurable functions has global sections (one per microstate, eq. 3.7), yet a macrostate R ⊆ S yields a sieve-valued valuation ν^R(A ∈ Δ) = {f_M | B̄(R) ⊆ f(Δ)} (eq. 3.10) with the same FUNC property as the quantum ν^ρ. (2) A proof that a sieve-valued valuation on any presheaf G obeys generalised FUNC, ν(B, G(f)(d)) = f*(ν(A, d)), iff it is a natural transformation G → Ω, i.e. a subobject of G (Theorem 4.1). (3) An informal argument, explicitly “not a genuine deduction” (§5.2), that if partial truth is fixed by which weakenings of a proposition are totally true, the truth value is a sieve. | Active — read in full 2026-10-01 (NOTE-329); worth reading as the series’ conceptual companion and the only place it states the classical analogue and the partial-truth motivation; the mathematics is light (one short theorem), and the sieve semantics it motivates is carried on in part III, part IV and Döring–Isham (LIT-343, LIT-335). |
| LIT-382 v2 | A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations | Butterfield & Isham (2001), arXiv:quant-ph/0107123. Butterfield & Isham show, over the poset V of commutative von Neumann subalgebras, that sieve-valued valuations and “interval valuations” (assignments of support projectors, or of subsets of Gelfand spectra) determine each other under stated conditions. If supports s(α, V) = inf{P̂ | α_V(P̂) = true_V} form a global element of the coarse-graining presheaf G, then α_{V₁}(P̂) = {V₂ ⊆ V₁ | G(s(α, V₁)) ≤ G(P̂)} is a sieve-valued valuation obeying FUNC (Theorem 3.1). There is an analogue for “tight” subobjects of Σ (Theorem 3.2). Both are satisfied by every state’s valuation ν^ρ. The ordering ≤ (or ⊆) is shown to be the natural relation securing sievehood, null, monotonicity, exclusivity and unit (§4). The review §2.4 also supplies the proof, via state extension, that G ≅ CloΣ: restricting the clopen set of P̂ to V₂ gives the clopen set of its coarse-graining. Part III had only asserted this. | Active — read in full 2026-10-01 (NOTE-328); worth reading as the finished statement of the series’ von Neumann-algebra framework. It proves the G ≅ CloΣ isomorphism (coarse-graining = direct image of clopen regions) that Döring–Isham’s daseinisation relies on, and it shows that a state’s sieve-valued truth values are fixed by its support, a global element of G. Its two theorems are correct but close to definitional, so its value lies in the framework rather than in depth. |
| LIT-389 v2 | Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity | Isham & Butterfield (1999), arXiv:gr-qc/9910005. Isham & Butterfield argue, without proving anything new, that presheaf toposes over a base of “contexts” suit physics because the subobject classifier Ω is fixed by the base. They say this answers the objection that many-valued logics are arbitrary (§2.2), and that it keeps the logic distributive (§1.1). They then sketch seven applications. Five are speculative one-paragraph proposals (frames, observers, causal sets, TQFT, synthetic-differential-geometry time in continuous histories). One (§3.7) recaps their Kochen–Specker results: KS ⇔ the dual presheaf D(Â) = Hom(W_A, {0,1}) on the operator category O has no global sections (dim H > 2), and every state ψ gives a FUNC-obeying sieve-valued valuation ν^ψ(A ∈ Δ) = {f_O : Â → B̂ | Ê[B ∈ f(Δ)]ψ = ψ}. | Active — read in full 2026-10-01 (NOTE-334); worth reading only as the programme’s short, informal manifesto as of 1999, written before parts III–IV appeared. It joins the Kochen–Specker valuations to the continuum scepticism (real-valued quantities, space and probabilities) that the Döring–Isham programme later develops, and the Literalism/Extra-Values framing appears nowhere else in the record. Its one technical section is a recap of KS I–II, and a reader of LIT-325, LIT-381 and LIT-378 loses little by skipping it. |
| LIT-390 v2 | Topos Theory and Consistent Histories: The Internal Logic of the Set of All Consistent Sets | Isham (1996), arXiv:gr-qc/9607069. Isham takes the poset B of all Boolean subalgebras (“windows”) of the history orthoalgebra UP, ordered by coarse-graining, and shows that the presheaf A_d(W) = {α | some d-consistent W′ ⊇ W contains α} is a subobject of the constant presheaf ΔUP, whereas the naive “realisable in W” assignment is not a presheaf at all (§4.1–4.2). The probability claim ⟨α, p⟩ then gets the sieve-valued semantic value {W ⊆ W₀ | ∃ d-consistent W′ ⊇ W with α ∈ W′} if d(α, α) = p and ∅ otherwise (4.16), a global element 1 → Ω in Set^B, so the logic of “all consistent sets at once” is the distributive but non-Boolean Heyting algebra of sieves on B. | Active — read in full 2026-10-01 (NOTE-333); worth reading as the origin of the Isham school’s sieve-valued, coarse-graining truth values, which the Kochen–Specker series names as its motivation (KS I, ref. [13]); it is the only place the construction is applied to consistent histories, it is short and elementary, and its constructions are correct as stated, but read it knowing that the Heyting values measure only whether α can be embedded in a consistent set (the probability d(α, α) is the same in every window), so it says nothing about the conflicts between consistent sets that motivate it. |
