CORRECTION IN v3 (2026-07-28). Versions 1 and 2 cited the companion flagship paper as "Preprints.org 224406 (2026)". That citation was premature: the companion was submitted to Preprints.org in July 2026 but was not posted, so no public record under that identifier exists. The reference is corrected to the companion's Zenodo concept DOI, 10.5281/zenodo.21488473. Every self-citation across all papers in the program was audited at the same time, and stale or missing identifiers elsewhere were corrected. No mathematical claim in this paper depended on the erroneous reference. Companion paper to "Exact rigidity certificates for Kochen-Specker sets, the graded flex dictionary, and the Peres-Penrose circle" (https://doi.org/10.5281/zenodo.21488473). The Gould-Aravind circle - the unique positive-dimensional flexible family of 33-ray Kochen-Specker sets, with Peres-33 and Penrose-33 as antipodes - is studied as a geometric object in its own right. Results: (1) an arithmetic classification of the circle's exact ring points (ring points exist iff 2 ramifies; exactly 8 special phases: Z[sqrt2] x2, Z[i] x4, Z[sqrt-2] x2; dimension 7 excluded by an exact denominator-4 obstruction); (2) the Berry/Wilczek-Zee holonomy of the KS loop: exact rational per-ray Berry phases summing to zero, trivial abelian holonomy, and the non-abelian eigenphase identified and proved in closed form, phi = 2*pi*sqrt(1867)/33 (rotating-frame reduction to a constant connection; 1867 = 43^2+3^2+3^2), with its section dependence characterized; (3) exact imaginarity profiles along the circle (Im v(theta) = sin(theta) y identically; total l1 imaginarity (8+20*sqrt2/3)|sin theta|); (4) hardware experiments on IBM Quantum devices: witness invariance confirmed on ibm_marrakesh (grand mean 11.049, +0.45% of the exact prediction 11, 297 circuits), and the holonomy eigenphase measured on two devices (Hadamard test and a constant-connection shortcut circuit), consistent with sqrt(1867)/33 to about 2%. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN); each label is backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity, with the executed hardware notebooks in rigidity/hardware_results/). Methods note: large-language-model assistants (Anthropic Claude) were used substantially in drafting and in developing the verification suite; every mathematical claim is machine-verified independently of that assistance, and the author takes full responsibility for the results.
CORRECTION IN v3 (2026-07-28). Versions 1 and 2 cited the companion flagship paper as "Preprints.org 224406 (2026)". That citation was premature: the companion was submitted to Preprints.org in July 2026 but was not posted, so no public record under that identifier exists. The reference is corrected to the companion's Zenodo concept DOI, 10.5281/zenodo.21488473. Every self-citation across all papers in the program was audited at the same time, and stale or missing identifiers elsewhere were corrected. No mathematical claim in this paper depended on the erroneous reference. Companion paper to "Exact rigidity certificates for Kochen-Specker sets, the graded flex dictionary, and the Peres-Penrose circle" (https://doi.org/10.5281/zenodo.21488473). The Gould-Aravind circle - the unique positive-dimensional flexible family of 33-ray Kochen-Specker sets, with Peres-33 and Penrose-33 as antipodes - is studied as a geometric object in its own right. Results: (1) an arithmetic classification of the circle's exact ring points (ring points exist iff 2 ramifies; exactly 8 special phases: Z[sqrt2] x2, Z[i] x4, Z[sqrt-2] x2; dimension 7 excluded by an exact denominator-4 obstruction); (2) the Berry/Wilczek-Zee holonomy of the KS loop: exact rational per-ray Berry phases summing to zero, trivial abelian holonomy, and the non-abelian eigenphase identified and proved in closed form, phi = 2pisqrt(1867)/33 (rotating-frame reduction to a constant connection; 1867 = 43^2+3^2+3^2), with its section dependence characterized; (3) exact imaginarity profiles along the circle (Im v(theta) = sin(theta) y identically; total l1 imaginarity (8+20*sqrt2/3)|sin theta|); (4) hardware experiments on IBM Quantum devices: witness invariance confirmed on ibm_marrakesh (grand mean 11.049, +0.45% of the exact prediction 11, 297 circuits), and the holonomy eigenphase measured on two devices (Hadamard test and a constant-connection shortcut circuit), consistent with sqrt(1867)/33 to about 2%. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN); each label is backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity, with the executed hardware notebooks in rigidity/hardware_results/). Methods note: large-language-model assistants (Anthropic Claude) were used substantially in drafting and in developing the verification suite; every mathematical claim is machine-verified independently of that assistance, and the author takes full responsibility for the results.
