CHSH-Form Values Without Bell Nonlocality: Inapplicability of the Tsirelson Bound on a Non-Factorized Fibonacci Fusion Space

Author: Berkay Yüksel Sayim ORCID: 0009-0004-4993-7352 DOI (concept, all versions): 10.5281/zenodo.19601998

Abstract

We construct a sequential measurement protocol on the n = 8 Fibonacci-anyon fusion space of dimension D = F₇ = 13 whose CHSH expectation value attains |S| = 3.406, above the value 2√2 ≈ 2.828 that bounds CHSH correlations under the standard setting-locality precondition, which this construction does not satisfy. The value is reproduced to all 16 displayed digits by an independent implementation and is consistent with the Bell-operator norm ‖B‖ = 3.670 computed directly from the four setting-pair unitaries.

Two interpretive caveats apply. First, |S| = 3.406 is not a bipartite Bell violation in the EPR sense, but a structural fact about CHSH-form expressions on a nonfactorized Fibonacci fusion space: Bob's observable is constant on the vacuum sector, and Bob's b′ setting contains the cross-bipartition braid generator σ₄, which acts on Alice's measurement edge — the operational sense in which setting-locality, a precondition of Tsirelson's bound, fails by construction. Removing σ₄ recovers ‖B‖ = 2.000 exactly in 8,000 of 8,000 random samples. Second, |S| = 3.406 is not a supremum on D = 13.

The construction is compared to the independent bipartite fusion-sector construction of Xu, Ye, and You, placing the values into a single Bell-operator-norm framework: from the bipartite Xu value |S| = 2.633 through the companion d₁ᵦ value |S| = 2.811 to the nonfactorized |S| = 3.406 reported here.

Contents

This record is compiled from main_v1.9.tex (RevTeX 4-2). The compiled main_v1.9.pdf is included.

Reproduction code:

• p2_chsh_reproduction.py — deterministic reproduction of |S| and the Bell-operator norm in double precision, with a nine-part consistency suite (unitarity, Yang–Baxter, distant commutators, D = 13, M_B = −I, the σ₄ commutator anchor, the σ₄-free bipartite control ‖B‖ = 2, kinematic-vs-operator equivalence, and the Xu–Ye–You three-copy control). Writes p2_chsh_reproduction_results.json. Requires NumPy.
• p2_highprec_verify.py — arbitrary-precision (mpmath) recomputation that reproduces all 16 displayed significant figures of |S| exactly (double precision reaches the last figure only to floating-point noise) and reconfirms unitarity and the Yang–Baxter relation at the working precision. Requires mpmath.
• p2_bipartite_control_sampling_validator.py — seeded sampling validator for the controlled bipartite test (Sec. IV): draws N = 8,000 random setting quadruples from the restricted generator alphabet (Alice: σ₁,σ₃; Bob: σ₅,σ₇) and confirms ‖B‖ = 2.000 in all 8,000 samples, to machine precision. Writes p2_bipartite_control_sampling_results.json. Of the nine braid-group consistency tests of Appendix A, this deposit backs the four generator-level and kinematic–dynamic checks (tests 1–3 and 8); the five word-level checks specific to the setting unitaries (tests 4–7 and 9) are not part of this record.

License

This work is released under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.