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
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
Preprint. Third paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), Paper D (the observer-context category) and a synthesis note. The slogan "contextual holonomy encodes quantum contextuality" is attractive and, as usually stated, false. This paper gives it a sharp form. On the context stack of maximal abelian subalgebras the canonical degree-2 class has order exactly n, and its cocycle holonomy along closed context loops detects EXACTLY the all-versus-nothing sector of contextuality for closed-triple Weyl families at even local dimension. Both containments are strict, each by an explicit certificate: at d=3 the class is nontrivial of order 3 while the achievable state-independent AvN value is 0, and KCBS exhibits state-dependent contextuality where the class shadow vanishes. The Wigner-friend loop generated by H and S splits by level: its projective (Pancharatnam) holonomy is +1 while its determinant-section holonomy is exactly -i, so any interpretational claim about "the" holonomy of that loop must state which one it means. ATTRIBUTION. The classification of state-independent values for qudit Weyl families, the even/odd dichotomy and the fixed value d/2, on which the state-independent side of the equivalence rests, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024). Attainment at general even d is the Raussendorf-Okay-Zurel-Feldmann construction. This paper uses both and rediscovers neither. A RETRACTION IS CARRIED IN THIS PAPER, NOT BURIED. An earlier draft presented the tier-two facets of the d=4 census on a (1/(2*sqrt2))Z grid with a single coefficient magnitude 3/(2*sqrt2), and read the resulting unit-coefficient bounds 4*sqrt2 and 14*sqrt2/3 as placing Tsirelson's sqrt2 inside the tier-two geometry. The whole presentation was a normalization artifact and is withdrawn (Retraction 1): the stored lift has coefficients in {0,+-1} with integer bounds 6 and 7, every facet functional in the census is rational, and Q(sqrt2) is not among the eighteen canonical arithmetic fields. Certificate: d4_arithmetic_profile.py. WHERE THE EVIDENCE IS THIN, STATED IN THE PAPER. Two sweeps quoted here drew no positive instances at all: the d=4 half of the 2,100-family equivalence test, and the 6,500-trial false-certificate sweep, which draws no contextual family at any of d=2,4,8 and therefore establishes the absence of false certificates only vacuously. Both are disclosed where they appear. Theorem 2 is labelled machine-validated rather than proved: the closed form is established end to end for the deletion families treated, while the general closed-triple d=2 statement in compressed coordinates remains open and is recorded as open in the verification ledger. HARDWARE CAVEAT, STATED UP FRONT. The quoted significances, about 56 sigma for the ray-level Wigner-friend holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented witness model and conditional on compatibility and nondisturbance assumptions. They are not systematics-aware and neither measurement is loophole-free; a roughly 3.5 sigma preparation offset on one strand is itself evidence of unmodelled systematics. We also record that the pinned value S = 4.6125 cannot be recomputed from the stored counts (a naive recomputation gives 4.5662, a gap of about 2.7 standard errors); the archived value 4.558 +- 0.018 is the one an independent reader can reproduce from the deposited data. Read these as consistency checks on real hardware, not as precision measurements or certifications. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. It was held back from the first publication batch and released only after three adversarial review passes; what each pass found, and how each item was closed, is recorded in verification/INDEX.md rather than quietly fixed. One defect found there is NOT closed and belongs to the companion Paper B, which cites a false-certificate artifact that is not in fact in the repository; it is recorded rather than left for a reader to discover. 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 . Note: https://doi.org/10.5281/zenodo.21630739 . Code archive: https://doi.org/10.5281/zenodo.21498498 AI-ASSISTANCE DISCLOSURE. 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 by the scripts named in the text, and the author takes full responsibility for the results, including any errors. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Preprint. Fourth paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), and a note on local validity. This paper makes "observer" a mathematical object: an isometric enlargement of the system together with a measurement context on the enlarged algebra, organized into a category whose invertible switches form a subgroupoid and which carries the U(1) cocycle of the companion papers. Four results follow. (1) Relative facts. Every observer admits an exact local classical section, while no global value assignment exists for the Peres-Mermin family, nor for any enlarged family containing its image under a Wigner enlargement. (2) A gauge-robust Z2 invariant of switch loops: +1 on all loops of Weyl switches, -1 on the oriented mutually-unbiased triangle. Nontriviality requires