Within the program of identifying rigid Kochen-Specker (KS) sets (Xu-Saha-Bharti-Cabello, PRL 132, 140201; Trandafir-Cabello, PRA 111, 052204; Kernaghan, arXiv:2603.16988), this paper contributes exact-arithmetic certificates, one unification, and one correction. The two flexible islands of Kernaghan's landscape (Peres-33 and Z[sqrt(-2)]-33) are the two antipodes of a single exact circle of complex 33-ray KS sets, the closed form of the Gould-Aravind phase family. We certify flex=1 exactly at both islands, prove local completeness, and correct Kernaghan's Observation 28: the flex is finite, not merely infinitesimal. The circle contains no degenerations (proved exactly) and its Sum P = 11 I mechanism is resolved via hyperoctahedral orbit tight frames. Exact rigidity certificates across the full known census make Peres-Penrose-33 the unique flexible KS set. The Hermitian rigidity matrix block-diagonalizes at real points (flex_C = flex_R + flex_skew) with the graded cycle split proved, and a torsion layer is proved flex-invariant. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN). The attached verification suite (35+ self-checking scripts) reproduces every certificate; companion hardware notebooks (IBM Quantum, two devices) confirm the witness-invariance prediction Sum P = 11 to 0.45% and measure the loop holonomy eigenphase to ~2%, consistent with the exact value 2*pi*sqrt(1867)/33. Note on methods: large-language-model assistants (Anthropic Claude) were used substantially (drafting, verification-script development, literature search, proof proposals); all mathematical claims are machine-verified by the attached suite, and the author takes full responsibility. Code and data: https://github.com/manuflog/contextuality-obstructions
Fourth paper of the rigidity program (companions: https://doi.org/10.5281/zenodo.21488473, https://doi.org/10.5281/zenodo.21498342, https://doi.org/10.5281/zenodo.21498385). In 2010, Blanchfield asked how common parameterized generalisations are among Kochen-Specker sets, and found every d=4 set she examined fully determined by its orthogonalities; the only known continuous family lived in dimension 3 (the Gould-Aravind circle, whose uniqueness among two-symbol d=3 alphabets the companion papers prove). This paper answers her question in dimension four. Results: (1) an anchor theorem: every two-symbol d=4 mechanism-stable graph contains Peres-24 and is therefore KS-uncolorable, so the d=3 uniqueness argument structurally cannot run in d=4; (2) the M9 family: for the unimodular mechanism |x|^2=1, an 89-ray, 35-basis KS core with entries {0, +-1, +-e^(i theta)} whose 433 orthogonalities hold identically in theta, KS-uncolorable at every point of the circle, with exact flex = 1 - to our knowledge the first continuous fixed-graph KS family in dimension 4; (3) a complete triage of all 16 two-symbol d=4 mechanisms: exactly two carry families - the M9 circle (flex 1) and the M10 line Re(x)=0 (exact flex 2, 21 rays) - while the off-center-circle mechanisms are proved rigid by a Moebius-stability method; (4) the geometry of the M9 circle: its Wilczek-Zee connection is exactly constant, with eigenphase quartic irreducible over Q with Galois group S4 (so no closed form of the d=3 type 2*pi*sqrt(1867)/33 exists - an honest structural contrast), and det W = exp(2*pi*i*23/35), nontrivial where d=3's is trivial; (5) a structure card: all four conjugation-fixed points of the circle degenerate and are exactly rigid while generic points flex - the reverse of the d=3 pattern. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / CONJECTURE / OPEN), each backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity). A final adversarial literature guard (including the Pavicic-Megill coordinatization program, the Waegell-Aravind catalogs, and the 2022 RMP review) found no prior continuous fixed-graph d=4 KS family; the claim is stated as first known. Methods note: large-language-model assistants (Anthropic Claude) were used substantially in drafting and in developing the verification suite; every mathematical claim is machine-verified independently of that assistance, and the author takes full responsibility for the results. v2 (2026-07-23): adds the refined reading of M10's flex (Remark 7.x): the second deformation is an exactly-integrable core-local twin-ray rotation, not portable to the full stable pool, so the mechanism-level flex of M10 is 1 = dim V; adds the conditional d=4 uniqueness results (decoupling certificates for all 8 rigid mechanisms; integer-pool sweep: 2082 exact flex computations, all critical {0,+-1}^4 cores rigid) and the guard-mandated citations (RMP 2022 review; Trandafir-Cabello rigid-sense disambiguation; Pavicic-Megill coordinatization distinguished; Waegell-Aravind catalogs). Verification suite updated with branch_d4unique.py, branch_intflex.py, branch_m10x.py. v3 (2026-07-25): structural pass on Sections 5-7. v4 (2026-07-27): the stale "SCAFFOLD DRAFT" banner and a stale Discussion paragraph removed; and, importantly, an attribution remark added -- the anchor property used here is NOT ours, it is due to Pavicic and co-authors (Pavicic/Waegell, Pavicic/Megill), and earlier versions did not say so. This deposit is document version v4, and the Zenodo version counter now reads v4 to match; do not read the two as independent.
