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) resolves braid-representation density across the SU(2)_k family. At d = 2 (the spin-½ 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 (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 = ẑ; 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. We frame the resulting no-go question — whether every such witness is density-blind at d ≥ 3 — as an open problem. Version notes (v1.0 → v1.1, following a comprehensive internal review of the full series): • A one-sentence correction note is added at the end of Sec. III.B: the closed-form passage stating that ẑ is a 5-fold axis at k = 8 (σ₁ = 144°) is corrected — σ₁ (144° about the measurement axis ẑ) gives K₃ = 1, not 3/√5; the value 3/√5 is realized by a distinct 72° rotation about a different tilted 5-fold axis. Fixed in both Sec. III.B and Appendix C, in the same in-text "Correction to v1.0:" format as the companion papers. No numerical value changes (3/√5 was and remains correct); only the angle/element attribution is corrected. • Attribution and scope: the {2,4,8} finite-image classification is now attributed to Freedman–Larsen–Wang (the k = 8 icosahedral borderline case via Kuperberg), with Tuba–Wenzl and Rowell–Tuba added as the underlying B₃-finiteness references; the {2,4,8} finite set is qualified throughout (abstract + 8 sites) as specific to the minimal-rank, one-qubit (d = 2) fusion space, distinct from the all-rank finite set k ∈ {1,2,4}; the FLW citation is completed. • The geometric discriminator distinguishing binary (SU(2), |G| = 48/24/120) from projective (SO(3), order 24/12/60) group orders is added, together with the general sector-phase family δ*(k) = πk/(k+2) unifying the σ₁ angles at k = 2, 4, 8. • Correction (internal review pass): a statement that the state-optimized value reaches 3/2 "for any nontrivial dynamics at every d ≥ 2" contradicted the paper's own Table I (at d = 2 the k = 4 and k = 8 values are 1.0 and 1.427, not 3/2); it is corrected to d ≥ 3 in Sec. III.D and Appendix C, matching the section's own title. Several smaller consistency and wording items from the same pass are corrected; no result, figure, or table value changes. • The d ≥ 5 scope statement is sharpened from "implied, not verified" to structural: no finite d = 5 anyon representation exists to test for k ≤ 16 (verified via an internal level sweep, k ≤ 16, not included in this deposit; the in-deposit sweep artifacts cover k ≤ 10). • The reported witness value is defined explicitly as the supremum over the braid-group image; the d ≥ 3 no-resolution result is scoped to odd d (neutral reading) with the even-d ≥ 4 neutral case noted as open. • A companion-work citation to the Fibonacci-only (k = 3) Leggett–Garg Letter (Concept-DOI 10.5281/zenodo.20372744) is added in the Introduction; a one-sentence limitation on weak/nonprojective measurement protocols is added to the Discussion; series-convention items (section numbering depth, PDF-metadata subject field) are applied. • An AI-use disclosure ("Use of AI tools") is added at the end of the paper. Four scripts that previously wrote their results only to stdout now also write JSON result files (no computation changed); the full exhaustive search additionally confirms that exactly three finite d ≥ 3 representations exist in the whole k ≤ 12 search space. • The Code and data section is qualified: the internal k ≤ 16 level sweep behind the d ≥ 5 scope statement (Appendix C) is not part of this deposit, matching the existing README disclosure. • Release date set (2026-07-22). 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 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. 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.