This record contains the preprint “Operational Auditability in Physics III: Auditability, No-Signaling, and the CHSH–Tsirelson Bound” (Version 1.0). The paper presents a compact operational framework based on explicit “held distinctions”, verification contours, and auditability requirements. Within this frame, it provides self-contained derivations of the CHSH classical bound and the Tsirelson bound, emphasizing which assumptions are operational (and therefore auditable) versus model-dependent. The manuscript also discusses no-signaling within the same contour-explicit approach and proposes a reproducibility template intended for heterogeneous checks and independent verification. Series note: Part III of a three-part series “Operational Auditability in Physics”.• Part I: Lorentz Symmetry as an Operational Invariant Under Contour Changes• Part II: Fluctuation Theorems as Conditional Holding-Equalities (Jarzynski, Crooks) Files:• PDF (readable preprint)• LaTeX sources (for reproducible rebuild) Keywords: Bell inequalities; CHSH; Tsirelson bound; no-signaling; reproducibility; operational foundations.
This record contains the preprint “Operational Auditability in Physics III: Auditability, No-Signaling, and the CHSH–Tsirelson Bound” (Version 1.0). The paper presents a compact operational framework based on explicit “held distinctions”, verification contours, and auditability requirements. Within this frame, it provides self-contained derivations of the CHSH classical bound and the Tsirelson bound, emphasizing which assumptions are operational (and therefore auditable) versus model-dependent. The manuscript also discusses no-signaling within the same contour-explicit approach and proposes a reproducibility template intended for heterogeneous checks and independent verification. Series note: Part III of a three-part series “Operational Auditability in Physics”.• Part I: Lorentz Symmetry as an Operational Invariant Under Contour Changes• Part II: Fluctuation Theorems as Conditional Holding-Equalities (Jarzynski, Crooks) Files:• PDF (readable preprint)• LaTeX sources (for reproducible rebuild) Keywords: Bell inequalities; CHSH; Tsirelson bound; no-signaling; reproducibility; operational foundations.
This paper presents a structural analysis of Bell–CHSH correlations using a hierarchy of geometric hidden-variable models defined on the unit circle. Bell’s theorem constrains hidden-variable models under the joint assumptions of locality, deterministic response functions, and measurement independence. The paper examines how Bell–CHSH behavior changes when these assumptions are modified one at a time. A minimal local model with setting-independent preparation obeys the classical Bell–CHSH bound |S| <= 2. A setting-dependent update can produce Bell violation but introduces operational signalling. Direct imposition of the cosine correlation is used as a structural reference, reproducing Tsirelson’s bound while preserving no-signalling marginals. The paper then shows that Tsirelson-level correlations can also arise from a geometric, context-dependent sampling rule without inserting the cosine function directly into the preparation law. In that construction, local deterministic response functions and operational no-signalling are preserved, while measurement independence is relaxed. The models are presented as structural illustrations rather than physical hidden-variable proposals. Their purpose is to make the assumption-dependence of Bell correlations explicit in a transparent geometric framework. Internal reference: CGI-RSR-000021.
Following the operative pattern of the absorption of E₀ = m₀c² as basal rest sector of the Sistema Vectorial SV (Lloret Egea, 2026g), this document proposes and proves the absorption of Bell's local bound CHSH ≤ 2 and the quantum bound of Tsirelson 2√2 as two operative manifestations of one and the same coupled angular factual apparatus of the Sistema Vectorial SV. The construction articulates the CHSH regime in three structurally distinct sub-regimes (R₀ uncoupled, R₁ weakly coupled, R₂ strongly coupled) parametrized by the structural coupling coefficient χ_c. On R₀, the Bell bound holds as exact algebraic invariant T ≡ 0 (mod 2) on the finite field 𝔽₂, with exact compliance verified on 78 772 cases without a single counterexample. On R₂ at full saturation (χ_c = 1), the Tsirelson bound 2√2 emerges as structural saturation of the unique coupled angular correlator admitted by the SV, with numerical verification at machine precision. The technical core of the document is the introduction of the Coupled Angular Factual Calculus of the Sistema Vectorial SV as a new mathematical apparatus, whose central result is the Theorem of Uniqueness of the Coupled Factual Cosine: under nine structural axioms of the SV apparatus — angular dependence, full anticorrelation in basis coincidence, full correlation in opposite basis, parity, polar complementarity, normalized factual conjugate component, standard SO(2) composition law on the auxiliary function C̃ = −C_SV, structural continuity, and minimal angular fidelity of the CHSH-SV apparatus — the only admissible coupled angular correlator is C_SV(δ) = −cos δ. The Tsirelson bound 2√2 emerges from this theorem as canonical saturation of the unique angular factual representation admitted by the apparatus, without invocation of Hilbert formalism, complex probabilistic amplitudes, or probability as truth criterion. The construction is compatible with operational compatibility with the experimental record, measurement freedom, and the experimental record cited herein. The Sistema Vectorial SV is positioned as a fourth technical reading of the Bell apparatus, complementary to and distinct from superdeterminism, post-quantum boxes, and generalized probabilistic theories, with three differentiating properties: deterministic discipline without grounding probability, structural derivation of the Tsirelson bound without Hilbert importation, and simultaneous compatibility with structural non-locality, operational compatibility with the experimental record, and measurement freedom. --- Note: Canonical material source on GitHub: https://github.com/juantoniolloretegea/SV-matematica-semantica/tree/main/documentos/adendas/matematica-fisica-factual-contemporanea-sv/de-bell-a-tsirelson-sin-formalismo-hilbert. Supplementary reproducible laboratory deposit on Zenodo: https://doi.org/10.5281/zenodo.20020201. These canonical sources are provided to facilitate direct consultation of the living textual record, reproducible laboratory verification and reader-side translation through standard browser translation tools when required.