Companion paper to "Exact rigidity certificates for Kochen-Specker sets, the graded flex dictionary, and the Peres-Penrose circle" (https://doi.org/10.5281/zenodo.21488473). A complete algebraic theory of Kernaghan's two-symbol alphabet landscape (arXiv:2603.16988). Results: (1) the exhaustive 80-case d=3 classification: exactly FOUR independent cancellation mechanisms (rational, modulus-2, phase, golden), not two - a fifth candidate is proved identical to the phase mechanism under the alphabet's own x<->1/x gauge duality - and any generator avoiding all four provably yields a 3-colorable pool (structure theorem via Godsil-Zaks); (2) the d=4 extension: a term-alphabet lemma, 16 new mechanisms from the 280-case 4-term classification, and no colorable regime (Peres-24 anchors every pool); (3) the mechanism-moduli audit across the census, proving the forward flex bound in general; (4) the uniqueness theorem: a machine-certified KS-coloring of the full 145-ray M3-line-stable graph proves the Gould-Aravind circle is the unique positive-dimensional flexible family of two-symbol d=3 KS sets - including the honest retraction of a tempting arithmetic explanation that failed direct verification; (5) the M2 mirror: the maximal circle-stable graph is KS-uncolorable, compactness pins the M2 circle to exactly five integer-trace exact points, and the Gaussian points x=+-(1+i) carry a NEW 49-ray KS core (SAT-certified with a three-anchor-validated encoder, critical, and exactly rigid) distinct from all six census islands and from the survey's own Gaussian island (shown here to reduce to Peres-33). Flexibility tracks the circle's four conjugation-fixed points exactly (Hypothesis L), and a J-splitting construction makes the real/skew flex decomposition well-posed at conjugation-symmetric non-real points. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / CONJECTURE / OPEN), each backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity). Methods note: large-language-model assistants (Anthropic Claude) were used substantially in drafting and in developing the verification suite; every mathematical claim is machine-verified independently of that assistance, and the author takes full responsibility for the results. Version 4 (July 2026) — status of the Observation 28/29 point. This paper extends and in one place corrects Part II of Kernaghan's survey (arXiv:2603.16988). The specific disputed point, the reported non-persistence of the Peres/Penrose flex, was conceded by its author in v8 of that preprint, in a labelled correction note that also publishes the cause. References are updated to v8 and the passage now describes the matter as settled. As elsewhere in this series, v8 is not treated as independent confirmation of finiteness: its continuation is double precision and its modular rank computation bounds flex only from above, as its author states. Version 5 (July 2026) — a proof step is withdrawn and replaced. The theorem is unchanged. Versions up to v4 justified the last step of the structure theorem (necessity of a mechanism for the raw pool) by invoking Godsil–Zaks, paraphrased as “rational-coordinate projective ray sets in R^n are always 3-colorable”, and concluded that the x=3 graph is 3-colorable. Both halves are false and are withdrawn. The paraphrase is false: the x=2 raw pool is a rational-coordinate ray set (49 rays, 138 orthogonal pairs) with chromatic number 4. The application is false at the pool it was applied to: the x=3 graph is not 3-colorable either, also chromatic number 4. Godsil–Zaks concerns rational points of the unit sphere, and a rational ray such as (1,1,1) does not normalize to one, so the theorem never applied to these pools at all. The theorem itself is unaffected. Graph 3-colorability was only ever a sufficient route to KS-colorability, and KS-colorability at x=3 is established directly by exhaustive certificate — 31488 distinct valid KS colorings, recomputed independently for this version. Both chromatic numbers were verified twice, by DSATUR backtracking and by SAT (Glucose3). An earlier section of the paper had already retracted this same invocation for the Re(x) = -1/2 line; that retraction should have been applied to the structure theorem at the same time and was not. It is now. Also corrected: earlier versions described cross-product completion as “the closure step that rescues the Golden and CK-31 islands from initial colorability”. That is wrong for CK-31, whose raw pool at x=2 is already KS-uncolorable (49 rays, 138 pairs, 26 triads, no valid coloring). Completion enlarges it but is not needed for uncolorability. Only the Golden island is genuinely rescued by completion. Version 6 (July 2026) — a theorem is scoped, its closed-pool reading withdrawn, and the resolution identified. The structure