| LIT-777 | Context–content systems of random variables: The Contextuality-by-Default theory | Dzhafarov & Kujala (2016), J. Math. Psychol. 74:11–33. Contextuality-by-Default for finite systems of categorical random variables, indexed by content and context. A system is noncontextual when some coupling of its context bunches makes every connection a maximal coupling; the test is a linear program, the binary cyclic criterion is stated (proved elsewhere), and the minimal total variation of a signed maximally connected quasi-coupling, minus 1, measures the degree of contextuality. | Active — read 2026-10-09 (NOTE-600); worth reading as the reference statement of Contextuality-by-Default: each measurement gets a separate random variable in each context, and a system is noncontextual when some coupling of its contexts makes every content’s copies coincide as often as their own distributions allow (maximal couplings). This separates context-dependent marginals (“direct influences”) from contextuality, turns the test into linear-programming feasibility (Theorem 4.1), and, since a signed such coupling always exists (Theorem 6.1), measures contextuality as the least total variation above 1. The definition was already being replaced by “multimaximal” couplings when it appeared (footnote 10). |
| LIT-788 | The Geometry of Causality | Gogioso & Pinzani (2023), arXiv 2303.09017, Part 3 of the trilogy. Empirical models on any cover of a space of input histories are the points of a causaltope, a product of simplices cut by linear causality equations, so causal fractions are linear programs. Causal separability relative to a given ambient space lets entangled or contextually controlled quantum switches witness indefinite order once known no-signalling constraints are imposed, with the causally separable fraction bounded below by the separable local fraction. | Active — skimmed 2026-10-09 (NOTE-624); worth reading as the geometric half of the trilogy and the successor of the withdrawn LIT-813’s polytope and causal-fraction results: the empirical models on any cover of a space of input histories are exactly the points of a “causaltope”, the product of simplices sliced by linear causality equations derived from the space’s topology, so supported fractions and causally separable fractions are linear programs. Causal separability is defined relative to an ambient space rather than to the indiscrete one, which lets quantum switches with entangled or contextual control witness indefinite causal order once known no-signalling constraints are imposed (and not otherwise); the causally separable fraction is bounded below by the separable local fraction, and in the switch examples it tracks, and sometimes equals, the local fraction (“contextual causality”). The BFW model is 0% supported by every causally complete 3-event space and 100% by each of three indefinite orders. Read in part: introduction, causaltopes, standard causal separability and the examples. |
| LIT-790 | Contextuality in Canonical Systems of Random Variables | Dzhafarov, Cervantes & Kujala (2017), Phil. Trans. R. Soc. A 375:20160389. Proposes that contextuality be judged only on a system’s canonical representation: every variable, and every joining or coarsening of interest, replaced by its binary splits, with connections coupled multimaximally (pairwise maximally). For two content-sharing k-valued variables with all splits kept, the canonical system is noncontextual if and only if one variable nominally dominates the other (Pr below the other’s for at most one value; Theorem 4.6). For continuous densities with all splits, any difference in distribution is contextual. | Active — read 2026-10-09 (NOTE-640); worth reading as the step that makes Contextuality-by-Default representation-dependent. Every variable is replaced by binary splits (dichotomizations) and connections are coupled multimaximally, so a verdict belongs to a chosen set of splits, not to the measurements alone. Its one worked result is striking: if every split of two content-sharing k-valued variables is kept, that pair alone is noncontextual exactly when one variable “nominally dominates” the other, i.e. its probabilities fall below the other’s for at most one value (Theorem 4.6). For continuous variables the analogue makes any difference in distribution count as contextuality. The authors present this as opening behavioural contextuality, not as a reductio. Proofs are in the supplement and contain small slips that do not affect the results. |
