The textbook CHSH measurement configuration {a, a', b, b'} = {0, pi/2, pi/4, -pi/4} with Form I evaluates to S = 0 on all four Bell states in the spin-1/2 convention used by gate-based quantum processors. This "failure mode", documented as a state-formula-angle mismatch in a companion pedagogical note (doi:10.5281/zenodo.20146219), is shown here to have a clean algebraic origin: the corresponding Bell operator reduces to sqrt(2) * (sigma_z (x) sigma_x + sigma_x (x) sigma_z), whose spectrum is {-2*sqrt(2), 0, 0, +2*sqrt(2)}, with the kernel spanned by {|Phi+>, |Psi->} and the non-kernel pair {|Phi->, |Psi+>} mapped to each other. Main results. The note establishes three properties of the failure-mode configuration: (1) The CHSH value S is preserved at exactly zero under arbitrary symmetric local Pauli channels (depolarizing, dephasing, bit-flip) and under local Rx and Rz rotations on either qubit, for every Bell state. Verified numerically with 1000 random Pauli channels per state and across the full range of standard noise channels. (2) Local Ry(theta) rotations break S = 0 only for the pair {|Phi->, |Psi+>}, with closed form S = 2*sqrt(2)*|sin(theta)|, reaching the Tsirelson bound at theta = pi/2. (3) The asymmetry has a direct algebraic origin in the correlation tensor: the kernel pair has equal XX and ZZ moments (t_1 = t_3), while the broken pair has opposite signs (t_1 = -t_3). This connects to and gives algebraic meaning to the Bell-state taxonomy of the companion paper. Diagnostic interpretation. The same CHSH measurement that fails in textbook form becomes a no-extra-resource diagnostic for coherent Ry-type errors restricted to a specific Bell-state family. If S > 0 in this configuration, then (i) the state is in the {|Phi->, |Psi+>} family, and (ii) a coherent Ry error of magnitude theta_err = arcsin(S/(2*sqrt(2))) is present. Incoherent Pauli noise alone cannot produce this signal. Calibration of novelty. The note explicitly searches the self-testing literature (Supic-Bowles 2020 review) and the CHSH-under-noise literature (Kofman 2012, Kofman-Korotkov 2008) and reports that this specific asymmetric breaking pattern was not located in those communities. Priority is explicitly disclaimed with respect to gate-set-tomography and randomized-benchmarking literatures, which were not exhaustively surveyed. The result is presented as either a novel observation or, if previously catalogued elsewhere, an independent algebraic rederivation with closed-form expression in terms of the correlation tensor. Reproducibility. The repository contains the LaTeX source, the bibliography, both figures, and seven Python scripts that reproduce every numerical claim from scratch using only NumPy and Matplotlib. Each script is self-contained and runs in seconds. Companion paper. This work extends the pedagogical note "State-dependent CHSH inequality on gate-based quantum processors" (Perez-Eugenio 2026, doi:10.5281/zenodo.20146219), which documents the original state-formula-angle compatibility issue and the corrected configurations. The present note can be read independently.
