We establish a formally verified three-layer separation theorem demonstrating that contextual information residue—the non-trivial Čech cohomology class of a Boolean 1cochain on a finite Z/2 cycle graph—is structurally invisible to both Shannon mutual information and bounded-degree polynomial computation. Specifically: (1) Shannon mutual information of a binary symmetric channel achieves AUC = 0.500 for contextual/noncontextual classification, exhibiting perfect orthogonality; (2) multilinear polynomials of degree d < n on n-bit cochains produce FN = 4088/4092 on the exhaustive corpus, a consequence of the Minsky–Papert degree barrier; and (3) exact rational symbolic evaluation Q via the parity formula parity(b) = (1 − i (1 − 2bi ))/2 ∈ Q achieves perfect classification with zero error. The algebraic core—including H 1 (Cn ; Z/2) ∼ = Z/2 and the non-coboundary witness for the triangle and square parity cochains—is verified in Lean 4 with Mathlib. The corpus comprises all 2n Boolean 1-cochains on cycles Cn for n ∈ {3, . . . , 12}, totaling 8,184 cochains exhaustively enumerated. All proofs, experiments, and data are publicly available. Published by The Institute for Applied Ontological Mathematics.
We establish a formally verified three-layer separation theorem demonstrating that contextual information residue—the non-trivial Čech cohomology class of a Boolean 1cochain on a finite Z/2 cycle graph—is structurally invisible to both Shannon mutual information and bounded-degree polynomial computation. Specifically: (1) Shannon mutual information of a binary symmetric channel achieves AUC = 0.500 for contextual/noncontextual classification, exhibiting perfect orthogonality; (2) multilinear polynomials of degree d < n on n-bit cochains produce FN = 4088/4092 on the exhaustive corpus, a consequence of the Minsky–Papert degree barrier; and (3) exact rational symbolic evaluation Q via the parity formula parity(b) = (1 − i (1 − 2bi ))/2 ∈ Q achieves perfect classification with zero error. The algebraic core—including H 1 (Cn ; Z/2) ∼ = Z/2 and the non-coboundary witness for the triangle and square parity cochains—is verified in Lean 4 with Mathlib. The corpus comprises all 2n Boolean 1-cochains on cycles Cn for n ∈ {3, . . . , 12}, totaling 8,184 cochains exhaustively enumerated. All proofs, experiments, and data are publicly available. Published by The Institute for Applied Ontological Mathematics.
We extend the Shannon-blind contextual information residue separation theorem fromcycle graphs to arbitrary finite graph families and introduce a partition-function formalismfor predicting error-regime boundaries. Our contributions are threefold. (1) We generalize theČech cohomological obstruction to parameterized n-cycles (proved for all n ≥ 3), completegraphs (K4 , with dim H 1 = 3), and trees (negative control: dim H 1 = 0, separation collapses).The polynomial degree barrier persists across all tested cycle lengths n ∈ {3, 4, 5, 7, 10, 15, 20}.P(2) We define a finite Čech Hamiltonian and partition function Z(β) = ω exp(−β · H(ω))on the measurement-cover graph, whose heat-capacity peak identifies a critical inversetemperature β ∗ predicting the error regime. (3) We present a falsifiability surface: the regimeclassifier (exact-decoder-required vs. surrogate-sufficient) is correct for all 8 tested graphfamilies, while the quantitative threshold predictions are falsified for 5 of 8 families undertolerance 0.03. All proofs are verified in Lean 4 with zero sorry declarations; all experimentsare exhaustive finite-graph enumerations reproducible from a single command. The separationis binary: any non-trivial cycle topology (H 1 ≥ 1) activates the full polynomial barrier,making exact symbolic computation the only reliable decoder for topologically structuredworkloads.
Abstract Relative-entropy measures of contextuality quantify the statistical distance between an empirical model and the noncontextual set. We study how this established quantity changes when an experiment is observed through a characterized loss mechanism. Each attempted trial is retained in the analysis, with missed outcomes represented by context-labeled no-click symbols, and the same stochastic observation channel is applied to the target model and every noncontextual comparator. Under this common-channel transportability assumption, the data-processing inequality gives an observed-data contextual divergence that cannot exceed its complete-data value. We define the absolute contextual-information loss and its normalized ratio, the fraction of missing contextual information (FMCI). We prove general reference-set contraction, exact scaling under missing completely at random loss, a cellwise erasure sandwich, a no-exact-hiding result under strictly positive retention, and additive and multiplicative decompositions for sequential design and detector channels. We also separate response dependence from disturbance of shared marginals and clarify why outcomes of unchosen incompatible contexts are not model-neutral missing entries. The theory is evaluated on the perfect-anticorrelation three-cycle, the Klyachko–Can–Binicioğlu–Shumovsky pentagon, and the Peres–Mermin magic square. The hidden-variable and inequality-constrained optimizations agree to within 10 − 12 nats. Exhaustive enumeration of all 104 context subcovers shows that, for these three canonical benchmark points, every proper subcover is noncontextual. A 51 × 51 scan of a tunable outcome-dependent loss family confirms that exact population hiding occurs only on the zero-retention boundary, while finite-sample detectability can deteriorate well before that boundary. For a known finite-alphabet erasure channel with cellwise retention probabilities we give an exact identification criterion, regular asymptotic and Fieller inference for FMCI, and diagnostics that withhold scalar ratio inference near the noncontextual boundary. Unknown outcome-dependent loss is handled by a calibration-constrained sensitivity workflow and union confidence sets. FMCI is therefore a conditional diagnostic of information attenuation under a declared observation channel, not a replacement for loophole-free contextuality certification.
