CHSH-Form Values Without Bell Nonlocality: Inapplicability of the Tsirelson Bound on a Non-Factorized Fibonacci Fusion Space
We construct a sequential measurement protocol on the n=8 Fibonacci-anyon fusion space of dimension D = F_7 = 13 whose CHSH-form 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 construction and is consistent with the Bell-operator norm ‖B‖ = 3.670. 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 setting choice enters through braid words acting on Alice's measurement edge via the cross-bipartition generator σ4, the operational sense in which setting-locality fails by construction. Removing σ4 recovers ‖B‖ = 2.000 exactly in 8,000 of 8,000 samples. Second, |S| = 3.406 is not a supremum on D = 13: random braid quadruples reach the algebraic maximum |S| → 4.000, and Haar-random U(13) unitaries reach |S| = 3.927. We compare to the independent fusion-sector construction of Xu, Ye, and You, who report |S| = 2.633 on a bipartite 3⊗3 graded fusion-sector space, and present a Bell-operator-norm diagnostic placing the values into a single framework: from Xu's |S| = 2.633 through the companion d_1b value |S| = 2.811 to the nonfactorized |S| = 3.406 reported here. Version notes (v1.9, following a comprehensive internal review of the full series): • The title is updated from "Setting-Locality Violation and a Trans-Tsirelson CHSH Value on a Fibonacci Fusion Space" to reflect the paper's actual finding: CHSH-form values without Bell nonlocality, and the inapplicability of the Tsirelson bound on a non-factorized fusion space — the reported value is not a "trans-Tsirelson" violation, because the Tsirelson bound does not apply on this space. • Additional corrections from a review pass: no numerical values changed. • The data/code availability DOI placeholder is set to this record's own concept DOI (10.5281/zenodo.19601998). Source hygiene: internal build annotations removed from the LaTeX source; only printed change is that DOI (PDF text layer verified — sole diff). • The from-scratch reproduction of the main value is now correctly attributed to this paper's own deposited reproduction script rather than to the companion landscape analysis (test 8 of the Appendix-A consistency suite). • Source hygiene: the file header comment block is removed for series consistency with the companion papers (print-identical). • Deposit completeness: a deterministic reproduction script is added, containing the explicit braid words of the four settings, the reference-state definition, and a nine-part consistency suite; it reproduces |S| = 3.406357867517455 and the Bell-operator norm 3.670 to machine precision. The reference state used in the CHSH evaluation is now stated explicitly in Appendix B. • Correction: a corrupted footnote macro is repaired. • Release date set (2026-07-17). Abstract/README wording harmonized to the body: the |S| = 3.406 value is reproduced to all 16 displayed digits (not "16 decimal places"). • An AI-use disclosure ("Use of AI tools") is added at the end of the paper, following the Data and Code Availability section. A norm in Eq. (5) is relabeled from Frobenius to operator norm; the numerical value (1.572) is unchanged. • The Data and Code Availability section now states precisely which of the nine Appendix-A consistency tests are backed by deposited scripts. A seeded sampling validator for the N=8,000 controlled-bipartite-test claim (Sec. IV) is added to the deposit. The sentence citing three reproduced |S| values is corrected to attribute the companion value to its own companion record rather than to this deposit. • Corrections (pre-publication verification pass): the availability statement now declares explicitly that the random-braid and Haar-unitary context values of Caveat (ii) are not part of this record. The three generator-level entries of the Appendix-A test table are re-anchored to the deposited JSON values (5.6e-17, 3.2e-16, 1.1e-16, correcting 1.7e-16, 3.3e-16, 1.5e-17; the reproduction script now also records the unitarity maximum, and all entries remain far below the 1e-10 acceptance threshold). Three hyphenation instances of Leggett-Garg are set to the series-wide en-dash form. • Release date set (2026-07-23). An author's note on the version history of this record is included as a separate file (author_note_version_history.pdf). 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 what it means for the CHSH form that the fusion space does not factorize, and evaluates it on the full, non-factorized space, where values above the Tsirelson bound appear although that bound does not apply there, because setting-locality fails; the correlations are better understood as contextuality than as nonlocality, and within the series the paper marks the limit, where the witness still fires but its usual meaning no longer holds. 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.