CHSH-Form Values Above 2 for Standard Fibonacci Anyons, and the Non-Topological Origin of Finite-Size Sector–CHSH Mutual Information in Z2 Lattice Gauge Theory

Abstract

We quantify the mutual information I between topological sector labels T and CHSH values S in two-dimensional lattice models. Throughout, I denotes a classical Shannon mutual information between a global discrete label and a scalar measured value; it is not an entanglement entropy between spatial regions. In the Z₂ lattice gauge theory on a 16× 16 torus at inverse temperature β = 2.0 we measure I = 0.0173 bits after subtracting the null level, 95% interval [0.0159, 0.0187], significant above 3σ in all 21 seeds against a circular-shift null whose false-positive level we measure at 0.05 of 21. Decomposing the label shows the association is carried entirely by its excitation-number component (+0.0174 bits on the gauge-invariant route, 21 of 21 above 3σ); the Wilson-loop component never dominates at any coupling examined, and its own contribution is bounded at 1.2×10⁻⁴ bits at β = 1.0, where the label carries 1.9995 of 2 bits of entropy and 3940 of 4000 measurements are effectively independent. The origin of this mutual information is therefore not topological. A finite-size scaling analysis indicates that it decays toward zero with system size. In the quantum toric code via exact diagonalization, abelian Z₂ anyons produce no local CHSH correlations at system sizes L ≥ 3, consistent with the inaccessibility of abelian topological entanglement to local Pauli operators. We systematically test three classical mechanisms for CHSH values above 2; none exceeds |S| = 2 in the implementations studied. Most significantly, within an idealized TQFT model we find that the two-generator d₁ᵦ circuit model of Fibonacci anyons produces CHSH-form values above 2. With the standard Fibonacci F and R data (δ = 0), the maximum over all 4096 sequences of length L = 12 is |S| = 2.7334 (96.64% of the Tsirelson bound), for the sequence ABBAABABBAAB. Under a phase deformation δ of these data, the maximum on a 50-point δ-grid is |S| = 2.8114 (99.40%) at δ ≈ 1.44π, for the sequence ABABABABABAB; as a grid maximum it is a lower bound on the maximum over δ. Generic δ ≠ 0 breaks the Yang–Baxter/hexagon consistency of the braid representation, so that value characterizes a deformed representation rather than the standard category. The CHSH-form values depend strongly on δ: with fixed measurement settings (from a numerical search at δ ≈ 1.44π), |S| for that sequence ranges from near zero to 2.72 (above 2 at only 14% of the 200 δ values sampled), while settings adapted to each δ give |S| > 2 at 99.5% of them. This paper asks whether CHSH-form values above 2 arise for standard Fibonacci anyons and whether a sector label in a Z2 lattice gauge theory carries information about the CHSH value, where the Fibonacci values above 2 come from a two-qubit circuit built from the anyon data, and the lattice mutual information is measurable at finite size but comes from the number of excitations rather than from the topological sector, and decays with system size; within the series it is the starting point, where a standard witness first meets a topological system. 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 (v2.3 → v2.4). • Title, abstract, and framing. The title is changed, and "CHSH violation" for the Fibonacci results becomes "CHSH-form values above 2". The abstract now states that I is a classical Shannon mutual information, not an entanglement entropy, with a non-topological origin; the v2.3 "information source crossover" near β ~ 2.0 is withdrawn. • Z2 lattice gauge theory: engine correction. The legacy engine's fixed order and deterministic acceptance of energy-neutral flips make its chain reducible; the corrected engine accepts them with probability 1/2 and passes four validation points within 1.2 standard errors (without it: 4.3 to 52.6). The 0.1% vectorized-engine claim is removed (+0.66% at L = 16). • Z2 results: seeds and gauge-invariant measurement. At β = 2.0, gauge-invariant observables give 0.0173 bits, above 3σ in 21 of 21 seeds (v2.3: 0.018 ± 0.011 bits); gauge-fixed Wilson lines give 0.0058 bits, above 3σ in 4 of 21 (v2.3: 0.015 ± 0.011, 8 of 10). The "same level" comparison is removed. • Z2 results: decomposition of the sector label. Recomputed on the corrected engine (21 seeds; v2.3: five): at β = 2.0 the excitation-number component carries the association (+0.0174 bits, 21 of 21 seeds above 3σ; Wilson-loop alone +0.000033 bits), and the Wilson-loop component never dominates at any coupling (v2.3: it dominates at β = 2.5). • Z2 results: null calibration, frozen regime, and literature. Measured false-positive rates of the permutation null: 5.0 of 5 above 2σ at β = 2.5 and 1.0 ± 0.2 of 21 above 3σ, versus 0.2 of 5 and 0.05 of 21 (circular-shift null). The "frozen regime" is now a property of the superseded engine. • Finite-size scaling. The scaling tables and figure are labeled as original analysis on the superseded engines; the 1000 permutation shuffles per seed are corrected to 2000. On the corrected engine: 0.0248 bits at L = 8, 0.0051 bits at L = 16, with 95% intervals including zero from L = 32 on. • Fibonacci anyons: values and the δ sweep. v2.3's |S| = 2.811 is replaced by 2.7334 (96.64%), the maximum over all 4096 sequences of length 12 of the d1b circuit model at δ = 0; 2.8114 is the maximum under a phase deformation at δ ≈ 1.44π. The sweep is a deformation, not sectors. • Fibonacci anyons: circuit model and attribution of the effect. The two d1b gates are described as circuit gates built from Fibonacci F- and R-matrices (AM and MB, formerly σ1 and σ2), not braid generators. Braiding alone does not exceed |S| = 2, so the excess is attributed to the circuit gates, not to braiding. • Other statements corrected. The Howard–Vala result now carries its premise; the prediction S(δ) = 2√2 cos²(δ/2) for Fibonacci anyons is stated as not derived; the Xu–Ye–You reference becomes Eq. (23); K3 values reach 99.998% of the Lüders bound 3/2. Section titles, the conclusion, and the history table are revised accordingly. • Deposit and license. New in the deposit: the corrected-engine per-seed values and validation, and the full Fibonacci enumeration for L = 3–12. The code is now released under the Apache License 2.0 instead of the MIT License of v2.3; paper, figures, and data remain under CC BY 4.0. • Bibliography, references, and notation. Seven references are added; three companion-record references now give concept DOIs (10.5281/zenodo.19601998, 19601352, 20372744) in place of 21282566, 21327673, 21327799. A notation section and a table of the |S| values of all tested models are added. • Deposit packaging. The code archive and the paper PDF are named sayim-2026- - -v (record: 1a, 1b, p2, p3 or p4). The paper PDF