We develop an exact structural reduction for a characterized class of three-setting, binary-outcome Bell functionals, transforming the operator optimization into an alternating one-dimensional Jacobi variational problem with a Bellman storage dual. The framework includes square-scalar operator welds, a classification of sixteen universal weld branches, two-matching coupling graphs, and endpoint-padded path-completeness rigidity. As an application, it contains I3322 and the Gigena–Kaniewski family and gives the exact rational bound ω(I3322)≤0.250875384626, yielding a certified two-sided interval of width 1.1202346452×10−10 when combined with an independent exact lower certificate. The work is structural and self-contained; critical attainment and dimension-growth questions are treated separately in a companion paper.
We develop an exact structural reduction for a characterized class of three-setting, binary-outcome Bell functionals, transforming the operator optimization into an alternating one-dimensional Jacobi variational problem with a Bellman storage dual. The framework includes square-scalar operator welds, a classification of sixteen universal weld branches, two-matching coupling graphs, and endpoint-padded path-completeness rigidity. As an application, it contains I3322 and the Gigena–Kaniewski family and gives the exact rational bound ω(I3322)≤0.250875384626, yielding a certified two-sided interval of width 1.1202346452×10−10 when combined with an independent exact lower certificate. The work is structural and self-contained; critical attainment and dimension-growth questions are treated separately in a companion paper.
We determine the quantum value of the three-setting, two-outcome Bell functional I3322. The value is the unique intersection q† of an analytic A-high stable curve of the exact Jacobi characteristic map with a central reverser curve. A computer-assisted proof isolates this intersection in a rational interval of width 10−55, beginning 0.2508753845139765356173366109459512989013219739714764755. We prove that the supremum over finite-dimensional tensor-product strategies, the unrestricted spatial value and the commuting-operator value all equal q†. No finite-dimensional strategy attains the value, whereas a normal infinite-dimensional spatial strategy and a commuting GNS strategy do. The proof combines an exact Bellman/path variational reduction, extraction of a positive bilateral calibrated Jacobi fibre from a maximizing weak-star state, a reflected-fibre central-incidence theorem and interval certificates for the unique stable/central intersection. A separate operator-positive three-remainder decomposition gives the commuting upper bound. The complete verification record is archived at DOI 10.5281/zenodo.22160086.
This preprint constructs finite rational proof objects for both sides of the quantum value of I3322 Bell inequality. Let 𝑄t denote the supremum over finite-dimensional tensor-product binary measurements and𝑄c the commuting-operator value. On the complete word space of total length at most four, we construct a rational Gram certificate. The full signed setting-relabeling stabilizer is the dihedral group of order eight, reducing the certificate to five rational blocks. Ex-act affine reconstruction and a uniform diagonal shift of certified positive interior prove: 𝑄c (𝐼3322) ≤ 62718846253/250000000000 = 0.250875385012. Independently, a local-dimension-499 strategy specified by rational projectors and a nonzero unnormalized rational Schmidt vector has exact normalized value 𝐿499 = 0.25087538451397653548 . . .. Con-sequently, 𝐿499 ≤ 𝑄t(𝐼3322) ≤ 𝑄c (𝐼3322) ≤ 0.250875385012, with endpoint difference below 5 × 10−10. The lower certificate also proves 𝑄t(𝐼3322) > 1/4. Every theorem is checked using exact rational or integer arithmetic; floating-point computation is used only to discover candidates. Complete certificates, source code, tests, and reproducibility materials are archived at: doi:10.5281/zenodo.21462892.
A post-release exact-arithmetic audit found a load-bearing global-amplitude gap in the Bellman upper-bound assembly of the earlier releases (v1.0.0-v1.2.0). Those releases are preserved as historical records; their exact-optimum, nonattainment, spatial-separation, and nonclosure claims are under correction and are not asserted by this version. What this version rigorously and unconditionally certifies: 0.2508753845015185 < omega_tensor <= omega_commuting <= 0.250875388108398 (window width 3.61e-9), with the lower endpoint an explicit 255-dimensional rational tensor-product strategy verified in exact arithmetic and independently reconstructed at 160-digit interval precision, and the upper endpoint an exact rational Bellman subsolution with an endpoint-clustered 25,601-knot witness, independently reconstructed cell by cell. The certificate stack, the audit that found the gap, a structural explanation (Bellman contact versus amplitude-normalization duality), and one recorded failed quantitative prediction are all included. Every load-bearing claim maps to replayable source and a frozen receipt.