The textbook CHSH measurement configuration {a, a', b, b'} = {0, pi/2, pi/4, -pi/4} with Form I evaluates to S = 0 on all four Bell states in the spin-1/2 convention used by gate-based quantum processors. This "failure mode", documented as a state-formula-angle mismatch in a companion pedagogical note (doi:10.5281/zenodo.20146219), is shown here to have a clean algebraic origin: the corresponding Bell operator reduces to sqrt(2) * (sigma_z (x) sigma_x + sigma_x (x) sigma_z), whose spectrum is {-2sqrt(2), 0, 0, +2sqrt(2)}, with the kernel spanned by {|Phi+>, |Psi->} and the non-kernel pair {|Phi->, |Psi+>} mapped to each other. Main results. The note establishes three properties of the failure-mode configuration: (1) The CHSH value S is preserved at exactly zero under arbitrary symmetric local Pauli channels (depolarizing, dephasing, bit-flip) and under local Rx and Rz rotations on either qubit, for every Bell state. Verified numerically with 1000 random Pauli channels per state and across the full range of standard noise channels. (2) Local Ry(theta) rotations break S = 0 only for the pair {|Phi->, |Psi+>}, with closed form S = 2sqrt(2)|sin(theta)|, reaching the Tsirelson bound at theta = pi/2. (3) The asymmetry has a direct algebraic origin in the correlation tensor: the kernel pair has equal XX and ZZ moments (t_1 = t_3), while the broken pair has opposite signs (t_1 = -t_3). This connects to and gives algebraic meaning to the Bell-state taxonomy of the companion paper. Diagnostic interpretation. The same CHSH measurement that fails in textbook form becomes a no-extra-resource diagnostic for coherent Ry-type errors restricted to a specific Bell-state family. If S > 0 in this configuration, then (i) the state is in the {|Phi->, |Psi+>} family, and (ii) a coherent Ry error of magnitude theta_err = arcsin(S/(2*sqrt(2))) is present. Incoherent Pauli noise alone cannot produce this signal. Calibration of novelty. The note explicitly searches the self-testing literature (Supic-Bowles 2020 review) and the CHSH-under-noise literature (Kofman 2012, Kofman-Korotkov 2008) and reports that this specific asymmetric breaking pattern was not located in those communities. Priority is explicitly disclaimed with respect to gate-set-tomography and randomized-benchmarking literatures, which were not exhaustively surveyed. The result is presented as either a novel observation or, if previously catalogued elsewhere, an independent algebraic rederivation with closed-form expression in terms of the correlation tensor. Reproducibility. The repository contains the LaTeX source, the bibliography, both figures, and seven Python scripts that reproduce every numerical claim from scratch using only NumPy and Matplotlib. Each script is self-contained and runs in seconds. Companion paper. This work extends the pedagogical note "State-dependent CHSH inequality on gate-based quantum processors" (Perez-Eugenio 2026, doi:10.5281/zenodo.20146219), which documents the original state-formula-angle compatibility issue and the corrected configurations. The present note can be read independently.
This work provides an explicit, reproducible demonstration of how theClauser-Horne-Shimony-Holt (CHSH) inequality depends jointly on thechoice of Bell state AND measurement configuration in the spin-1/2qubit convention used by gate-based quantum processors. Main result. The combination of CHSH Form I, S = | E(a,b) - E(a,b') + E(a',b) + E(a',b') |, with the commonly-cited angular configuration {0, pi/2, pi/4, -pi/4}yields S = 0 for ALL FOUR Bell states in the spin-1/2 Bloch-sphereconvention, not selectively for one. This frequently misleadspractitioners who prepare |Phi+> on gate-based hardware and applytextbook angles, obtaining S = 0 and misdiagnosing the issue as ahardware fault, code bug, or absence of entanglement. Corrections provided. For the two states whose correlation functionin the XZ plane has argument theta_a - theta_b (|Phi+> and |Psi->),two minimal one-parameter fixes recover the Tsirelson bound 2*sqrt(2): 1. Retain the angles and apply CHSH Form II: S = | E(a,b) + E(a,b') + E(a',b) - E(a',b') |. 2. Retain Form I and replace b' = -pi/4 with b' = 3*pi/4. For the remaining two Bell states (|Phi-> and |Psi+>, whose correlationargument is theta_a + theta_b), neither single-parameter fix suffices;the optimal angles b and b' must be exchanged. A complete optimal-angletable for all four states is given in Section 4 of the manuscript. Contents. - paper.pdf / paper.tex: 8-page manuscript with full derivation, tables, code listing and appendix. - chsh_verification.py: self-contained Python script (NumPy only) reproducing all numerical claims of the paper. Its standard output is the verbatim text reproduced in Section 5, and the script additionally cross-checks the closed-form correlation expressions against explicit 4x4 matrix calculations on random angles (azimuthal angles included) to machine precision. - CHANGELOG.md: complete history of corrections relative to earlier versions of this deposit. Reproducibility. Running 'python3 chsh_verification.py' produces theverification tables displayed in Section 5 of the paper, byte for byte.