complementary switching; the converse fails, and the counterexample is pinned. The invariant counts switching parity at the tau level, the same object as the Wigner-friend -i. (3) The movable cut. The comparison map between cut placements is fixed only up to a choice of section: its ray-level content is invariant, the section-level bookkeeping shifts by the loop holonomy, and the update rule available to each observer is forced. (4) The Frauchiger-Renner chain is localized. Its observer cycle has trivial class and holonomy exactly +1, which refutes the natural conjecture that such paradoxes measure holonomy. Its contradiction is instead state-sector, with contextual fraction exactly 1/6 and the CHSH inequality at value 7/3 as the unique separating facet of the codeword polytope. ATTRIBUTION. The classification of value assignments on the Peres-Mermin square that this program builds on is due to Abramsky, Cercelescu and Constantin (FSCD 2024). Their priority is acknowledged in the text and here. HARDWARE CAVEAT, STATED UP FRONT. One result is anchored by data taken on ibm_fez. The quoted significance of roughly 56 sigma is shot-noise statistical only, computed under the implemented interferometric model. It is not a systematics-aware figure, and the experiment is not loophole-free. Read it as a consistency check of the ray-level prediction on real hardware, not as a precision measurement. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. Results are tagged in the text as Complete, Sketch, or Gap. The categorical and cohomological layer awaits specialist review. The paper presents 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 paper, and the author takes full responsibility for the contents. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
CORRECTION IN v2 (2026-07-28). Proposition 9 of version 1 is withdrawn. It stated that the mod-2 (odd-Q) shadow of the exact Z_d criterion can produce a false certificate, quoted 28.6 percent as the rate at d=8, and cited an explicit example in the repository. All three are wrong. For even d, solvability of Mx = gamma mod d implies solvability mod 2, so any lambda closing mod 2 pairs evenly with gamma: the shadow can never fire on a noncontextual system. The false-positive rate is exactly 0 at every even d, provably, and no such certificate exists, which is why the cited artifact was never committed. The criterion is instead incomplete in the opposite direction: a Z_d-cycle can certify with a value that is nonzero mod d but even, and the mod-2 reduction of that value vanishes, so the shadow stays silent on a genuinely contextual system. The v1 error made the criterion look weaker than it is while naming the wrong failure mode. What survives is the point on certificate minimization: a minimal odd-Q support need not be a Z_d-cycle, so it can be a false witness while the verdict is always right. Carried in the paper as a labelled Remark; no other result depends on the withdrawn claim. Certificate: verification/shadow_soundness_exact.py. State-independent contextuality of qudit stabilizer measurements is not just present or absent: it comes in a definite strength. For m-qudit Weyl-Heisenberg context families in local dimension d, the achievable all-versus-nothing obstruction value is exactly {0, d/2} for every even d and {0} for odd d. ATTRIBUTION: this value classification, both the even/odd dichotomy and the fixed value d/2, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024), whose parity obstruction 2k = 0 (mod d) and normal-form classification settle it. Our Obstruction Spectrum Theorem recovers their result in explicit Weyl-certificate language; it is not an independent discovery, and the paper says so in its abstract, its related-work section and a theorem-by-theorem comparison table. On that basis we develop three further layers. (i) A closed-form calculus: contextuality is decided by an integer commutator carry b(C), computed from the symplectic form with no phases or matrices, together with an exact value-bit formula and a cycle criterion. The carry invariant has the algebraic form of a Pontryagin-square quadratic refinement; we do NOT construct the cohomology operation, and the language is hedged accordingly. (ii) A 2-adic tower: the obstruction lives at the top layer and no single lift carries it (Depth Rigidity), with a fiber doubling law governing propagation d -> 2d. The doubling law is stated as a conjecture with an explicitly open step. (iii) A three-device superconducting replication of the d=2 witness. HARDWARE CAVEAT, STATED UP FRONT. The combined figure of 80 sigma is SHOT-NOISE ONLY and overstates the combined significance. The three devices fail a homogeneity test (chi-squared about 266 on 2 d.o.f.), so a systematics-aware combination gives approximately 7 sigma, not 80. Each device individually violates the bound by 32 to 61 sigma on shot noise, so the qualitative violation is robust, but this is NOT a loophole-free certification and the 80 sigma number should not be read as a single-S significance. This is a preprint, not a peer-reviewed article. The obstruction's geometric home, a canonical degree-2 class on the stack of maximal abelian subalgebras, 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.
