Correlation Witnesses versus Magic: A k-Resolved Nonstabilizerness Map for SU(2)_k Anyon Fusion Spaces

Abstract

Standard correlation witnesses — spatial Bell/CHSH inequalities, temporal Leggett–Garg K₃, and KCBS contextuality — are known to detect nonstabilizerness ("magic") in some settings. We report a worked example, a k-resolved nonstabilizerness map of the SU(2)_k anyon braid-representation family, in which a standard temporal witness is instead systematically blind: at k=4 the three-time Leggett–Garg witness saturates its macrorealistic bound exactly (K₃=1.000000 over every state, braid element, and measurement axis, in the equal-time-step protocol of Table 1 with V₁=V₂) while the single-qubit fusion channel carries near-maximal nonstabilizerness (M₂=0.5585, 95% of the finite-dimensional ceiling log₂(3/2)). We prove this blindness as a structural theorem: the witness depends only on the Bloch-sphere Gram geometry of the braid orbit, not on nonstabilizerness, and k=4 happens to align the fixed measurement axis with a threefold orbit symmetry (Bloch dot products all −1/3) that caps the witness at 1; a finite, Clifford-generating group — the chiral octahedral group, the k=2 braid image — with a misaligned axis reaches K₃=3/2 under a two-propagator protocol. Neither finiteness of the braid image nor the Clifford property is by itself the mechanism. We complement this temporal certificate with three independent certificates of genuine nonstabilizerness — two of them long-range (a doubled-Fibonacci mutual-information witness, H=1.700979, and a gauge-invariant minimum ground-space stabilizer Rényi entropy, ≥6.5 against an exact-zero toric-code control), the third a complementary gate-based non-Cliffordness measure — an honest negative control confirming that leakage-free constructions remain strictly additive, and a hardware-oriented single-qubit signal-to-noise prediction. We further probe the k=4 dissociation at its shared d=3 interface with Kochen–Specker–Klyachko contextuality, where the KCBS witness shows only a weak, nongeneric correlation with magic. Two reconciliations with the literature are made explicit: the present result contradicts neither the equivalence of maximal nonlocality and maximal magic established for optimized magic states in a different game, nor the contextuality-supplies-magic theorem; both concern a different object than the fixed, per-k braid generator studied here.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 asks whether the inert blind spot at k=4, where the temporal Leggett-Garg witness stays exactly at the classical bound, means that the quantum resource is absent or only invisible to that witness, and it finds the nonstabilizerness nearly maximal, confirmed by three independent certificates, while proving the blindness to be geometric: the witness depends on the geometry of the braid orbit rather than on the resource, and a finite Clifford-generating group with a misaligned axis reaches the quantum bound, so neither finiteness nor the Clifford property is the mechanism; within the series it is the resource explanation, where blindness is a matter of alignment, not absence. 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 (v1.0 → v1.1, following a comprehensive internal review of the full series): • No previously reported value changed. Two quantities are reported for the first time (the Monte-Carlo standard errors of the sampled representatives); the remaining new numerals in the text are equation references to the cited literature. • Accuracy of description: the braid generator used in the computation is now named precisely rather than described as "the same" generator; the fusion-channel statement no longer claims that the same qutrit is used throughout the paper; the correlation figure is no longer expressed as a percentage "of the way to a perfect correlation"; the Table I caption now states that "maximized over braid words" applies to three of the nine rows rather than to all of them; and the Figure 1 caption is brought into agreement with the table it describes. • Correction (attribution of prior work): the work of Zhang et al. was described as a survey and is now described as a theorem, and the two long-range certificates reported here are explicitly no longer offered as a priority claim — that statement belongs to Zhang et al., whose theorem is now quoted. The citation to Korbany et al. now points to Eq. (3.30) and the bibliography entry is pinned to arXiv:2605.22424v2 — the previously cited "Eq. (35)" is correct for v1 only, and the unpinned link resolved to v2, where the equation is numbered differently. The Ding et al. entry carries its full title, and the Xu et al. reference now carries its journal publication (Nat. Phys. 20, 1469 (2024)) alongside the preprint. • Scope of the Leggett-Garg result — sharpened: the no-go statement following Howard and Vala is given with its source qualifier ("stabilizer operations"), matching the companion paper; and the sufficient condition for witness blindness is stated in terms of the orbit Gram structure rather than the order of the symmetry axis, with a deposited spot-check (icosahedral k=8 orbit) as an aligned, higher-order counterexample to the previous phrasing. • Scope of the witness evaluation — made explicit: a scope remark states which theorem of the cited reference applies to this model (Theorem 3, formulated for string-net ground states), what it does and does not require of the regions, and why the lattices used here are too small to permit a direct evaluation of the mutual information. Correction: the surrounding text cited Theorem 1 and now cites Theorem 3. • Added: the eleven representatives underlying the reported M2 range are now identified by construction type (condensate, random, and eigenbasis), and their Monte-Carlo standard errors (0.05-0.53) are reported. • Figure 2 — legibility: the in-panel annotation is re-wrapped so that it no longer crosses the k=4→k=8 arrow or the legend; wording, data, and markers are unchanged (p5a_fig2_orbit_geometry.py, one text call). • Newly deposited: the ground space of the restricted sector (p5a_V_gs_2x2.npy), so that the values reported in p5a_tsre_robust_m2_range.json can be recomputed from deposited data. It spans the same subspace as the array deposited in the companion record (doi:10.5281/zenodo.21362245) to machine precision, while the basis within that subspace differs; the README states both, with the agreement of the projectors as the evidence. • Provenance completed: the two lattice inputs whose md5 values are declared in p5a_korbany_lattice_vacuum.json are deposited in the companion record; the file now says so, and that record is cited at the point where its extraction method is used. • Licensing — now stated consistently in the paper, and the code license has changed: paper, fig