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 = F₇ = 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 d₁ᵦ circuit-model value |S| = 2.733 with the standard Fibonacci F and R data (2.811 under a phase deformation of the R-matrix) to the nonfactorized |S| = 3.406 reported here. 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. 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. 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. ------------------- Version notes (v1.9 → v2.0). This version changes the companion value, the Bell-operator norm, the companion descriptions, the attribution of Tsirelson's bound, the reproduction statement, the control test, citations, notation, licensing, the bibliography, and one consistency test. • Unchanged. The title of this paper and the headline values |S| = 3.406 357 867 517 455 and ‖B‖ = 3.670 are unchanged. • Companion value corrected. v1.9 gave |S| = 2.811 (99.4% of the Tsirelson bound) without noting it is not reached at the standard Fibonacci point; the text now gives 2.733 (96.6% of 2√2) there and names 2.811 as the maximum of a phase-deformation sweep. C = 0.988 is no longer printed. • Bell-operator norm of the companion column corrected. At the settings attaining the standard-point value, ‖B‖ = 2.827 (C = 0.932) for the companion, not 2√2 as in v1.9; for Xu's construction ‖B‖ = 2√2. The bipartite constructions (2.633, 2.733) now "stay within" the bound instead of "saturating" it. • Description of the companion papers corrected. Paper 1b reports |S| = 2.824 (99.84% of the Tsirelson bound) at L = 8, where v1.9 stated saturation |S| = 2√2; in its two-generator encoding braiding alone gives |S| = 2 exactly. The companion value is now described as a circuit-model value. • Attribution of Tsirelson's bound corrected. The text now follows Cirel'son (1980): the bound follows from the observable form of setting-locality with the standard C*-algebra, the four correlators taken in a single common state. v1.9 had stated the joint conditions of EPR-bipartite factorization and unitary-form setting-locality. • Reproduction statement corrected. v1.9 described Horodecki-optimized parameters on a two-state reduced density matrix, which fits the companion, not the present D = 13 construction. The sentence now lists four fixed setting words, fixed observables M_A and M_B, and |S| reproduced exactly in two independent codes; the setting-word search is not part of this record. • Control test and commutator relations restated. The σ₄-free control, called "the strongest direct evidence" in v1.9, is now a corollary of the commutator structure (B = −2M_A, ‖B‖ = 2 exactly); the 8,000-sample run is a numerical check. The commutator norm is ‖[σ₄, M_A]‖ = 2/√φ ≈ 1.572; vanishing for other generators is exact. • Citations and attributions corrected. Summers–Werner is now cited as context for observables in spacelike-separated regions of a local net of von Neumann algebras; the Fan–de Garis reference is removed (22 → 21 entries). Xu–Ye–You is no longer called "independent parallel" work, and the equation number changes from Eq. (9) to Eq. (23). • Terminology and notation. The description of the measurement edges now matches the deposited script: σ₁, σ₂, σ₃ and σ₅, σ₆, σ₇ leave x₃ unchanged, and σ₄ changes x₃. A footnote (2.000 exactly in 8,000 of 8,000 samples; |S| = 3.927 for Haar-random U(13); 4.000 algebraic maximum) and a Notation section are added. • Licensing and deposit. v1.9 released the whole record under CC BY 4.0; paper, figures, and data stay CC BY 4.0, and the deposited source code is now under the Apache License 2.0 (LICENSE-CODE added). The README carries the revised companion value (2.733; 2.811 under a phase deformation of the R-matrix). • Bibliography. Titles of the companion records for Papers 1a, 1b, and 3 are updated, and the "[Concept-DOI, always points to latest version]" annotation is removed from three entries. Full author lists are given for Hensen et al. (19 authors), Giustina et al. (22), and Shalm et al. (34). • Consistency test 8 corrected. v1.9 gave 1.3 × 10⁻¹³ as the maximum error of test 8 (S_kin = S_dyn); the deposited script returns a difference of exactly 0 in double precision, and the table and sentence now state this. Tests 1–3 and 4–7, 9 are unchanged. • Deposit packaging. The code archive and the paper PDF are named sayim-2026- - -v (record: 1a, 1b, p2, p3 or p4). The paper PDF is also deposited as a separate file next to the archive. The author's note is updated and deposited as a separate PDF. • Release date: the deposited build is dated 2026-10-03; the Zenodo publication date is set to the same day. An author's note on the version history of this record is included as a separate file (author_note_version_history.pdf). The complete version notes of this version are given under "Additional descriptions" (type Notes) of this record. Version notes of earlier versions remain in the records of those versions, which stay listed in the version history of this record.