Extension des cadres de contextualité : du bruit à la séquentialité La recherche d'une meilleure compréhension de la nature est une quête éternelle pour les physiciens. Si la théorie quantique prédit la plupart des résultats expérimentaux obtenus à ce jour, son interprétation reste controversée. Au cours du siècle dernier, cela a maintenu occupés les scientifiques du monde entier, conduisant à deux théorèmes fondamentaux en information quantique : le théorème de Bell et le théorème de Kochen-Specker. D'une part, le théorème de Bell prouve que toute théorie doit être non locale et d'autre part, le théorème de Kochen-Specker examine la contextualité, une notion similaire à la non-localité sous d'autres hypothèses. La contextualité est très prometteuse et au cœur de nombreux avantages quantiques, tels que l'accélération du calcul et la communication restreinte. Cependant, fournir une vérification expérimentale convaincante de la contextualité reste complexe, car les tests sont sujets à des imperfections, appelées failles, qui affaiblissent l'argument logique. On peut citer la faille de précision finie : toute mesure réaliste est intrinsèquement bruyante ou la faille de compatibilité, où deux mesures consécutives ne sont pas mesurables conjointement. De plus, la contextualité apparaît dans divers cas, notamment dans des scénarios séquentiels où les mesures sont effectuées en séquence sur le même système. Dans ce cas, la notion de contextualité existante devient obsolète, et les hypothèses doivent être adaptées. Afin de mieux appréhender la contextualité, nous proposons deux cadres. Tout d'abord, nous assouplissons le déterminisme et l'indépendance des paramètres, qui constituent les hypothèses fondamentales de la contextualité définie par Kochen et Specker. Nous élaborons un cadre pour traiter les écarts par rapport à ces hypothèses, ce qui permet de définir une nouvelle notion de contextualité adaptée aux imperfections expérimentales. Cette notion de contextualité s'exprime par une limite sur la fraction contextuelle (une mesure de la contextualité) dont nous prouvons la continuité. Nous introduisons également la fraction de signalisation. Enfin, nous démontrons comment ce cadre peut s'appliquer à des données expérimentales avec quelques exemples. Le deuxième cadre que nous présentons concerne les scénarios séquentiels, dans lesquels les dispositifs sont placés l'un après l'autre. À la différence des scénarios de mesure standards, chaque dispositif interagit avec le système et le perturbe. C'est pourquoi, dans ce cadre, nous formalisons les caractéristiques et la contextualité des scénarios séquentiels. Aussi, nous relions la non-contextualité des scénarios séquentiels à des hypothèses physiques, de la même manière que Kochen et Specker. Sous certaines conditions, nous établissons une correspondance entre les scénarios séquentiels et les scénarios de mesure qui préserve la non-contextualité. La contextualité une caractéristique propre à la théorie quantique. Mais pour être sûr de son intérêt, il faut s'assurer de la véracité de la théorie quantique. C'est dans cette optique que le théorème de Pusey, Barrett et Rudolph (PBR) a montré que, sous des hypothèses raisonnables, chaque état pur de la théorie quantique est un élément distinct de la réalité. Ainsi, nous menons une étude préliminaire du théorème PBR et lui fournissons une structure. Ce travail se concentre également sur un élément central du théorème, la notion de chevauchement. À partir de cette notion, nous parvenons à de nouvelles définitions et à une nouvelle compréhension du chevauchement. Cette étude ouvre la voie à d'autres directions de recherche à partir du théorème PBR. Enfin, nous complétons cette thèse par un paquet Python open source, disponible sur GitHub et PyPi. L'objectif de ce paquet est de fournir un outil accessible à tous et toutes dans le domaine de la contextualité. Nous présentons donc son utilisation et son utilité au travers d'exemples et de documentation.
Contextualité et non-localité dans les systèmes décrits par des variables continues La mécanique quantique présente des propriétés étonnantes qui n'ont pas d'équivalent en physique classique. Ces propriétés sont au cœur des applications possibles de la mécanique quantique. Le thème principal de cette thèse est l'étude de deux des propriétés fondamentales de la mécanique quantique: la non-localité et la contextualité. Dans ce cadre, nous poursuivrons deux objectifs: premièrement, nous étudierons comment certains résultats obtenus pour les systèmes discrets peuvent être étendus aux systèmes décrits par des variables continues; deuxièmement nous étudierons comment il est possible de tester ces deux propriétés dans les systèmes quantiques décrits par des variables continues.Dans une première partie, nous étudions l'ensemble des distributions de probabilités locales et ``no-signaling'', c'est à dire qui ne permettent pas de transmettre d'information. Nous commençons par traduire le problème en terme de contraintes sur des espaces de mesures de probabilité. Nous introduisons ensuite un ensemble de mesures de probabilité qui sont les analogues en variables continues des probabilités découvertes par Popescu et Rohrlich dans le cas discret. Enfin, nous caractérisons l'ensemble des mesures de probabilité ``no-signaling''. Plus précisément, nous montrons que les mesures introduites sont des points extrémaux de l'ensemble des mesures de probabilité ``no-signaling'' et que leur enveloppe convexe est dense dans l'ensemble des mesures de probabilité ``no-signaling''. Dans une seconde partie nous nous intéressons à une preuve de la contextualité de la mécanique quantique dans une formulation qui ne dépend pas de l'état. Plus particulièrement, concernant l'inégalité de non-contextualité de Peres-Mermin, nous montrons qu'il est possible de la généraliser pour des observables définies sur des espaces de Hilbert de dimension arbitraire, voire infinie. Cette généralisation nous permet d'identifier les propriétés communes des observables qui conduisent à une violation maximale de l'inégalité de Peres-Mermin.En dernier lieu, nous nous intéressons à des états intriqués du champ électromagnétique de deux cavités. Ces états sont non-locaux et violent une inégalité de Bell formée de mesures de la parité déplacée. Nous étudions comment ces états peuvent être préparés et mesurés expérimentalement. Enfin, nous analysons l'effet des imperfections expérimentales et des pertes
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.