theorem (“no mechanism, no KS set”) was stated through v5 without an alphabet hypothesis. For Hermitian-closed complex pools it is FALSE, and the cause is a term the classification missed: with x* as an entry, (x*)*·x = x² appears, so the closed term set is {±1, ±x, ±x*, ±N, ±x², ±(x*)²}. Counterexample, verified twice in independent exact implementations: x = √3 + i avoids every locus M1–M5, yet its closed pool (145 rays / 390 pairs / 30 triads) is KS-uncolorable; unclosed it is colorable. The identity x² + (x*)² = N holds along the whole lines arg x ≡ ±30° (mod 180°), at every modulus — a positive-dimensional uncolorable locus outside the classification. The resolution, also in this version: the new lines carry no new mechanism at core level and no new flexible family. Along them the ratio x/x* is a primitive sixth root of unity at every modulus, and projective rescaling identifies the lines’ uncolorable content with the known Eisenstein island: the line-stable graph is constant in the modulus away from a finite exceptional set, all 23 of its minimal uncolorable cores (complete enumeration) consist of modulus-independent rays, every core has mechanism flex exactly 0, and all 43 rays of the certified core embed projectively into the closed ω-pool. The pool’s one genuine deformation (flex exactly 1) lives entirely on colorable decoration rays. The uniqueness theorem — the Gould–Aravind circle as the unique positive-dimensional flexible family, scoped to M1–M4 — is unaffected. A complete x²-extended classification of closed-pool identities is recorded as open. The theorem is restated with the unclosed-alphabet hypothesis (under which it is true, and true for all real x); the term-alphabet lemma, correct as literally stated, now carries its scope explicitly. Found by the program’s own adversarial verification gate; every number above is from exact arithmetic with SAT and backtracking engines agreeing. Version 7 (July 2026) — the open problem left by v6 is closed, and the two-mechanism dichotomy is restored. v6 corrected a scope error (with x* as an entry the product x² appears, so the published 80-case classification is incomplete for Hermitian-closed pools) and recorded the complete x²-extended classification as open. This version carries it out. The classification. The closed term set has six types, giving 224 three-term cases; 126 are impossible for every nonzero x, and the rest reduce to the published loci M1–M5 together with exactly four new positive-dimensional loci — x²+(x*)² = ±N (the lines arg x = ±30° and ±60°) and x²+(x*)² = ±1 (two rectangular hyperbolae) — plus eight isolated points, all of which lie on the 60° line. Restricted to the 80 published cases the same computation returns exactly the 54 impossible cases and exactly M1–M5, reproducing the original theorem. The corrected dichotomy, now a theorem rather than an open problem. The closed raw pool is KS-uncolorable if and only if (A) |x|² ∈ {2, 1/2} or x ∈ {±2, ±1/2}, or (B′) x/x* is a primitive third or sixth root of unity. Established by an audit of all 58 realizable locus-membership classes, using the fact that the index-labelled closed graph depends only on which loci x belongs to. Why this is deflationary, and the point of the version. A ratio lemma explains the new uncolorable loci rather than adding a mechanism: projective rescaling by 1/x* carries the degree-one sub-pool isomorphically onto the pool over {0, ±1, ±r} with r = x/x*, multiplying every inner product by 1/N and so preserving orthogonality exactly. Since |r| = 1, the uncolorable cases are precisely those where r is a primitive third or sixth root of unity, and there the sub-pool is the unclosed Eisenstein pool at every modulus. The two hyperbola loci are genuinely new identities but carry no uncolorable content at any point tested; the arg = 45° locus, where r = i has order four and the pool is colorable, shows that being a root of unity is necessary but not sufficient. Consequence for the two-mechanism reading. Hermitian closure does not break the dichotomy reported in arXiv:2603.16988; it relocates one half of it. The literal rule “|x|² ≥ 3 and x not a root of unity implies colorable” fails — x = √3 + i is a counterexample — but that x has x/x* = e^{iπ/3}, and the corrected statement is again exactly two mechanisms: modulus, acting on x; phase, acting on x/x*. We read this as identifying the correct invariant for the phase mechanism rather than as a counterexample to the pattern. Verification scripts x2class_derive.py, x2class_pools.py, x2class_scan.py, x2class_deflate.py and x2class_closure.py accompany this version in the public repository; every colorability verdict was decided by SAT and by an independent backtracker in agreement, in exact arithmetic in the appropriate number field.