This record contains the manuscript and complete verification artifact for the Bell inequality J62_4422 of Pál and Vértesi.It certifies the exact rational bracket750075557472719639 / 10^18 ≤ β_Q ≤ 750091086083765047288227625 / 10^27,and exact attainment of 3/4 by the stated two-qubit strategy. The lower endpoint is an exact rational strategy. The upper endpoint is certified by a sum-of-squares chain for the retained 365-word relaxation level.The Krawczyk certificate localizes the underlying KKT zero and establishes a strict local maximum, including negativity of the border form in all 256 boundary strata; it does not establish global optimality of that localized point. Accordingly, this work does not claim the exact quantum value, qubit-restricted optimality, minimum optimizer dimension, or global optimality of the localized KKT point.The archive includes SHA-256 manifests, standard-library replay programs, independently written certificate checks, mutation controls, and pinned full Krawczyk replays. The acceptance programs are fail-closed under optimized Python. The manuscript and certificates are licensed CC BY 4.0; verification code is licensed MIT.Version 1.1 is a hygiene-only update: it replaces one machine-local default path in a helper with the portable repository root ".", updates the corresponding checker and certificate source bindings, and leaves the mathematical calculation, inputs, claims, and replay outputs unchanged.
We give exact rational certificates for two results in the bipartite Bell scenario with three two-outcome measurements per party. First, the maximum of the I3322 (Froissart) functional over the almost-quantum set -- the 1+AB level of the Navascues-Pironio-Acin (NPA) hierarchy -- is certified to lie between 0.2514708838 and 0.2514709016 (width <2×10−8), in the normalization with classical bound 0 and quantum maximum ≈0.2508754; to our knowledge no value for this landmark had been published. Second, we exhibit a single exact rational behavior T∗ carrying certificates in both directions: a rational moment matrix proving T∗ almost-quantum, and a rational dual object proving that every NPA level-2 completion of T∗ requires slack at least q=4.5934×10−4, so that T∗ is not quantum. The strict inclusion Q⊊Q~ in the minimal three-setting scenario is thereby established end-to-end with no reliance on floating-point computation. Anatomically, the witness is locally quantum: all nine of its 2×2 correlator subtables pass the Tsirelson-Landau-Masanes criterion, so its non-quantumness is irreducibly a three-setting phenomenon, concentrated in ordering-dependent pseudo-moments coupling each party's third setting to the other's first two; the level-2 certificate is equivalently an exact sum-of-squares identity that any quantum state values nonnegatively and the witness is forced to value at −q. The certificates follow the rational rounding-projection-lifting programme recently developed for noncommutative optimization; for NPA moment matrices in canonical form the required projection reduces to a per-moment uniform shift, making certification essentially free at these sizes, and we work the programme in both the feasibility and infeasibility directions on one witness. Machine-checkable certificates and verifiers requiring only the Python standard library accompany the paper. Files in this record: the paper (PDF); exact rational certificates for both theorems (i3322_1AB_certificate.json, i3322_level2_infeasibility_certificate.json); standalone verifiers requiring only the Python 3 standard library (verify_i3322_certificate.py, verify_level2_certificate.py); and the generating scripts (certify_landmark.py, certify_level2.py). See README_ancillary.txt for which artifact proves which theorem and how to run the verifiers. v2: corrects the mechanism attribution in Sec. 5 (anatomy) and adds the archive reference to Data Availability; theorems, certificates, and verifiers unchanged.