| LIT-808 | The Topology of Causality | Gogioso & Pinzani (2023), arXiv 2303.07148, Part 2 of the trilogy. Extends Abramsky–Brandenburger from no-signalling to arbitrary, possibly dynamical or indefinite, causal structure: causality is continuity in the lowerset topology, causal functions form a presheaf that fails to be a sheaf exactly on spaces with “solipsistic contextuality”, empirical models live on any open cover, and standard empirical models on causal switch spaces (total orders included) are always local, so the Leggett–Garg model is local for its own total order. | Active — read 2026-10-09 (NOTE-612); worth reading as the direct successor of the withdrawn LIT-813’s sheaf construction, now over spaces of input histories (LIT-800) rather than a “locale of inputs”: a joint input–output function is causal when each event’s output depends only on the histories with that event as a tip, causal functions are exactly free assignments of outputs to tips (up to tip-equivalence in non-tight spaces), and an extended function is causal iff it is continuous for the lowerset topology. Causal functions form a separated presheaf that is a sheaf exactly when the space admits no “solipsistic contextuality”, so tight spaces give a sheaf and some dynamical causal structures do not, letting deterministic local data fail to glue. Empirical models are compatible families of distributions over causal functions on any open cover, from the fully solipsistic to the classical; standard empirical models on causal switch spaces, total orders included, are always local. Source of THEORY-171. |
| LIT-813 v2 | The Sheaf-Theoretic Structure of Definite Causality | Gogioso & Pinzani (2021), QPL 2021, EPTCS 343:301–324; withdrawn by the authors in 2024. Extends Abramsky–Brandenburger to a finite causal order. Inputs live on lower sets, sections are causal functions (an output depends only on inputs in its past), and an empirical model is a causal conditional distribution. Locality is a global section, equivalently a mixture of deterministic causal functions (Proposition 21). The models form a polytope cut out by linear causality equations, and pre-orders sketch indefinite causality. The construction’s “locale of inputs” is not a locale, as the authors’ withdrawal says. | Superseded — by LIT-800, LIT-808, LIT-788; read 2026-10-09 (NOTE-629), and superseded by its authors’ trilogy: LIT-800 (The Combinatorics of Causality, arXiv 2206.08911), LIT-808 (The Topology of Causality, 2303.07148) and LIT-788 (The Geometry of Causality, 2303.09017). The authors withdrew it: the arXiv record’s v3 (3 April 2024) states that the paper “has been superseded by arXiv:2206.08911v4, arXiv:2303.07148 and arXiv:2303.09017”, that its Definition 3 (the locale of inputs) “is not fit for purpose”, and that it is “unlikely to be the right reference”. The reading confirms the defect: the meet given in Proposition 5 can leave the poset, and the poset is not distributive, so it is not a locale (NOTE-629 gives a two-event counterexample). The successors replace the locale of inputs with spaces of input histories under the lowerset topology, a genuine topology; the sheaf of causal functions and locality as a global section are redone in LIT-808, and the polytope and the BFW causal fractions are proved in LIT-788. |
| LIT-831 | Contextuality-by-Default 2.0: Systems with Binary Random Variables | Dzhafarov & Kujala (2017), Quantum Interaction 2016, LNCS 10106:16–32. CbD 2.0 for binary random variables: a system is noncontextual when its bunches are compatible with the multimaximal couplings of its connections, couplings in which every subset of a connection is maximally coupled. For binary variables that coupling exists, is unique and has an explicit staircase form (Theorem 1), so subsystems of a noncontextual system are noncontextual (Theorem 3). Cyclic and consistently connected systems are judged as in CbD 1.0, and the total-variation measure carries over. For non-binary variables the construction fails, and dichotomization is proposed. | Active — read 2026-10-09 (NOTE-643); worth reading as the paper that defines “CbD 2.0”: for systems of binary random variables, the copies of a content across contexts are coupled multimaximally (every subset, equivalently every adjacent pair in the order of their means, maximally coupled) rather than maximally as a whole, and such a coupling exists and is unique (Theorem 1). The change buys heredity (a noncontextual system’s subsystems are noncontextual, Theorem 3) and removes the partial/complete distinction, and it leaves cyclic systems, consistently connected systems and the negative-probability measure as they were. It shows by example that for non-binary variables multimaximal couplings may fail to exist, may not be unique, and are not stable under coarse-graining, and ends by proposing to dichotomize every variable, the step the canonical-systems paper (LIT-790) takes. |
| LIT-836 | Probabilistic Foundations of Contextuality | Dzhafarov & Kujala (2017), Fortschr. Phys. 65:1600040. Being jointly distributed is transitive in Kolmogorovian probability. So if a measurement were the same random variable in every context, overlapping contexts would force a joint distribution, and Kochen–Specker and Bell contradictions would show that such variables do not exist. The assumption to drop is “Noncontextual Identification”. Contextuality becomes the impossibility of a coupling of context-indexed variables in which same-content copies satisfy a property C. For inconsistently connected binary systems, C is multimaximality (Definition II.1). | Active — read 2026-10-09 (NOTE-623); worth reading as the short foundational argument for Contextuality-by-Default. The paper argues that the traditional reading of a Kochen–Specker or Bell-type contradiction (no joint distribution exists) cannot be right within Kolmogorovian probability, because being jointly distributed is transitive, so overlapping contexts already force one. The assumption a reductio refutes is “Noncontextual Identification”, that a content is the same random variable in every context. The paper then restates contextuality as the impossibility of a coupling with a specified property C. It gives multimaximality as C for binary measurements, with the pairwise characterization (Theorem II.3) that CbD 2.0 (LIT-831) cites. It ends with Specker’s three boxes treated without assuming consistent connectedness. |
| LIT-841 | Semantic Unification: A Sheaf Theoretic Approach to Natural Language | Abramsky & Sadrzadeh (2014), ARXIV-1403.3351, in the Lambek Festschrift (LNCS 8222, pp. 1–13). Models basic Discourse Representation Structures as sections of a presheaf of consistent literals over (vocabulary, variables), an anaphoric resolution as a cover by variable maps, and the meaning of a discourse as the gluing of its sentences’ sections, unique when it exists. Failure to glue is inconsistency, not contextuality in the Abramsky–Brandenburger sense; competing resolutions are competing covers, ranked in the probabilistic section by corpus counts. | Active — read in full 2026-10-09 (NOTE-644); worth reading as the first paper to model discourse meaning as sheaf-theoretic gluing, and as the record’s earliest bridge between the Abramsky–Brandenburger framework and natural-language semantics. A basic Discourse Representation Structure is a section of a presheaf of consistent literals; a choice of anaphoric resolution is a cover, a jointly surjective family of variable maps; and the meaning of the discourse is the gluing, which is unique when it exists (Proposition 1, its only result). Its “contextuality” is context dependence, not the failure of a compatible family to glue: the only obstruction it exhibits is inconsistency of literals. Its probabilistic section ranks candidate covers by summed corpus counts rather than gluing distributions. Short and preliminary by its own account. |
| LIT-842 | On the Quantum-like Contextuality of Ambiguous Phrases | Wang, Sadrzadeh, Abramsky & Cervantes (2021), ARXIV-2107.14589, SemSpace 2021 (ACL Anthology, pp. 42–52). Treats each word of a two-word ambiguous phrase as a measurement whose outcome is the selected meaning. One hand-specified support model is possibilistically contextual; its BNC/ukWaC probabilities, like all the corpus models, are signalling. Under Contextuality-by-Default two subject–verb/verb–object pairs come out contextual, but the paper’s own bootstrap leaves both plausibly noncontextual. | Active — read in full 2026-10-09 (NOTE-645); worth reading as the first paper to test natural-language data for contextuality in the formal sense rather than the loose