CHSH-Form Values Above 2 in Fibonacci Anyon Braiding: A Complete Landscape of Braid Words up to Length 12

We report a complete sequence-by-sequence landscape of CHSH-form values for six Fibonacci anyons in two encodings, covering all words of length L=3–12 with two generators and L=2–9 with three. In the two-generator d₁ᵦ encoding, braiding alone cannot exceed the classical value: the two generators act on different anyons and commute, and all 8190 braid words of length L=1–12 give |S|=2 exactly on the leakage-free fusion code. A two-qubit circuit built from the same Fibonacci F- and R-matrices, which does not satisfy the braid relations, reaches |S|=2.733 at L=12 with the standard Fibonacci data (96.6% of the Tsirelson bound 2√2) and |S|=2.811 at the optimum over a phase deformation; this excess is therefore not a braiding effect. Throughout, “CHSH violation” denotes a CHSH value exceeding 2 on the Fibonacci fusion space, which does not factorize into spatially separated Alice/Bob subsystems; the reported values are signatures of nonseparability and single-algebra contextuality, not Bell violations in the Einstein–Podolsky–Rosen sense. In the three-generator encoding, the generator σ₃ that acts across the bipartition is the only source of entanglement: without it, the state stays in the fusion code as a product state, for all sequences and deformation phases. Under four-dimensional computational-subspace projection the maximum is |S|=2.824 at L=8, and 91.8% of the words exceed 2 at L=9, including 510 words without σ₃ that do so only through the projection; a Horodecki-type score on the native five-dimensional observables exceeds 2 for 42.8%. Sequential braiding on a shared fusion space produces a positive lag-1 autocorrelation of the simulated CHSH value, ACF(1)=+0.35 at 11.22σ, because the state is carried from step to step; a null test with independently drawn phases shows none. This paper maps CHSH-form values for six Fibonacci anyons over all braid words up to length 12 in two encodings, where braiding alone in the two-generator encoding stays exactly at 2, the values above 2 come from a circuit model built from the same anyon data and, in the three-generator encoding, from the one generator that acts across the bipartition, with the dependence on the readout projection stated next to each number; it adds the autocorrelation of the simulated CHSH value under sequential braiding with a null test and a robustness check; within the series it turns a single finding into a complete landscape and locates where the values above 2 come from. 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 (v1.5 → v1.6). This version changes the title and framing, the d1b braiding result, the headline sector-phase value, the σ3 result, Protocol A and B evaluations, sweeps and figures, appendices, sequential braiding, deposit, bibliography, and license. • Title and framing: title changed from "Generic CHSH Violation in Fibonacci Anyon Braiding: A Landscape Analysis" to "CHSH-Form Values Above 2 in Fibonacci Anyon Braiding: A Complete Landscape of Braid Words up to Length 12"; abstract, introduction, headings and conclusion revised to match, priority statements removed. • New result: in the two-generator d1b encoding, braiding alone cannot exceed the classical value (σ2 and σ4 commute); all 8190 braid words of length L = 1–12 give |S| = 2 exactly. The d1b values above 2 refer to a two-qubit circuit model whose gates AM and MB do not satisfy the braid relations. • Sector phase and headline value: δ is now a deformation parameter, not a "topological sector phase". The headline |S| = 2.811 becomes 2.733 (96.6%, C = 0.932) at δ = 0, with 2.811 the optimum over δ; Table I rows L = 6 and 9 change from 2.398/0.662/84.8 and 2.624/0.850/92.8 to 2.433/0.693/86.0 and 2.639/0.861/93.3. • The σ3 result and the companion analysis: C = 0 on blocking σ3 is now stated only for the three-generator encoding from the product input state, in the assignment |11⟩ ← |4⟩. Fig. 3 shows the σ3-active curve up to |S| = 2.808 and the blocked curve at |S| = 2. • Protocol A projection: at L = 9, 510 of 18,064 words exceeding |S| = 2 contain no σ3. The L = 8 maximum 2.824 corresponds to about 2.59 (C = 0.824) with |11⟩ assigned to |4⟩; the L = 2 row changes from 2.000 to 2.029; enumeration now covers L = 10 and 12. • Native five-dimensional evaluation (Protocol B): with the threshold |S| > 2 + 10⁻¹⁰, 64.6% above 2 at L = 9 becomes 42.8%, with 28.5% exactly at the classical value; L = 3 changes from 25.9% to 7.4%. The "majority under both protocols" statements are replaced and Protocol B is renamed a "Horodecki-type score". • Sector-phase sweeps and figures: sweeps now use the first δ = 0 maximizers under Protocol A for L = 3, 6, 8, 10 and 12 (243, 232432, 23343322, 2322343322, 232222343322), with maxima 2.804, 2.798, 2.824, 2.824 and 2.824. All four figures are regenerated; new Fig. 2 maps |S| over L and δ. • Fourier appendix: the v1.5 six-term expansion (constant term 1.931, R² = 0.38 at L = 12) is replaced by least-squares fits of the deposited sweeps at L = 3, 8 and 12. With N = 4 harmonics, R² = 0.997, 0.837 and 0.914; no physical meaning is attached to the harmonic content. • Braid-relations appendix: a new paragraph states the braid relations hold on the 50-point grid only at δ/π = 0, 0.6 and 1.2, with Yang–Baxter residuum at least 0.196 elsewhere. Table V is corrected: 0.473 becomes 0.474 at δ/π = 0.5, and ≤ 8.2×10⁻¹⁶ becomes < 10⁻¹⁵ at δ = 0. • Sequential braiding and autocorrelation: the ACF(1) result is now a hypothesis ("Proposed experimental test"), not a "falsifiable prediction", and uses Protocol A, not B. In Table IV, Model A changes from +0.343 (10.85σ) to +0.346 (11.22σ), quoted as +0.35, and Model C from 0.000 to +0.035 (1.13σ). • Deposit and README: new scripts d9, d10, d11 and make_1b_fig4_map.py are added; d4, d5, d6 and d8 are corrected for the omitted 1/√2 projection factor, and N2_max (up to 1.918) is now at most 1. The Data and Code Availability section and the README are rewritten. • Bibliography and cited sources: statements about cited work are corrected (Bonesteel; Hormozi; "Eq. (9)" to "Eq. (23)"; Xu, Ye, and You; Summers and Werner), Fan and de Garis is removed, Shapourian, Mong, and Ryu (2020) is added (44 entries), and journal data and both companion-record titles are updated. • License and form: paper, figures and data are released under CC BY 4.0 and the deposited source code under the Apache License 2.0 (v1.5 stated CC BY 4.0 for scripts and JSON files); `LICENSE-CODE` is added. A notation table is added, and the acknowledgments environment is replaced by an unnumbered section. • Deposit packaging. The code archive and the paper PDF are named sayim-2026- - -v (record: 1a, 1b, p2, p3 or p4). The paper