A Kernel-Role Determinacy Theorem for Bell-Witness Fixation, No-Signaling, and Tensor-Level Compatibility This manuscript develops a fixed-domain AASC analysis of Bell-nonlocal quantum entanglement as a theorem of reusable witness determinacy. It does not introduce hidden variables, modify quantum mechanics, derive Hilbert-space formalism from below, or propose a new signaling mechanism. Instead, it asks a prior structural question: once a Bell-type operational family is used as the same non-degenerate witness across settings, lawful transformations, retained records, comparison, and later classification, what admissible same-domain standing form is forced? The governing claim is that AASC is not an optional interpretive overlay on physics. Once an operational Bell family is asserted as a determinate reusable witness, admissibility, standing, reference, and irreversibility are already active as minimal necessity conditions of same-witness objecthood. Target instantiation therefore carries governance occupancy: an alleged determinate Bell witness with no admissible governance realization is not an empty alternative, but a refusal or retyping of the target. The manuscript defines finite operational Bell objects with explicit setting carriers, outcome families, channel/interface data, standing-token valuations, continuation and report/reuse structure, comparison profiles, and act-time status. It then defines object-indexed query sorts, dependent response families, exactly recoverable operational simulations, lawful Bell groupoids, six governance-effect functors, and target-active extensional governance equivalence. The result is a typed interface in which arbitrary native formalisms can be compared by their target-bearing governance effects rather than by their internal syntax. The central Bell-specific theorem states that every same-target candidate-admissible Bell-governance construction is forced component by component into the certified boundary-compatible standing form on the complete target-active practice. Candidate admissibility does not assume the canonical form at the outset. Instead, target occupancy, lawful transport, standing-flow exclusion, compatibility certification, coordinate-deviation exhaustion, and componentwise closure force exact target-bearing realization and governance equivalence to the canonical representative CP∗C^{*}_{P}CP∗. The target-induced governance quotient therefore has exactly one element. The Bell carrier is treated as a compatible empirical model on a finite measurement cover. Bell-local completion is identified with the existence of a global section, and Bell nonlocality is the obstruction to such a section. The manuscript separates operational marginal invariance, absence of controllable communication, and exclusion of same-domain standing transport. No-signaling is therefore preserved without retyping Bell nonlocality as superluminal communication, metric transport, distant repair, or hidden standing transfer. The finite-dimensional quantum specialization uses the standard bipartite tensor-product and local-POVM carrier. Every separable realization in that carrier is Bell-local; therefore, if the fixed Bell-nonlocal empirical model has such a finite-dimensional quantum realization, every realizing state in that carrier is entangled. Post-quantum Bell-nonlocal no-signaling models remain within the general witness theorem but outside the quantum-entanglement corollary. The accompanying Lean 4 development, release v0.8.0, directly verifies the finite Bell-side witness carrier: denominator-32 vector/Born weights, structural no-signaling, CHSH numerator 808080, all sixteen deterministic local extrema, finite denominator-32 Bell-local-envelope exclusion, and conditional endpoint routing. Its direct witness certificate is vectorBorn_flagship_spine, and its deepest released endpoint is reasonableDeepestFiniteEntanglement_spine. The manuscript-level typed governance interface, empirical lawful groupoid, compatibility equivalence, coordinate-deviation exhaustion, completed role collapse, candidate-to-canonical derivation, and finite-dimensional quantum specialization are not attributed to the released Lean layer. This paper is the Bell-witness member of the paired AASC quantum determinacy arc. The record-side manuscript, Admissible Record Construction in Physically Non-Selective Quantum Regimes, fixes when local physical record occurrences stand as the same evidential record objects. The present manuscript fixes when the Bell operational family sampled by such records stands as the same reusable witness. The paired endpoint is therefore: record-role fixation+Bell-witness compatibility.\text{record-role fixation} + \text{Bell-witness compatibility}.record-role fixation+Bell-witness compatibility. The manuscript is classificatory and eliminative rather than mechanistic. It does not claim that AASC generates the Bell probabilities or supplies a hidden physical influence. Its claim is that reusable Bell-witness determinacy strongly constrains how Bell-nonlocal correlations may be coherently classified on a fixed domain: not as independently complete local states later connected by a mechanism, not as Bell-local hidden-variable value assignments, not as superluminal signaling, and not as same-act repair, but as boundary-level compatibility carried by one standing-preserving Bell witness.