one, and as the paper that names the obstacle the test meets: corpus estimates of meaning selection are almost always signalling. In the sheaf-theoretic framework one hand-specified support model ({tap, box} × {pitcher, cabinet}) is possibilistically contextual (a Hardy-type failure), but its corpus probabilities are signalling, so the framework cannot judge them. Under Contextuality-by-Default the Bell-type corpus systems are noncontextual, and two noun–verb pairs read as subject–verb and verb–object (adopt/boxer, throw/pitcher) are CbD-contextual, by 1/30 and 7/30. Its own parametric bootstrap then gives those two a probability above .56 and above .08 of being noncontextual, on fractions with denominators between 3 and 30, so the contextual verdicts are not statistically secured. |
| LIT-843 | Combining contextuality and causality: a game semantics approach | Abramsky, Barbosa & Searle (2024), ARXIV-2307.04786, Phil. Trans. R. Soc. A 382:20230002. Adds an enabling relation to a measurement scenario and reads contextuality as a game: the Experimenter chooses measurements, Nature chooses outcomes, and Nature’s deterministic strategies replace the event sheaf. Recovers flat contextuality and Gogioso–Pinzani causal Bell scenarios, and expresses adaptive MBQC by Experimenter strategies. The strategy presheaf is not a sheaf: compatible deterministic strategies can fail to glue or glue non-uniquely. | Active — read in full 2026-10-09 (NOTE-646); worth reading as the version of sheaf-theoretic contextuality in which measurements are performed in a causal order and the choice of the next one may depend on earlier outcomes. A causal measurement scenario adds an enabling relation (Kahn and Plotkin’s information matrices renamed); Nature’s deterministic strategies over it replace the event sheaf, and the Abramsky–Brandenburger definitions of empirical model and contextuality then go through unchanged. Flat scenarios and Gogioso and Pinzani’s Bell scenarios with a causal order come out as special cases, and Experimenter strategies express adaptive measurement, as in the Anders–Browne GHZ OR-gate. Its sharpest point is that the strategy presheaf is not a sheaf: deterministic, compatible local strategies can fail to glue (Example 7.2) or glue non-uniquely (Example 7.3). A short paper; several results are deferred to an extended version. |
| LIT-844 | Quantum-Like Contextuality in Large Language Models | Lo, Sadrzadeh & Mansfield (2024), ARXIV-2412.16806, Proc. R. Soc. A 481:20240399 (2025). Instantiates a three-sentence anaphora schema with the support of a PR prism 51,966,480 times from Simple English Wikipedia and takes referent probabilities from BERT: 0.148% of the models are sheaf-contextual by a signalling-corrected test and 71.1% are CbD-contextual. The support is fixed by the schema, so each verdict turns on how uncertain BERT is in three sentences. | Active — read in full 2026-10-09 (NOTE-648) from the arXiv preprint (the journal version could not be reached); worth reading as the large-scale test of language-model output for contextuality, and for what its numbers rest on. A PR-prism schema of three sentences (“It is X1 and the same one is X2”, ”… X2 … same … X3”, ”… X3 and the other one is X1”) is instantiated with 866,108 Simple English Wikipedia noun pairs and their adjectives; BERT gives one referent probability per sentence, and the support is fixed by the schema. Of 51,966,480 models, 77,118 (0.148%) pass the signalling-corrected sheaf test (SF < 1/6) and 36,938,948 (71.1%) the CbD test (Δ < 2). Both verdicts are fixed by three numbers per instance; the CbD one holds whenever BERT is uncertain and roughly consistent across the three sentences. The link to embedding distance is a softmax identity plus weak correlations (R² ≤ 0.08). |
| LIT-845 | Closing Bell: Boxing black box simulations in the resource theory of contextuality | Barbosa, Karvonen & Mansfield (2021), ARXIV-2104.11241, in Samson Abramsky on Logic and Structure in Computer Science and Beyond (Springer, 2023). Characterises which convex maps between the empirical models of two scenarios are induced by classical (non-adaptive) procedures: exactly those given by a non-contextual model of a “hom” scenario [S, T] whose outcomes are procedures. Non-local games are procedures into a one-bit scenario, and the hom construction makes scenarios with predicates a closed category. | Active — read in full 2026-10-09 (NOTE-650); worth reading as the result that says which transformations between the empirical models of two different scenarios are classical. A probabilistic procedure S → T (a mixture of deterministic procedures, each answering every measurement of T by a context of S and a post-processing of outcomes) induces a convex map EMP(S) → EMP(T). A convex map is so induced exactly when it comes from a non-contextual model, satisfying a simpliciality predicate, on a new “hom” scenario [S, T] whose outcomes are procedures. Compatible local procedures that do not glue into one global procedure are contextual models of [S, T]: “contextual simulations”. The construction makes scenarios with predicates a closed category. Non-adaptive only; the arXiv v2 corrects Theorem 44 of the published chapter. |
