This paper uses the standard triangular odd-cycle anticorrelation obstruction as a finitebenchmark for Virtual Probability Theory. The obstruction itself is classical: three binary paircontexts can be locally overlap-compatible while admitting no positive global gluing, becauseperfect anticorrelation around an odd cycle violates the parity constraint. The contributionis therefore not a new contextuality obstruction, but a complete VPT fiber-and-certificateaudit of this minimal benchmark.On the declared triangular source architecture, the paper proves the benchmark member-ship profileP△ /∈ C+△, P△ ∈ Clin±△ , P△ ∈ CQ2,R△ ,and also proves the fixed-source nonnegative scalar Q2 obstruction and the same-sourcecoherent-versus-incoherent scalar-square control separation. It characterizes the signed-linearfiber, the architecture-relative hidden-negativity minimum, the exact coherent scalar Q2-fiber for the perfect target, the closed noisy scalar realization profile, and a finite-unionaggregate-cone description for noisy targets.All conclusions are finite, architecture-relative, readout-relative, and algebraic. ScalarQ2 is used as a score-generating parametrization, not as a new observable probabilitycalculus. The finite-code certificate is context-outcome-address-free and target-entry-value-free, but explicitly benchmark-pattern-aware; its code-length field is a syntactic audit, not aKolmogorov-complexity, minimum-description-length, or compression theorem
This paper uses the standard triangular odd-cycle anticorrelation obstruction as a finitebenchmark for Virtual Probability Theory. The obstruction itself is classical: three binary paircontexts can be locally overlap-compatible while admitting no positive global gluing, becauseperfect anticorrelation around an odd cycle violates the parity constraint. The contributionis therefore not a new contextuality obstruction, but a complete VPT fiber-and-certificateaudit of this minimal benchmark.On the declared triangular source architecture, the paper proves the benchmark member-ship profileP△ /∈ C+△, P△ ∈ Clin±△ , P△ ∈ CQ2,R△ ,and also proves the fixed-source nonnegative scalar Q2 obstruction and the same-sourcecoherent-versus-incoherent scalar-square control separation. It characterizes the signed-linearfiber, the architecture-relative hidden-negativity minimum, the exact coherent scalar Q2-fiber for the perfect target, the closed noisy scalar realization profile, and a finite-unionaggregate-cone description for noisy targets.All conclusions are finite, architecture-relative, readout-relative, and algebraic. ScalarQ2 is used as a score-generating parametrization, not as a new observable probabilitycalculus. The finite-code certificate is context-outcome-address-free and target-entry-value-free, but explicitly benchmark-pattern-aware; its code-length field is a syntactic audit, not aKolmogorov-complexity, minimum-description-length, or compression theorem
Virtual Probability Theory represents observable finite probability laws as normalized readoutsof latent virtual contribution objects. VPT1 [1] established the declared readout architectureand the admissible-score-to-probability theorem. VPT2 [2] classified the readout fibers andobservable quotient structures induced by aggregation, scoring, normalization, and observation.This paper develops the third foundational layer: the admissibility geometry and invariancetheory of fixed declared linear and quadratic score mechanisms.The linear regime is modeled on a finite-dimensional ordered real vector space (V, K), whereK is a closed convex pointed cone and each declared score functional belongs to the dual cone,ℓO,y ∈ K∗.The paper distinguishes cone-positive aggregate domains, visible nonnegative-score domains,and fully admissible domains. It proves that cone-positive aggregate domains are inverse-imagecone domains, develops visible-cone gap theory for nonseparating score channels, formalizesconstrained reachability of visible gaps, identifies hidden cone-violation cancellation, and recoversordinary finite source probability and its deterministic contextwise pushforwards as the positivescalar-linear sector.The quadratic regime is modeled on real or complex finite-dimensional mode spaces withfixed positive semidefinite forms GO,y . The paper proves the null-channel characterizationsquad,AO (y) = 0 ⇐⇒ AO,y A ∈ ker GO,y ,and describes the admissible domain as the complement of contextwise null sets. It then studiesreadout-kernel invariance, real and complex scale invariance, global phase invisibility, localrelative-phase diagnostics, declared component interference, and partition-coarsening behavior.The contribution is not new cone duality or new quadratic-form theory in isolation. Thecontribution is a regime-specific admissibility and invariance calculus for fixed VPT readoutmechanisms. On the nonempty admissible domain, observable probabilities remain ordinaryfinite probability laws throughout.
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.