ON THE VERSION NUMBER. "v3" is the version of the NOTE, not of this Zenodo deposit. Versions 1 (February 2026) and 2 were circulated privately and never deposited; v1 contained four errors of substance, which are stated with their repairs in the "What changed" section rather than quietly removed. This is the first Zenodo deposit, so if you see a deposit counter it will not match the document version. The document version is the one that matters for citation. Preprint, version 3 (complete rewrite). This note is the synthesis of the contextuality-obstructions program; the proofs live in the companion papers. The proposal is that the Copenhagen viewpoint, stripped of its rhetoric, is a set of structural statements about one geometric object: the transformation groupoid of measurement contexts (maximal abelian subalgebras) with its canonical U(1) 2-cocycle. In that formulation: local validity, the exact classicality of every single context, is a theorem with an explicit witness; complementarity is the nonvanishing holonomy of the cocycle along closed context loops, which detects exactly the all-versus-nothing sector of contextuality; collapse is pinned by statistics, repeatability and covariance to a one-parameter family per degenerate block whose unique efficient member is the Lueders rule, with a classical twin theorem identifying the same axioms' classical solution as Bayesian conditioning; observer-relativity of facts and the movable cut are theorems of an enlarged observer-context category; and the Frauchiger-Renner argument localizes to an ordinary state-dependent violation, with contextual fraction exactly 1/6, carried by a single CHSH facet, the loop holonomy of the observer switch being provably trivial. ATTRIBUTION. The classification of achievable state-independent values, the even/odd dichotomy and the fixed value d/2, is due to Abramsky, Cercelescu and Constantin (FSCD 2024). It is recast here in certificate language; the priority is theirs and is stated in the text. HARDWARE CAVEAT, STATED UP FRONT. Two signatures were measured on superconducting hardware. The quoted significances, about 56 sigma for the triangle holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented model. They are not systematics-aware, and neither experiment is loophole-free. Read them as consistency checks on real hardware, not as precision measurements. ERRATA. Version 1 of this note contained four errors of substance. Section "What changed" states each of them and its repair. They are recorded rather than quietly removed. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. Each tenet is either a theorem of the companion papers, a machine-verified certificate in the public suite, or is explicitly labeled a conjecture. The paper offers structural characterizations within its stated axioms; it is not an interpretation-free derivation of Copenhagen and it does not claim to resolve the measurement problem. AI-ASSISTANCE DISCLOSURE. This work was carried out with substantial AI assistance. Every load-bearing claim is machine-verified by scripts that ship with the papers, and the author takes full responsibility for the contents. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Paper C: https://doi.org/10.5281/zenodo.21633744 (posted 2026-07-28, after three adversarial review passes; it carries a labelled retraction of an earlier sqrt(2) reading). VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Files of this version are restricted (18 October 2026) because it was released under earlier licence terms: the source code was under the MIT License. All current versions use one clear licensing: paper, figures and data under CC BY 4.0, source code under the Apache License, Version 2.0. Please use the latest version: https://doi.org/10.5281/zenodo.21134454 Standard correlation witnesses — spatial Bell/CHSH inequalities, temporal Leggett–Garg K₃, and KCBS contextuality — are known to detect nonstabilizerness ("magic") in some settings. We report a worked example, a k-resolved nonstabilizerness map of the SU(2)_k anyon braid-representation family, in which a standard temporal witness is instead systematically blind: at k=4 the three-time Leggett–Garg witness saturates its macrorealistic bound exactly (K₃=1.000000 over every state, braid element, and measurement axis, in the equal-time-step protocol of Table 1 with V₁=V₂) while the single-qubit fusion channel carries near-maximal nonstabilizerness (M₂=0.5585, 95% of the finite-dimensional ceiling log₂(3/2)). We prove this blindness as a structural theorem: the witness depends only on the Bloch-sphere Gram geometry of the braid orbit, not on nonstabilizerness, and k=4 happens to align the fixed measurement axis