| LIT-846 | Categories of Empirical Models | Karvonen (2018), ARXIV-1804.01514, QPL 2018, EPTCS 287:239–252 (2019). Defines simulations between empirical models on different scenarios: a simplicial relation from target measurements to sets of source measurements, and a stochastic outcome map natural in the context. Non-contextual models are those simulable from the terminal model, the non-contextual fraction is monotone along simulations, strong contextuality is reflected, and contextual models cannot be cloned. | Active — read in full 2026-10-09 (NOTE-651); worth reading as the paper that first gives empirical models on different measurement scenarios a notion of morphism: a simulation d → e answers each measurement of e by a set of jointly measurable measurements of d (a simplicial relation, so the cover may change) and a stochastic map of outcomes natural in the context. With it, non-contextual models are exactly those simulable from the terminal (empty) model, the non-contextual fraction can only grow along a simulation, strong contextuality is reflected backwards, Graham reductions are simulations (Vorob’ev’s theorem), and no contextual model can be cloned. It is restricted to no-signalling models and has no preprocessing. |
| LIT-847 | A comonadic view of simulation and quantum resources | Abramsky, Barbosa, Karvonen & Mansfield (2019), ARXIV-1904.10035, LICS 2019. Measurement protocols form a comonad on empirical models; a simulation of e by d is a co-Kleisli map from d ⊗ c (c non-contextual) to e. Simulability coincides with convertibility by a term in the free operations of the resource theory of contextuality, including a new conditional measurement; the non-contextual fraction is monotone, and contextual models cannot be cloned. | Active — read in full 2026-10-09 (NOTE-647); worth reading as the paper that makes simulation between empirical models adaptive and shows it is the same thing as convertibility by free operations. Measurement protocols (decision trees of compatible measurements) form a comonad MP on scenarios and on empirical models; a simulation of e by d is a deterministic map MP(d ⊗ c) → e with c non-contextual. Such a simulation exists exactly when e is obtained from d by a term in the free operations (relabelling and translation of measurements, coarse-graining, mixing, choice, tensor, and the new conditional measurement), with an equational theory and normal forms. Along any simulation the non-contextual fraction cannot fall, and no contextual model can be cloned. |
| LIT-851 | Analysing Ambiguous Nouns and Verbs with Quantum Contextuality Tools | Wang, Sadrzadeh, Abramsky & Cervantes (2021), Journal of Cognitive Science 22(3):391–420. Proves that in a binary cyclic system CbD’s signalling quantity Δ is twice the summed minimal direct influences of context in Jones’s causal canonical models, and measures it in 90 rank-2 noun–verb systems from British corpora: Δ is about 1.35 whatever the words’ ambiguity, and homonymous verbs carry about 70% of it against about 50% for polysemous ones, on small counts and a marginal test. | Active — read in full 2026-10-09 (NOTE-654); worth reading as the paper that measures how much signalling there is in natural-language meaning selection, and that gives signalling a causal reading. Its Proposition 1 shows that in a binary cyclic system the CbD quantity Δ is twice the sum, over contents, of the least direct influence of context that any of Jones’s canonical (causal) models of the data must posit, each the total-variation distance between a content’s two marginals. Applied to 90 rank-2 noun–verb systems from the BNC and ukWaC, Δ averages about 1.35 in every class of ambiguity, and homonymous verbs carry about 70% of it against about 50% for polysemous verbs (14 against 55 systems; my recomputation from its appendix gives a Welch t of 2.0, two-sided p ≈ 0.06). It tests no system for contextuality. Read with the caveat that most systems rest on very few corpus occurrences: 51 of the 90 have Δ exactly 0, 2 or 4. |
