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
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
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
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
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.
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.
CT 4.0.2 is the foundational bundle for Complementarity Theory (CT): physical updating modeled as finite committed record crossing an interface aperture under a least-incoherence selection rule, with mathematical setting in synthetic differential geometry (SDG). The paper presents five contributions: (1) A conditional derivation of the local Minkowski interval at a stable aperture in the first-order single-aperture regime, from CT primitives together with named structural inputs (a smooth aperture model, an adapted-frame choice, a posited seam-budget axiom, and a saturating-history existence hypothesis). Every non-theorem input is stated where it enters. (2) A complex-amplitude structure for the soft-commit regime: composition and superposition as operational axioms; the Born probability map definitional, with a named, independently testable interference-order axiom (after Sorkin) as the stated route to its quadratic form; a finite-dimensional Hilbert structure following from the axioms. (3) A multi-subject geometric proposal: intersection-network gluing with subjects anchored at the primitive substrate. The transported Lorentz triangle product is classified as connection loop holonomy — identity marks a flat face — distinct from Čech patch consistency; the U(1) face holonomy is conditionally realized as the normalized third-order Bargmann invariant. (4) A research program for deriving the residue role H from τ₀-substrate structure in complexified SDG, with the Residue-to-Phase problem restated precisely (the in-paper result is narrowed to a valid minimal-compact-factor theorem) and a conjectural horizon-phase quantization ansatz. (5) A conditional phase-to-spin and two-spinor bridge: three orthogonal noncommuting phase flows on a distinguished C² amplitude sector generate SU(2) ≅ Spin(3), with the exact 3-2-3 correspondence and the standard Penrose two-spinor route to Lorentzian geometry — established mathematics whose CT premises are recorded as named open problems. Version note: 4.0.2 is a status-correction erratum revision of the 15 May 2026 deposit (that record carries the Zenodo version label v4.0.1; there is no intermediate content revision). Claims are re-labeled to match what is proved, in nearly every case restoring the labels the supporting notes themselves use. A new supporting note, Identity as History, proves the record/history recurrence dichotomy behind the paper's arrow-of-time remark; the P-bound diagnostic is added with a reflexive audit caveat. The full changelog is Appendix A of the paper. The bundle contains the paper, twelve supporting notes (LaTeX and PDF), two H-program notes, the standalone SDG note, the Malus note, and a navigation README. An AI-assistance disclosure is included in the paper's acknowledgments.
A finite defect is never measured in the abstract. It is measured relative to a declared evaluation structure: what is compared, which repairs are admissible, which gauge measures the remaining defect, which transformations preserve the evaluation, and which terminal rule assigns status. This paper consolidates that structure into a single finite observer object. The observer object packages a carrier, local state assignment, comparison maps, defect map, repair coboundary, optional closure map, gauge, compatible transformation class, and terminalizer. It is a definitional companion to the finite obstruction calculus: it introduces no new tower and makes no physical, dynamical, empirical, or authority claim. The paper proves three elementary structural facts. First, the observer-relative obstruction is well defined on the quotient residual space when the admissible repair class is linear. Second, the observer triple already used in Papers 000 and 008 assembles into the larger observer-object packet once the carrier, repair class, closure map, and terminalizer are declared. Third, for fixed carrier and repair data, the residual class is independent of the gauge, while its measured magnitude is fixed by the declared observer gauge. The practical point is narrow: every reported measurement already assumes an observer. Without declaring that observer, the same defect can be reported as different numbers, or a proposal can be mistaken for an authorized judgment. This paper gives the corpus one finite place to ask: under what observer? Limits and scope. This paper does not claim observer-independent authority, derive quantum mechanics, derive the Born rule, solve the measurement problem, explain consciousness, or prove that every possible evaluation factors through this packet. Corpus-wide packet completeness remains an open bookkeeping obligation.