Whether a quantum proof outperforms a classical one — both checked by the same efficient quantum verifier — is a question answered, so far, only relative to oracles. Beneath that answer is a structure with no quantum content at all: the difference between a resource one can copy and one one cannot. We show that this difference is cohomological. A witness that carries a copy map collapses every verifier to a memoryless sampler, so its distinguishing power is a functional of k-wise marginals alone; the obstruction to copying is a Čech class on a cover of a state by its measurement bases, and the Forrelation function of Aaronson and Ambainis is exactly the transition cocycle of that cover. The separation holds precisely when the class is nonzero: invisible from inside any single copyable chart, where it is a coboundary, and visible to the linear witness that reads the whole cover at once. Four familiar facts turn out to be four faces of the same class. Its non-vanishing is the no-cloning theorem; the dimension of its kernel is the Landauer erasure cost of the witness register; its representative on the standard/Hadamard cover is Forrelation; and the collapse of the copyable verifier onto one chart is why a classical witness, for these problems, cannot win. We make each face a finite, exactly computed certificate: on a corpus of distinguishing tasks built from Heisenberg–Weyl orbits, every depth-two adaptive sampler has advantage exactly zero — a total quantification over the strategy space, not a sampled estimate — while the linear witness reads a squared-Forrelation gap the kernel computes from truth tables through the Hadamard transform. Two independent exact computations agree on every verdict, and the engine that results either returns a gap two exact paths both certify, or refuses — and a refusal carries its own evidence, a certificate of exactly the obstruction that stopped it. The asymptotic separation of the underlying family is not ours; we cite it. What is ours is the identification — that a separation of complexity classes is, at bottom, a hole. All results below are machine-checked in Lean 4; the formal record and its axiom footprint are described in Section 8. The paper is legible in mathematics alone. Published by The Institute for Applied Ontological Mathematics.
In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed as a compatible family of distributions, in order to define a cohomological obstruction for the family as a certain cohomology class. This class vanishes if the family has a global section. Thus the non-vanishing of the obstruction provides a sufficient (but not necessary) condition for the model to be contextual. We show that for a number of salient examples, including PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector configuration due to Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions, the obstruction does not vanish, thus yielding cohomological witnesses for contextuality.
Quantum measurements need not reveal preexisting values: the outcome assigned to an observable can depend on which compatible observables are measured with it. This contextuality underlies the Peres-Mermin proof of the Kochen-Specker theorem and, in quantum computation with magic states, tracks the even/odd parity of the local dimension separating the classically simulable from the genuinely quantum. ATTRIBUTION: the even/odd value classification of state-independent qudit contextuality (the parity dichotomy and the fixed obstruction value d/2) is due to Abramsky, Cercelescu and Constantin, FSCD 2024; this paper supplies its geometric and cohomological ACCOUNT, not a rediscovery of it. Organizing all measurement contexts of M_n(C) into the stack [X/PU(n)] of maximal abelian subalgebras, we show the lifting class of U(1) -> U(n) -> PU(n) is vacuous on the action groupoid U(n) x X but, on PU(n) x X, is a canonical degree-2 class of order exactly n. Its real shadow, through the signed-permutation group B_n, is a Z_2 extension splitting if and only if n is odd, with "-I is a rotation square iff n is even" as the mechanism; at n=4 it restricts to the real two-qubit Pauli class, whose pairing with the Peres-Mermin word is -1, exactly the obstruction to a noncontextual value assignment. We compute H^2 = Z/n (+) Z_2 for n >= 4 and identify the canonical class as a generator of the Z/n summand. STATUS AND SCOPE. Every step is tagged Complete, Sketch (reduced to a named standard theorem whose hypotheses are verified), or Gap, and a status table closes the paper. The finite and algebraic core is machine-checked by scripts in the public suite. The geometric layer rests on standard but non-elementary facts about differentiable-stack cohomology and the Lyndon-Hochschild-Serre spectral sequence; that bookkeeping is reduced to named standard theorems and AWAITS SPECIALIST REVIEW. This is a preprint, not a peer-reviewed article, and should be read at that strength. The finite Weyl value spectrum is developed in a companion paper. Code, data and verification suite: https://github.com/manuflog/contextuality-obstructions Note on methods: large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance, and the author takes full responsibility for the results, including any errors.
We place several known indicators of Kochen--Specker contextuality -- the KCBS correlator χ, the contextual fraction $\CF$, the Shannon-entropic n-cycle inequality of Chaves and Fritz, and the operational commutator witness D of Paper~I -- into a single projector-geometric framework organized around the overlap matrix $\Tcal_{ij} = d^{-1}\tr[(\hat P_i \hat Q_j)^2]$, where P^i and Q^j are the joint-eigenspace projectors of the two compatible observable pairs within a measurement context. The state-independent scalar content of $\Tcal$ is carried by two independent contractions: the mutual information energy $E = \sum_{ij}\Tcal_{ij}$ of Paper~I (equivalently, its logarithmic form S2=−log2E), and the Maassen--Uffink extremal overlap $c_\MU = \max_{i,j}|\langle a_i,b_i | c_j,b_j\rangle|$. We prove that S2 is non-increasing under coarse-graining, that $S_2(\Gcal) > 0$ is a necessary configuration-level condition for observable contextuality, and that the additive composition $S_2(\Gcal) = \sum_αS_2(\Gcal_α)$ is exact for the KCBS pentagon. We further show that in the spin-1 realization of the KCBS pentagon, a shared ms=0 eigenstate in each context forces $c_\MU = 1$, rendering every Maassen--Uffink-type bound trivial -- a structural mechanism that makes explicit why outcome-entropic uncertainty relations based on $c_\MU$ are silent on KCBS contextuality, while S2≈2.7266~bits throughout. Applied to KCBS and CHSH, the framework identifies regimes in which every state-dependent witness considered here is silent yet $S_2(\Gcal) > 0$ by an amount set by the projector geometry alone.