| LIT-945 v3 | Unscrambling the omelette of causation and inference: The framework of causal-inferential theories | Schmid et al. (2020). The paper introduces causal-inferential theories, which pair a theory of causal influences with a theory of inference and say how the two interact. In this frame it recasts operational and realist theories and characterizes the assumptions behind no-go theorems based on Bell locality and on generalized noncontextuality. It shows that the existence of a Leibnizian classical realist representation implies a generalized-noncontextual one; the converse is conjectured. Read in part. | Active — read 2026-10-14 (NOTE-737) in part, not in full: the sections that characterise generalized noncontextuality and Leibnizianity (§§V, VI, VIII, IX) were read closely, §II–IV and the proofs in Appendices C–G were not, so the NOTE is Skimmed. Worth reading as the current statement of what a generalized-noncontextuality no-go denies: a classical realist representation that preserves inferential equivalence. It also shows why a prepare-measure scenario, which allows no Bell-like no-go, can still show contextuality. Read it knowing that Leibnizianity implies generalized noncontextuality only in one direction (the converse is Conjecture 1′), that only classical inference is built, that the nonclassical realist theory it aims at is specified by desiderata only, and that it does not discuss the Kochen–Specker or sheaf notions. |
| LIT-980 v2 | What is nonclassical about uncertainty relations? | Catani, Leifer, Scala, Schmid and Spekkens (2022). Some operational theories with nontrivial uncertainty relations admit a generalized-noncontextual model, so such relations alone are not evidence of nonclassicality. For any state whose four-state reflection orbit is realizable (equal predictabilities with flipped signs, and one equivalence of equal mixtures), preparation noncontextuality bounds the predictabilities of two binary measurements by |⟨X⟩| + |⟨Z⟩| ≤ 1. In a theory where every state has the orbit, this bounds the uncertainty relation itself, and qubit theory’s circular trade-off violates it, by up to √2. | Active — read 2026-10-14 (NOTE-825) in full; worth reading as the clearest case where the form of an observed trade-off, not its existence, witnesses preparation contextuality, in a prepare-measure scenario with two binary measurements that are never made together, so that no sheaf or Kochen–Specker test applies. For a state with a realizable four-state reflection orbit, noncontextuality forces |⟨X⟩| + |⟨Z⟩| ≤ 1, with no perfectly distinguishable states needed. Read it knowing that the bound covers a whole uncertainty relation only in a theory where every state has the orbit, that the operational equivalence is used exactly and the claimed noise robustness is asserted, not shown, and that the three-measurement bound is verified by computer only. Filed for QUESTION-067; the account is THEORY-315. |
| LIT-1352 v2 | Towards a Mathematical Structure of Global Phenomenal Consciousness | Lee-Youngzie, Tsuchiya, Robinson, Dietz & Monti (2026), Entropy 28(6):615, a hypothesis article. Intra-phenomenal structures model the parts of one experience, and inter-phenomenal structures place an experience among others by similarity judgements. A presheaf maps each part, in a finite topology of parts, to the dissimilarity matrix measured for it. Agreement of narrow qualia on an overlap is read as phenomenal binding, and the presheaf’s being a sheaf as the broad quale’s being fixed by its parts. Worked for a two-hemifield visual field. A sheaf failure is read as disunity for binocular rivalry and as context dependence for the Ebbinghaus illusion. No data. | Active — read in full 2026-10-22 (NOTE-1059) from the version of record in PubMed Central (PMC13298370, CC BY 4.0, via the Europe PMC full-text XML); worth reading as the first worked proposal to join intra-phenomenal (part–whole) and inter-phenomenal (similarity) qualia structures by a presheaf, with phenomenal binding read as agreement of measured narrow qualia on overlaps and a sheaf as a broad quale fixed by its parts. It is a hypothesis article with no data. It does not identify unity with a global section: it assumes unity always holds, and it reads a sheaf failure as disunity for binocular rivalry and as context dependence for the Ebbinghaus illusion without a criterion between them. Its worked topology example is malformed, and by its own definition of gluing the rivalry case is not a sheaf failure. |