Bell’s theorem rules out a broad class of local hidden-variable explanations, but Bell violation, operational no-signalling, process-level locality, quantum admissibility, completion geometry, and the numerical quantum boundary are distinct statements. Conflating these levels obscures both what Bell establishes and what a proposed reconstruction must still explain. This paper applies a typed-boundary analysis to the Bell/CHSH setting. It does not derive a new Bell inequality, does not derive Tsirelson’s bound from a new physical principle, and does not claim that Bell’s theorem has been solved. Instead, it separates Bell-local factorisation, operational access, remote write-power, history-conditioned process closure, local admissibility-law ownership, parent-level completion, and scale/resource custody into distinct scientific addresses. The analysis is tested against established native results including Fine’s theorem, Tsirelson–Landau–Masanes correlator geometry, Information Causality, the NPA hierarchy, joint-measurability results, uncertainty and steering, relativistic independence, and dimension-witness frameworks. It also distinguishes the target-specific CHSH algebraic-to-quantum residual from quantum-bound slack, finite-address excess, numerical certification gap, and experimental uncertainty. Under the comparator family examined here, the independent Bell-local new-physics delta is null. The non-null contribution is a typed correspondence, custody, and falsification architecture that prevents constraints at different levels from being silently substituted for one another and supplies an explicit reopening rule for prospective tests at genuinely held-out addresses.
Abstract The violation of Bell inequalities by quantum systems is among the most precisely tested predictions in physics, yet the origin of the Tsirelson bound CHSH ≤ 2√2 is rarely stated in fully algebraic terms. We show that this bound is an instance of the stabiliser-range theorem first established for check-digit codes over finite alphabets. The three-element framework (G, F, E) of that work translates to the quantum bipartite setting as follows: the state group G is the unitary group U(HA ⊗ HB), the fold F is the singlet state |ψ−⟩, and the perturbation family E is the set of local unitary transformations. The product stabiliser Stab× (the classical, local-hidden-variable description) is a strict subset of the tensor stabiliser Stab⊗ (the quantum description); the difference is exactly the diagonal SU(2) subgroup {U ⊗ U : U ∈ SU(2)}. We prove a Lie-algebra stretch factor theorem: every generator X ∈ su(2) satisfies ‖dφdiag(X)‖HS = √2 · ‖dφprod(X)‖HS, where φdiag and φprod are the diagonal and product embeddings of SU(2) into U(4). This stretch factor is the algebraic source of the √2 in the Tsirelson bound. We further establish a Stab-Tsirelson equivalence chain: the diagonal stabiliser condition (U ⊗ U)|ψ−⟩ = |ψ−⟩ implies, through a single algebraic sequence, that the CHSH operator has eigenvalue −2√2 on |ψ−⟩. For N-particle systems we prove inductively that the MABK (Mermin-Ardehali-Belinskii-Klyshko) ratio satisfies M(N) = 2(N−1)/2, confirmed numerically for N = 1, …, 5. No-signaling emerges as a corollary: the diagonal stabiliser is inaccessible from either subsystem alone. The framework places the Tsirelson bound, the multiparticle scaling, and the no-signaling principle within a single algebraic structure, providing a unified account of quantum nonlocality that is traceable to the check-digit symmetry-matching condition Stab(G, F) ∩ E = {e}. Complete proofs are provided in the supplementary document.
Bell--CHSH is frequently paraphrased as ``local realism is impossible'' after experimentalviolations of CHSH. The theorem is correct, but the paraphrase hides a structural assumption:\emph{selection centrality} (fair sampling), i.e.\ that the detected sample is not asetting- and hidden-variable--dependent reweighting of the hidden prior.We give a precise measure-theoretic separation between (i) the \emph{central} sector, whereEobs=Efull and CHSH≤2 holds under measurement independence and locality,and (ii) the \emph{noncentral} sector, where the observed correlations are computed under areweighted measure dνa,b=w(a,b,λ)dρ.We prove a sharp CHSH inflation boundSobs≤2+4δ,δ:=supa,b∫∣w(a,b,λ)−1∣ρ(dλ),so reproducing Tsirelson's 22 by strictly local measurement-independent models via selectionrequires δ≥(22−2)/4≈0.2071.We provide (A) a canonical Radon--Nikod\'ym construction (piecewise constants on sign regions plusa small positive remainder), (B) an explicit local factorized detection-loophole construction thatreproduces arbitrary finite correlation tables, and (C) an implementable optics protocol to estimateor upper-bound δ from time-tag data via window dithering, threshold and spectral sweeps, andauxiliary-tag binning.This paper critiques not the validity of Bell's theorem, but the common overstatement of what CHSHviolations imply without an explicit, quantitative centrality audit.
We consider a subclass of bipartite CHSH-type Bell inequalities. We investigate operations which leave their Tsirelson bound invariant, but change their classical bound. The optimal observables are unaffected except for a relative rotation of the two laboratories. We illustrate the utility of these operations by giving explicit examples. We prove that, for a fixed quantum state and fixed measurement setup except for a relative rotation of the two laboratories, there is a Bell inequality that is maximally violated for this rotation, and we optimize some Bell inequalities with respect to the maximal violation. Finally, we optimize the qutrit to qubit ratio of some dimension witnessing Bell inequalities. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.