with a threefold orbit symmetry (Bloch dot products all −1/3) that caps the witness at 1; a finite, Clifford-generating group — the chiral octahedral group, the k=2 braid image — with a misaligned axis reaches K₃=3/2 under a two-propagator protocol. Neither finiteness of the braid image nor the Clifford property is by itself the mechanism. We complement this temporal certificate with three independent certificates of genuine nonstabilizerness — two of them long-range (a doubled-Fibonacci mutual-information witness, H=1.700979, and a gauge-invariant minimum ground-space stabilizer Rényi entropy, ≥6.5 against an exact-zero toric-code control), the third a complementary gate-based non-Cliffordness measure — an honest negative control confirming that leakage-free constructions remain strictly additive, and a hardware-oriented single-qubit signal-to-noise prediction. We further probe the k=4 dissociation at its shared d=3 interface with Kochen–Specker–Klyachko contextuality, where the KCBS witness shows only a weak, nongeneric correlation with magic. Two reconciliations with the literature are made explicit: the present result contradicts neither the equivalence of maximal nonlocality and maximal magic established for optimized magic states in a different game, nor the contextuality-supplies-magic theorem; both concern a different object than the fixed, per-k braid generator studied here. About this series: This record is part of a series of related works from my independent research on Fibonacci anyons, with Ising anyons as their natural counterpart. I started in April 2026, and it has been a long and insightful journey in which I learned a lot; the work uses different methods and stays within verifiable, nonspeculative physics. The common thread of the series is a split: Ising anyons are limited to Clifford operations, while Fibonacci anyons are computationally universal, and across the series I map what standard witnesses of nonclassicality can and cannot certify on such systems. I consider Fibonacci anyons a serious candidate for topological quantum computing, given their universality and their topological protection against local noise. A hybrid approach with Ising is conceivable, but problems such as instability and certification would have to be solved first, and each needs research of its own.Correction notice (2026-10-10). In the summary above and in the section on the certificates of the paper, item (2) is described as a second long-range certificate: “a gauge-invariant minimum ground-space stabilizer Rényi entropy, ≥6.5”. This description is too strong. The values reported for item (2) are Monte Carlo estimates of the total stabilizer Rényi entropy of individual sampled ground-space vectors on small lattices; they are neither a minimum over the ground space nor a lower bound. They show that these states are not stabilizer states. A value of the total magic at one fixed system size does not certify long-range magic, which is a property of families of states of growing size, defined by robustness under shallow local circuits (F. Wei and Z.-W. Liu, arXiv:2503.04566; D. A. Korbany et al., arXiv:2605.22424; Y. Zhang et al., arXiv:2605.15150). The long-range certificate of this work is item (1); that non-Abelian string-net ground states carry long-range magic is shown by Zhang et al. In the same section, the fraction of Pauli expectation values in {0, ±1}, measured for one ground-space state, is a nonstabilizerness indicator only; it is not a diagnostic proposed by Wei and Liu. No computed value changes. The paper text will be corrected in the next version. This paper asks whether the inert blind spot at k=4, where the temporal Leggett-Garg witness stays exactly at the classical bound, means that the quantum resource is absent or only invisible to that witness, and it finds the nonstabilizerness nearly maximal, confirmed by three independent certificates, while proving the blindness to be geometric: the witness depends on the geometry of the braid orbit rather than on the resource, and a finite Clifford-generating group with a misaligned axis reaches the quantum bound, so neither finiteness nor the Clifford property is the mechanism; within the series it is the resource explanation, where blindness is a matter of alignment, not absence. Use of AI tools: In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's. I run multiple verifications at the different stages of the work and one before release, including cross-checks with an unrelated AI model from a different company, and all references are checked against the original sources. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication.
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.