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

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 (v1.2 -> v1.3). This version states the scope of the k=4 resultexplicitly, aligns the series wording for K3, corrects one statement, andadds one reference; no computed value, figure, table, or title changed. • The abstract now states explicitly that the k=4 blindness concerns the single-qubit (d=2) fusion space. This was the scope of the analysis throughout — the theorem is stated "for any single-qubit dichotomic pair" and the k=4 result is called a single-qubit corollary — but the scope was not attached to the statement itself. This is a change of statement, not only of wording. • The abstract, the introduction and the theorem section now name the witness as K3 and note that it is one of several three-time Leggett-Garg witnesses, which differ in the signs of the three correlators; all Leggett-Garg results in this work refer to K3. This is the wording used across the series. In the introduction the sentence following the new passage was reworded from "This is not a new claim" to "The present work is not a new claim", so that it still refers to the reported result rather than to the inserted passage. • The theorem section now states the protocol explicitly: for each evaluation the two propagators are fixed group elements, each unchanged across the three correlator runs, and no post-selection is applied. This states what the equations already define; nothing in the derivation changed. • Certificate (1) now names its object in the statement itself: the vacuum minimal-entropy state in the S frame, as the scope remark already said. • Correction: earlier versions read Korbany's Theorem 3 in the reverse direction ("an integer value would indicate a stabilizer ground space"). The theorem is a sufficient condition; an integer value only gives no certificate. • Emary, Lambert and Nori, Rep. Prog. Phys. 77, 016001 (2014), is now cited as the source for the three-time Leggett-Garg forms. • The README abstract carries the same scope statement, and in addition the v1.2 reformulation of the protocol clause, which had not reached the README in v1.2. • Inside the deposit archive the paper source, the PDF and the notes file are named main_v1.3.*, matching the version of the record. • Release date: the deposited build is dated 2026-09-30; the Zenodo publication date is set at upload and may differ. Version notes (v1.1 -> v1.2), condensed. The complete notes for that version remain in the v1.2record: https://doi.org/10.5281/zenodo.22904849 • No value, claim, or citation changed. Bibliography completed (all 32 authors of Xu et al.; the journal publication of Byles et al. added); Cieśliński et al. (Phys. Rev. A 113, 052404 (2026)) discussed in the introduction as a related but distinct, spatial (Bell-type) line of work; the equal-time-step qualifier of the central K3=1.000000 sentence stated the same way as later in the paper; deposited build dated 2026-09-22. Version notes (v1.0 -> v1.1), condensed. The complete notes for that version remain in the v1.1record: https://doi.org/10.5281/zenodo.22698154 • No previously reported value changed. The corrections were to description and attribution, not to results: the work of Zhang et al. is a theorem rather than a survey, and the two long-range certificates are no longer offered as a priority claim; the citation of Korbany et al. points to Eq. (3.30), with the bibliography entry pinned to arXiv:2605.22424v2; the cited theorem of the mutual-information reference is Theorem 3, not Theorem 1; k=6 is listed with the dense levels rather than among those carrying a finite non-Clifford plateau; and the sentence on the contextuality interface records th