A dichotomous Leggett–Garg witness for braid-representation density in SU(2)_k: exact at d=2, dimension-limited at d ≥ 3

Whether a temporal (Leggett–Garg) measurement can certify the computational power of an anyonic braid representation—its density in the unitary group, the property underlying universal topological quantum computation—has, to our knowledge, not been studied. We give a complete operational characterization of when the dichotomous Lüders–Leggett–Garg witness K₃ = 2 C(B) - C(B²) (one of several three-time Leggett–Garg witnesses) resolves braid-representation density across the SU(2)ₖ family. At d=2 (the spin-1/2 fusion space) the witness resolves density exactly: it saturates the dimension-independent Lüders bound 3/2 on every dense representation and stays strictly below it on the finite ones, which on this minimal-rank, one-qubit space occur precisely at k ∈ {2,4,8}—a sharp threshold distinct from the all-rank finite set k ∈ {1,2,4} of Freedman–Larsen–Wang and Kuperberg. The k=4 representation is structurally inert under K₃ at d=2 (K₃ = 1 for every measurement axis) despite a larger quantum dimension than k=3, its braid image being finite; and k=8 is pinned at 3/√5 (at Q=z; 1.427 axis-optimized). Both follow from the underlying SO(3) geometry. At d ≥ 3 the same witness ceases to resolve density—it saturates 3/2 even on finite representations—and we trace this to the extra dimension itself, not to any particular braid angle. This is shown directly, in the state-optimized reading, on three finite d=3 families; in the neutral reading it is established for odd d≥3, and the neutral case at even d≥4 is left open. We frame the resulting no-go question—whether every such witness is density-blind at d ≥ 3—as an open problem. This paper resolves the split across the whole SU(2)_k family and characterizes when the temporal Leggett-Garg witness certifies the density of a braid representation, the property behind universal topological quantum computation: on the one-qubit fusion space it resolves density exactly, saturating the quantum bound on every dense representation and staying below it on the finite ones, whose inert behaviour follows from the underlying SO(3) geometry, while in higher dimensions the same witness ceases to resolve density, which is framed as an open problem; within the series it is the map that generalizes the split and supplies the mechanism behind it. 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 (v1.1 → v1.2). This version changes the scope wording of the structural-inertness label and the completeness statements, universal-computation wording, d≥3 statements, literature attributions, novelty statements, figures, licensing, deposited package, README, and bibliography. • Scope of the "structurally inert" label. The label now reads "under K₃" in four places, and the abstract and Appendix C heading add "at d=2", because the paper's table lists a k=4 representation at d=3 (targeted K₃ = 1.500); no number changed. • Scope of the completeness statements. Two introduction passages ("the complete", "entire") are narrowed to "across the SU(2)_k family". Abstract, conclusion, Sec. IV and Sec. V now state the d≥3 scope: three finite d=3 families (state-optimized), odd d≥3 (neutral), even d≥4 left open. • Universal-computation wording. The statements equating braid universality with density, and the "(universal)"/"(non-universal)" labels, are replaced by: projective density on the selected fusion space is the representation-theoretic criterion for braid universality on that encoded space. The figure legend reads "dense"/"finite". • Statement on the d≥3 results. "Under either state reading", "at all d≥3" and "for any nontrivial dynamics at every d≥3" are narrowed to targeted and odd-d neutral readings, "at every d≥3 we tested", and SU(2)_k braid dynamics. All 15 d≥3 entries (d=3–5, k≤10) lie within 0.01 of 3/2 (minimum 1.49990). • Attribution of the Lüders-bound literature. The claim that 3/2 can be exceeded only under the von-Neumann rule is limited to the framework of Budroni and Emary. Mal–Majumdar is now described as treating the four-term inequality (classical bound 2, reaching 2√2), and the trapped-ion (1.739 ± 0.014) and NV-center (1.625 ± 0.022) results are stated separately. • Attribution of the k=8 case. The resolution of the icosahedral k=8 (r=10) borderline case, finite at d=2 but dense at higher rank, is now attributed to Freedman–Larsen–Wang (r=10, n≥5 clause) instead of Kuperberg, in three places. • Novelty statements. "To the best of a full-text prior-art search" is replaced by "to our knowledge" in two places. • Added and revised figures. New Fig. 2 is a resolution map across the SU(2)_k family by level k and anyon spin j, each cell a value from the deposited files p6_gate1_sweep_results.json and p6_rho_closed.json. Fig. 1 is re-rendered at its printed size with two labels moved into the caption; no underlying data changed. • Added scope statement. The outlook now states that the reported values are those of the idealized sequential Lüders protocol and that an experimental test would additionally have to address the clumsiness loophole (Wilde–Mizel 2012; Emary–Lambert–Nori 2014). • Licensing, availability, and acknowledgments. The statement that all code and result files are CC BY 4.0 is replaced by a "Data and Code Availability" section: paper, figures, and data under CC BY 4.0, deposited code under the Apache License 2.0. An Acknowledgments section states no specific grant and no competing interests. • Deposited package. The sweep script p6_gate1_sweep.py is no longer part of the record, and the sentence stating that it covers k≤10 is removed. Reported optima are explicit braid words in p6_gate1_sweep_results.json, checkable with the new verifier p6_gate1_verify.py; LICENSE-CODE and the resolution-map PDF and PNG are added. • README. It lists the verifier and its results file in place of the sweep script, lists the new figure files, adds the resolution map (48 cells) to its key numerical claims, and carries the "under K₃" wording and license line. It gives the companion Letter and two record titles under their new titles. • Bibliography. Journal data and DOIs added for Tuba–Wenzl and Rowell–Tuba; issue, page range, and DOI added for Mal–Majumdar; DOI added to the 1985 Leggett–Garg reference; Wilde–Mizel 2012 and the data record added. The companion Letter is cited under its new title, and the "[Concept-DOI, always points to latest version]" annotation is removed. • Numerical results. No previously printed numerical result is changed. • 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 da