Files of this version are restricted (18 October 2026) because it was released under earlier licence terms: the source code was under the MIT License. All current versions use one clear licensing: paper, figures and data under CC BY 4.0, source code under the Apache License, Version 2.0. Please use the latest version: https://doi.org/10.5281/zenodo.21134322 We construct four spatially separated Majorana zero modes (MZMs) per party on a finite Kitaev honeycomb lattice in its non-Abelian phase, bound to well-separated Ising vortices, and use them to realize a local, noncommuting CHSH measurement algebra. Two static fusion-parity observables per party, Z_A=iγ_{A,1}γ_{A,2} and X_A=iγ_{A,2}γ_{A,3} (and analogously for Bob), commute exactly between parties (P₂=0) and fail to commute within a party (P₁=‖[Z_A,X_A]‖=2.000000); on the maximally entangled fusion-parity Bell state the CHSH expectation reaches the algebraic Tsirelson value S=2√2=2.828427. This value survives fermion-parity superselection exactly, because the Bell state is a definite-parity eigenstate with unit weight in its own sector — a statement specific to the present four-MZM, measurement-only construction, not to braid-then-fuse protocols, which are a different experiment. A commuting-only control setting returns K=√2 ≤ 2, and the finite-size Majorana footprint leaking across the bipartition decays monotonically with system size (isotropic scaling 17.0% → 10.2% → 2.3%). We stress the critical scope-limiting fact, following Howard and Vala [Phys. Rev. A 85, 022304 (2012)]: a physical Bell-inequality violation using only topologically protected Ising-anyon operations does not follow from this construction, since Ising braiding generates only the Clifford group; that step requires a non-Clifford ("magic") resource, the subject of a companion paper. What we report here is the local realization of the noncommuting Bell-operator algebra on a genuine microscopic lattice substrate, not a claim of Bell nonlocality. A loop-threading relative phase δ reproduces the analytic corollary S(δ)=2√2·cos²(δ/2) to a residual of 4.4×10⁻¹⁶, and an independent gauge-invariant flux-holonomy diagnostic confirms no adiabatic channel mixing at the quoted scale (‖[W,Z_A]‖_F=0.0132), with the sole open discrepancy being a 0.020-rad gap between the accumulated Berry phase at one full flux quantum and the target π, traced to genuine near-degeneracies along the loop rather than to a construction error.Correction notice (2026-10-23). The summary above states that the sole open discrepancy is a 0.020-rad gap between the Berry phase accumulated at one full flux quantum and the target π, “traced to genuine near-degeneracies along the loop rather than to a construction error”. This explanation is not correct. The single-mode phase summed along the loop, δ_full mod 2π = 3.1212, is not gauge invariant: the loop is closed on an eigenvector recomputed at Φ = 2π, whose phase the eigensolver fixes independently, so the quoted value is the gauge-invariant parallel-transport phase plus a closing phase that depends on the eigensolver’s start vector (here π). The gauge-invariant parallel-transport phases of the near-zero modes are −0.0204 and +0.0197 rad, all within 0.021 rad of zero; none is near π. The pre-registered single-mode closure criterion is therefore not met, and in this simulation the flux insertion does not generate δ dynamically. The “near-degeneracies” are particle–hole partner energies, equal by symmetry. The Bell-operator algebra, the gauge-invariant holonomy result and all other reported values are not affected. The paper text will be corrected in the next version. About this series: This record is part of a series of related works from my independent research on Fibonacci anyons, with Ising anyons as their natural counterpart. I started in April 2026, and it has been a long and insightful journey in which I learned a lot; the work uses different methods and stays within verifiable, nonspeculative physics. The common thread of the series is a split: Ising anyons are limited to Clifford operations, while Fibonacci anyons are computationally universal, and across the series I map what standard witnesses of nonclassicality can and cannot certify on such systems. I consider Fibonacci anyons a serious candidate for topological quantum computing, given their universality and their topological protection against local noise. A hybrid approach with Ising is conceivable, but problems such as instability and certification would have to be solved first, and each needs research of its own. This paper asks whether a real microscopic lattice can carry the Bell structure, and constructs four Majorana modes per party on a Kitaev honeycomb lattice in its non-Abelian phase, realizing the full CHSH measurement algebra locally, with observables that commute between the parties and fail to commute within a party; the algebra holds, reaching the algebraic Tsirelson value and surviving the fermion-parity superselection, but a physical Bell violation does not follow, because topologically protected Ising operations generate only Clifford gates and the missing step is a non-Clifford resource, which is the subject of a companion paper; within the series it is the substrate test, where algebra on real material is not yet a violation. Use of AI tools: In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's. I run multiple verifications at the different stages of the work and one before release, including cross-checks with an unrelated AI model from a different company, and all references are checked against the original sources. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication.