Port closure is not identifiable from a scalar trace alone. For every finite signed BV trace measure μ and load I>0, Theorem 1 characterizes the positive accounting fiber μ=I K+A−B and shows that its K-projection is the entire cone of finite nonnegative measures. The Jordan decomposition is classical; the contribution is the port-specific maximality statement: no nonconstant port property is informative before extra-trace restrictions are supplied. Relative to fixed Q and declared information C, a port predicate is certified only when constant over a nonempty compatible class; a property-changing admissible gauge exposes certificate insufficiency. Empirical validity of C and causal exhaustivity remain separate premises. Once a valid certificate fixes an exhaustive one-port allocation, Bᵢ=Dᵢ+I Kᵢ+Aᵢ. A uniform restorative bound yields Kᵢ≤(B*−Δ)/I and makes time above any certified coupling floor O(1/I). Proposition 7.4 supplies a complementary obstruction: in the pumped qubit, any fixed C¹ strictly increasing coordinate preserving fixed-depth return at unbounded load forces ess sup cᵢ≥Δ I, so P3 and L1 cannot hold together. The qubit witness has unit port density and closes only by duration collapse; an affine shelving witness closes through depletion of the pumpable population, G=O(1/I). Direct and displaced finite-rate counting demonstrate record ambiguity, while a frozen Morris-Lecar audit shows growing restorative capacity and eventual loss of fixed-depth return. These model results do not establish a distinct reduced-coupling mechanism in a real device, which still requires an independently produced certificate, empirical one-port exhaustivity, and measured exposure under a frozen episode rule. All conclusions are relative to (Q,C). Keywords: maximal port ambiguity; data informativity; certification; coordinate-relative identifiability; bounded variation; positive accounting fiber; gauge-insufficiency test; port closure; unknown-input reconstruction; port-Hamiltonian identifiability; finite-capacity exposure law; quantum trajectories; electron shelving; Morris-Lecar; hybrid systems
Here “operational objectivity” is used only in a certificate-relative sense: satisfaction of a selected public-audit standard, not quantum Darwinism, spectrum-broadcast objectivity, or complete intersubjectivity. A declared family of complete queries determines an access-relative public class. We first state an elementary fiber-stability proposition: when a certificate’s target dependence factors through the public evaluation map and its auxiliary records do not encode which member of a public fiber was realized, a pass cannot identify the underlying physical state if that map is noninjective. We then propose a typed audit certificate C = (W,I,Cal,G,D,R,U) with six requirements: declared access, calibrated interventions, relative gauge alignment, a dependence-valid evidence pedigree, reproducibility with independently generated error oppor- tunities, and minimal epochal public revision. Construction J1 tests both the joint consistency and the inferential limit of this architecture inside standard real-amplitude quantum theory with ordinary real tensor composition. Ideal real-trine effects are nominal targets, while symmetrically relabeled noisy trines are the implemented effects and span each full local rebit effect space. Nevertheless I4 ±(3/5)J⊗J ρ±= 4 give probability 1/9 for every outcome of every implemented epoch-0 product context. A separately declared and calibrated global projective measurement outside that interface distinguishes them with total-variation distance 3/5 and equal-prior success 4/5. The witness depends on an independently trusted, hidden-direction-sensitive calibration source χH = (I4 + J⊗J)/4; calibration-source access is not target-query access, and no preparation–measurement duality or joint-SPAM self-calibration is assumed. Supporting results record exact transcript invariance for finite adaptive protocols on fresh preparations, the bound 1−(1−ε)n for an ε single-round discrepancy, and a distribution-free finite-sample upper certificate. Thus the selected O1–O6 pass can coexist with noninjectivity of the full separated-product map in J1. The claimed contribution is the fiber-stability diagnostic, typed synthesis, and joint witness, not a new operational quotient, local-tomography theorem, rebit defect, general objectivity criterion, or selection of complex quantum theory.
Complementarity-First Foundational Release I is a ten-record research release consisting of nine technical papers and this overview. The technical corpus comprises 246 date-neutral public-manuscript pages and spans a conceptual-formal primitive, four quantum-foundations studies, a paired-incidence and transport architecture for finite/local gravity, two distinct local Regge descendants, and a source-controlled Unified Dynamics synthesis. This article supplies the release- wide paper map, dependency graph, claim/evidence classification, artifact register, and open-problem ledger. Its organizing principle is anti-flattening: conceptual proposals, conditional theorems, exact countermodels, finite computations, prospective tests, negative results, and source-controlled syntheses are not treated as interchangeable evidence. The foundational paper defines a completed relation with internal role distinction and shows by elementary independence models that the primitive does not itself determine probability, continuity, geometry, dynamics, or complex quantum theory. The quantum branch separates a conditional reconstruction audit from three focused problems: readout of the two-rebit tomographic defect, certificate-relative operational objectivity underanoninjectivepublicinterface, andtheresourcerequirementsforrecordsandfiniteautonomous maintenance without a free thermodynamic arrow. The gravity branch begins only after a rank-two paired-chiral carrier, Lorentzian real structure, orientation, affine displacement sector, and transport law are selected; common-coframe data then arise only conditionally through the admitted simplicity, gluing, and nondegeneracy chain, after which the branch reaches conditional Palatini–Regge structure and exact finite periodic endpoints but not arbitrary-mesh or continuum closure. Two later gravity papers address different questions: descriptor-local H2/H2R transfer and a conditional curved-Regge identity, versus an intrinsic six-sector Reynolds cancellation mechanism for local zero germs. Unified Dynamics organizes controlled quantum and gravity descendants around a finite dual-pair carrier and a common BF-type kernel while retaining a variation-space obstruction, open complement-operation coherence, no absolute scale, and no Level-4 prediction. The strongest defensible conclusion is therefore a disciplined finite/local reconstruction architecture with exact technical advances, explicit failures, and auditable selectors. Release I is not a completed theory of physics, a continuum unification, a derivation of the Born rule or complex scalars from primitive complementarity, a quantized-gravity result, or an externally peer-reviewed and independently replicated corpus.
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.