Leggett–Garg K₃ Values Above 1 in Fibonacci-Anyon Braiding: 99.998% of the Lüders Bound, and Exactly 1 for Ising 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. Two 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, an algebraic phase-deformation parameter δ tunes a singular point δ = 3π/5 at which the generator σ₁ collapses to a scalar to machine precision, so that every braid word reduces to a global phase. The Fourier-period analysis of the δ dependence reported in earlier versions of this letter is withdrawn. 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 for the undeformed generators (δ = 0). The closest previously published work we identified, “Leggett–Garg on a topological system,” is G\'omez-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. This paper tests the three-time Leggett-Garg inequality on braiding and compares Fibonacci with Ising anyons on the same fusion space, where Fibonacci braiding reaches 99.998% of the Lüders bound while Ising braiding stays exactly at the classical value 1; within the series it is the universality split seen on the time axis. 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 (v1.2 → v1.3). This version changes the title, the Fourier-period analysis, the deformed generator family, the envelope symmetry, the findings count, the Ising search description, attributions and scope, structure and licensing, the figure, bibliography and deposited package. • Title. Renamed from "Leggett–Garg saturation and structural signatures in Fibonacci-anyon braiding" to "Leggett–Garg K₃ Values Above 1 in Fibonacci-Anyon Braiding: 99.998% of the Lüders Bound, and Exactly 1 for Ising Braiding" (the Lüders bound 3/2 is not reached exactly: exhaustive L ≤ 11 gives 99.984%). • Withdrawal of the Fourier-period analysis. The v1.0 (k = 6) and v1.1/v1.2 (k = 3) Fourier-period results are withdrawn and the dependent text revised. Corrections: k = 6 reproduces at exact length L = 9; the envelope is not 2π/3-periodic (K₃,max = 1 at δ = 3π/5, 1.2047 at 3π/5 + 2π/3). • The deformed family of generators is not a braid representation at generic δ. The v1.2 checks refer to δ = 0; for δ ≠ 0 σ₁σ₂σ₁ = σ₂σ₁σ₂ fails even up to a global phase, holding in [0, 2π) only at δ = 0, 3π/5, 6π/5. "Sector phase" is renamed deformation parameter. • New result: mirror symmetry of the envelope. K₃,max(δ) = K₃,max(6π/5 − δ), i.e., symmetric about the singular point and about δ = 8π/5 (via complex conjugation). The deposited envelope satisfies it to within 7×10⁻¹⁵ (word by word at most 2.5×10⁻¹⁴); δ = 6π/5 reproduces K₃ = 1.499762. • Number of findings. v1.2 listed five findings and three structural signatures; the text now lists three findings and two structural signatures. Initial-state independence is now a consistency check, and the qualifier "not implied by universality alone" is dropped. • Description of the Ising δ search. v1.2 said "same exhaustive depth L ≤ 9"; the text now says all 4⁹ words of length L = 9 at each of 49 grid points δ ∈ [0, 2π]. The reported values (range [1.000, 1.500], 1.500 at δ/π ≈ 1.167) are unchanged. • Attributions and scope. Prior-art claims are reworded ("closest previously published work we identified") and the "first Leggett–Garg test" abstract sentence is removed. The Howard–Vala statements now give their premise in full (only topologically protected stabilizer operations), and density in SU(2) becomes necessary and sufficient at d = 2 (v1.2: necessary). • Added scope statement. The outlook states that an experimental test would additionally have to address the clumsiness loophole (Wilde–Mizel 2012; Emary–Lambert–Nori 2014). • Structure, licensing, and availability. A Notation section and an Acknowledgments section (no specific grant; no competing interests) are added; "Code and data" becomes "Data and Code Availability". Licensing: paper, figures and data CC BY 4.0, deposited code Apache License 2.0 (v1.2: all code and result files CC BY 4.0). • Figure. Fig. 1(b): the Fourier-period annotation box is removed, the axis label reads "deformation parameter δ/π" and the panel title "Deformation dependence (Fibonacci)"; curve and singularity line unchanged (same data). The caption now names the data files and the plotting script. • Bibliography. arXiv identifiers added for Emary–Lambert–Nori (1304.5133), Andersen et al. (2210.10255) and Xu et al. (2404.00091); full author lists for Iqbal et al. (18 authors) and Xu et al. (32 authors). The "[Concept-DOI, always points to latest version]" annotation is removed; Wilde–Mizel 2012 and the data record are added. • Deposited package. Added: lgi_envelope_word_audit.py and lgi_braid_relation_check.py with their results files (seed 20260928), and LICENSE-CODE (Apache License 2.0; LICENSE remains CC BY 4.0). In lgi_ising.py and lgi_fibonacci.py only the printed label "exhaustive L<=9" is corrected (results files byte-identical). • 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. • Release date: the deposited build is dated 2026-10-03; the Zenodo publication date is set to the same day. 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.