D2d-C3 covers the Bell inequality structure of the Triniton discrete framework. Results established: (I) R4 — CHSH ≤ 2 from local factorization, with explicit verification on the classical Triniton sector and the geometric note on spatial rotations; (II) dim=2 qubit from M2+PE; (III) su(2) algebra from step transitions in Im(Q); (IV) ∣Ψ𝑇+⟩from PE+unitarity+IV.N2+M1 — four-proposition derivation; (V) E(a,b) from su(2) — algebraiccomputation with declared convention; (VI) CHSH = 2√2 with explicit angle assignment; (VII) finite-N correction with conservative bound 4π/N. Zero free parameters. Zero ad hoc adjustments.
A bound on homodyne Bell violation set by the state class, not by the binning. For over two decades, attempts have been made to violate a CHSH inequality using homodyne detection with sign binning alone, since this measurement is nearly lossless and would permit a loophole-free test. The values achieved remain small: 2.048 and 2.076. Two causes are possible — the coarseness of sign binning, or the class of states used. This work separates the two exactly. Because the sign-binned quadrature measurement is a Hermitian operator with spectrum in [-1,1], the maximum over all states is the largest eigenvalue of the Bell operator, and the restriction to perfectly photon-number-correlated states |ψ⟩ = Σ cₙ |n,n⟩ is a projection onto the diagonal subspace. Both maxima are compared at identical angles, so no optimisation over states is involved. Result: the perfectly paired class saturates at S = 2.10198 ± 0.00001, established by convergence to Fock truncation N = 1150, while relaxed pairing reaches 2.22 under the same measurement prescription and continues to rise. The bottleneck is the pairing, not the binning. The gain originates from a targeted admixture of asymmetric photon number at |0,4⟩ and |4,0⟩. Three supporting computations explain the bound: the exact bound a₁ ≤ 0.811876 on the fundamental harmonic of the correlator, which the ideal paired state exhausts to 99.84 %; the completeness of the pairing criterion as an entanglement witness under local symplectic optimisation; and the ineffectiveness of complex amplitudes and Kerr nonlinearity. An attempted proof via subadditivity is documented as having failed, together with the reason no term-by-term estimate can succeed. The bound is presented explicitly as a numerically supported conjecture, not a theorem, and is falsifiable: a state of the form Σ cₙ |n,n⟩ with S > 2.1021 refutes it. Practical consequence for experiment: within the perfectly paired class, further optimisation cannot exceed 2.10198. Higher values require breaking the pairing deliberately. This record contains the manuscript and supplementary material in English and German (PDF and LaTeX sources), and the script reproduce_all.py, which regenerates every tabulated value in eight individually callable sections. Deutsche Fassung des Manuskripts und des Ergänzenden Materials sind in diesem Record enthalten.
A supplementary calculational note to the interpretational program "Spacetime as Register, Not Stage". The note reconstructs the full structure of spin-1/2 singlet correlations — the joint distribution P = 1/4(1−αβ·a·b), the correlation E(a,b) = −a·b, and CHSH S = 2√2 — from the program's picture of the "entangled" pair as one indivisible complement unit read out locally at two sites by independently oriented detector settings. The note proves sufficiency, not derivation, and keeps an explicit ledger of its epistemic status: two rules are added and underived (a half-angle spinor readout structure, and the square law for amplitudes), two are derived conditionally (antisymmetry, relative to an explicit phase bridge; normalization, fixed by completeness with no free parameter), and every remaining gap is recorded in a debt register. The model exits the factorizable (Bell) class exactly where quantum mechanics does: conditioned on the full preparation state, local marginals remain 1/2 (parameter independence holds per preparation state), while outcome independence fails — with no superluminal transport claim derived from statistics, no superdeterminism (the preparation state carries no free parameter that could co-vary with later settings), and operational no-signaling verified numerically. A deterministic hidden-orientation contrast model is included, reproducing the linear correlation −1+2θ/π and S ≤ 2, to show that the cosine requires a genuine exit from the factorizable class rather than a change of vocabulary. The English text is a full translation of the author's Polish source version (both files included; frozen 28.07.2026, changes tracked in the program's correction register, Series 33).
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.