Leggett–Garg saturation and structural signatures in Fibonacci-anyon braiding
We numerically test the three-time Leggett–Garg inequality K₃ ≤ 1 for the standard B₃ Fibonacci-anyon braiding representation on the two-dimensional fusion space of three τ anyons. Exhaustive enumeration over all 4^L braid words up to length L = 11 and random sampling to L = 40 show that K₃ saturates the Lüders bound 3/2 to 99.998%, with the first violation already at L = 3. Three structural signatures accompany the saturation. First, replacing the Fibonacci generators by the Ising-anyon generators on the same 2D fusion space gives K₃ = 1 exactly for every L ≤ 11 in the exhaustive search and every random word tested at even L ∈ {12, 14, …, 40} — a sharp split that mirrors the Howard–Vala no-Bell-violation result for Ising braiding in the spatial CHSH setting. Second, the sector phase δ tunes a singular point δ = 3π/5 at which the generator σ₁ collapses to a scalar to machine precision and braiding becomes impossible. Third, the Fourier spectrum of the envelope K₃,max(δ) = max_{|w| ≤ L} K₃(δ; w) is dominated by the k = 3 harmonic (period 2π/3), reflecting optimal-word reshuffling across the sweep (correcting the k = 6 envelope value reported in v1.0, which does not reproduce under larger search-space sanity checks at L_max ∈ {7, 8}). As a consistency check, we confirm that K₃ for the optimal L = 11 word is initial-state independent (every pure state and the maximally mixed state agree to ~10⁻¹⁵, machine precision), as required by a generic d = 2 trace identity for qubit observables. All Yang–Baxter, unitarity, and (σ₁σ₂)³-scalar sanity checks pass at machine precision. To our knowledge this is the first Leggett–Garg test for non-Abelian anyon braiding specifically, and for the Fibonacci model in particular. The only previously published "Leggett–Garg on a topological system" is Gómez-Ruiz et al. (2018), which differs in three ways: the system is abelian (Kitaev chain, not Fibonacci); the qubit basis is formed by paired edge Majorana modes rather than the fusion channel of three anyons; and K₃ is used as a probe of a topological phase transition rather than as a saturation test. Code, seeds, and data are released with the preprint. Version notes (v1.2, following a comprehensive internal review of the full series): • Bibliography and citation completeness: two orphan entries are resolved (Emary–Lambert–Nori 2014 is now cited for the moving-bound formula; Fine 1982 is removed, as no body citation existed for it); three citations are added (Fritz 2010, closed-form temporal-CHSH correlator; Emary 2013, decoherence/noise-threshold framework; Kofler–Brukner 2008, conditions for quantum violation of macrorealism); a companion-work citation to the SU(2)_k Leggett–Garg study (Concept-DOI 10.5281/zenodo.20531124) is added at Open Question O3; a one-sentence limitation notes that the result is for projective Lüders measurement (weak/non-projective protocols untested); a bare "saturates already at L = 9" table caption now carries the 99.998%/never-exact qualifier used elsewhere; the title hyphen is set to an en dash for series consistency; v1.1 in-document correction scaffolding is removed (its content is preserved in the version history). • Series-wide notation: "non-Abelian" capitalization is corrected to the series-wide target form throughout; two citation titles (Brennen 2009; Xu 2024) are corrected from "non-abelian" to "non-Abelian" to match their published titles. • Builder-fidelity corrections (no numerical result, table, or figure changes): the impossibility of a spatial-CHSH violation by Ising braiding alone is now attributed to its primary source, Howard and Vala (Phys. Rev. A 85, 022304, 2012), with Clarke, Sau, and Das Sarma (Phys. Rev. X 6, 021005, 2016) repositioned as supplying the enabling non-Clifford phase gate for Majorana wires; the Fibonacci CHSH-saturation statement is now carried by braid-representation density (Nayak et al.), with Brennen et al. cited for their explicit sub-Tsirelson CHSH-violating settings; Open Question O2 is restated in two stages (reachability geometry established; the deeper divisibility question open); two qubit involutions "have an anticommutator proportional to the identity" (wording precision; the formula was always correct); the scalar-collapse threshold δ* = 3π/5 is identified with the intrinsic σ₁ rotation angle δ*(k=3) = πk/(k+2) of the companion SU(2)_k analysis — both are the relative phase arg(R₁/R_τ) of the two braid eigenvalues. • An AI-use disclosure ("Use of AI tools") is added at the end of the paper. A new deposited script re-derives the 3.1×10⁻¹⁶ scalar-collapse value cited in Sec. IV.B from the paper's own generator construction. • The Code and data section's random-seed sentence and the README's sanity-check claims are corrected to match the deposited scripts exactly; the period-scan script's seed was misstated, and two of the four reproduction scripts do not carry the three named sanity checks (they were previously implied to). • Correction (pre-publication verification pass): the sector-dependence paragraph is re-anchored to the deposited period scan. The previous statement that K₃ = 1 is also reached at nonsingular sector values, and the quoted recovery K₃ = 1.4999996 at δ/π = 1.556 with L = 24, do not reproduce from the deposited data and are withdrawn; the deposited scan shows the K₃ = 1 endpoint only at the algebraic singularity itself, with the weakest nonsingular sectors at its grid neighbors (K₃ ≈ 1.051), and a newly deposited seeded validator records the budget recovery there (K₃ = 1.075 at L = 12, 1.148 at L = 24). The abstract's initial-state-agreement figure is restated as ~10⁻¹⁵ (machine precision), matching the deposited spread. The Budroni–Emary reference gains its arXiv identifier. README claims about sanity checks and fixed-word periodicity are aligned with the deposited scripts. • Restated for transparency (correction first made in v1.1): the dominant Fourier harmonic of the envelope K₃,max(δ) was corrected from the k = 6 value reported in v1.0 to k = 3 (period 2π/3); the v1.0 identification does not reproduce under larger search-space sanity checks at L_max ∈ {7, 8}, and the withdrawal is recorded openly in the version history. • Release date set (2026-07-23). 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 tests the three-time Leggett-Garg inequality on braiding and compares Fibonacci with Ising anyons on the same fusion space, where the universal model saturates the quantum bound while the Clifford-limited one stays exactly at the classical bound, accompanied by structural signatures of the saturation; within the series it is the universality split seen on the time axis, where the witness becomes a detector for computational universality. 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.