CORRECTION IN v2 (2026-07-28). Proposition 9 of version 1 is withdrawn. It stated that the mod-2 (odd-Q) shadow of the exact Z_d criterion can produce a false certificate, quoted 28.6 percent as the rate at d=8, and cited an explicit example in the repository. All three are wrong. For even d, solvability of Mx = gamma mod d implies solvability mod 2, so any lambda closing mod 2 pairs evenly with gamma: the shadow can never fire on a noncontextual system. The false-positive rate is exactly 0 at every even d, provably, and no such certificate exists, which is why the cited artifact was never committed. The criterion is instead incomplete in the opposite direction: a Z_d-cycle can certify with a value that is nonzero mod d but even, and the mod-2 reduction of that value vanishes, so the shadow stays silent on a genuinely contextual system. The v1 error made the criterion look weaker than it is while naming the wrong failure mode. What survives is the point on certificate minimization: a minimal odd-Q support need not be a Z_d-cycle, so it can be a false witness while the verdict is always right. Carried in the paper as a labelled Remark; no other result depends on the withdrawn claim. Certificate: verification/shadow_soundness_exact.py. State-independent contextuality of qudit stabilizer measurements is not just present or absent: it comes in a definite strength. For m-qudit Weyl-Heisenberg context families in local dimension d, the achievable all-versus-nothing obstruction value is exactly {0, d/2} for every even d and {0} for odd d. ATTRIBUTION: this value classification, both the even/odd dichotomy and the fixed value d/2, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024), whose parity obstruction 2k = 0 (mod d) and normal-form classification settle it. Our Obstruction Spectrum Theorem recovers their result in explicit Weyl-certificate language; it is not an independent discovery, and the paper says so in its abstract, its related-work section and a theorem-by-theorem comparison table. On that basis we develop three further layers. (i) A closed-form calculus: contextuality is decided by an integer commutator carry b(C), computed from the symplectic form with no phases or matrices, together with an exact value-bit formula and a cycle criterion. The carry invariant has the algebraic form of a Pontryagin-square quadratic refinement; we do NOT construct the cohomology operation, and the language is hedged accordingly. (ii) A 2-adic tower: the obstruction lives at the top layer and no single lift carries it (Depth Rigidity), with a fiber doubling law governing propagation d -> 2d. The doubling law is stated as a conjecture with an explicitly open step. (iii) A three-device superconducting replication of the d=2 witness. HARDWARE CAVEAT, STATED UP FRONT. The combined figure of 80 sigma is SHOT-NOISE ONLY and overstates the combined significance. The three devices fail a homogeneity test (chi-squared about 266 on 2 d.o.f.), so a systematics-aware combination gives approximately 7 sigma, not 80. Each device individually violates the bound by 32 to 61 sigma on shot noise, so the qualitative violation is robust, but this is NOT a loophole-free certification and the 80 sigma number should not be read as a single-S significance. This is a preprint, not a peer-reviewed article. The obstruction's geometric home, a canonical degree-2 class on the stack of maximal abelian subalgebras, is developed in a companion paper. Code, data and verification suite: https://github.com/manuflog/contextuality-obstructions Note on methods: large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance, and the author takes full responsibility for the results, including any errors.
CORRECTION IN v2 (2026-07-28). Proposition 9 of version 1 is withdrawn. It stated that the mod-2 (odd-Q) shadow of the exact Z_d criterion can produce a false certificate, quoted 28.6 percent as the rate at d=8, and cited an explicit example in the repository. All three are wrong. For even d, solvability of Mx = gamma mod d implies solvability mod 2, so any lambda closing mod 2 pairs evenly with gamma: the shadow can never fire on a noncontextual system. The false-positive rate is exactly 0 at every even d, provably, and no such certificate exists, which is why the cited artifact was never committed. The criterion is instead incomplete in the opposite direction: a Z_d-cycle can certify with a value that is nonzero mod d but even, and the mod-2 reduction of that value vanishes, so the shadow stays silent on a genuinely contextual system. The v1 error made the criterion look weaker than it is while naming the wrong failure mode. What survives is the point on certificate minimization: a minimal odd-Q support need not be a Z_d-cycle, so it can be a false witness while the verdict is always right. Carried in the paper as a labelled Remark; no other result depends on the withdrawn claim. Certificate: verification/shadow_soundness_exact.py. State-independent contextuality of qudit stabilizer measurements is not just present or absent: it comes in a definite strength. For m-qudit Weyl-Heisenberg context families in local dimension d, the achievable all-versus-nothing obstruction value is exactly {0, d/2} for every even d and {0} for odd d. ATTRIBUTION: this value classification, both the even/odd dichotomy and the fixed value d/2, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024), whose parity obstruction 2k = 0 (mod d) and normal-form classification settle it. Our Obstruction Spectrum Theorem recovers their result in explicit Weyl-certificate language; it is not an independent discovery, and the paper says so in its abstract, its related-work section and a theorem-by-theorem comparison table. On that basis we develop three further layers. (i) A closed-form calculus: contextuality is decided by an integer commutator carry b(C), computed from the symplectic form with no phases or matrices, together with an exact value-bit formula and a cycle criterion. The carry invariant has the algebraic form of a Pontryagin-square quadratic refinement; we do NOT construct the cohomology operation, and the language is hedged accordingly. (ii) A 2-adic tower: the obstruction lives at the top layer and no single lift carries it (Depth Rigidity), with a fiber doubling law governing propagation d -> 2d. The doubling law is stated as a conjecture with an explicitly open step. (iii) A three-device superconducting replication of the d=2 witness. HARDWARE CAVEAT, STATED UP FRONT. The combined figure of 80 sigma is SHOT-NOISE ONLY and overstates the combined significance. The three devices fail a homogeneity test (chi-squared about 266 on 2 d.o.f.), so a systematics-aware combination gives approximately 7 sigma, not 80. Each device individually violates the bound by 32 to 61 sigma on shot noise, so the qualitative violation is robust, but this is NOT a loophole-free certification and the 80 sigma number should not be read as a single-S significance. This is a preprint, not a peer-reviewed article. The obstruction's geometric home, a canonical degree-2 class on the stack of maximal abelian subalgebras, is developed in a companion paper. Code, data and verification suite: https://github.com/manuflog/contextuality-obstructions Note on methods: large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance, and the author takes full responsibility for the results, including any errors.
Preprint. Third paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), Paper D (the observer-context category) and a synthesis note. The slogan "contextual holonomy encodes quantum contextuality" is attractive and, as usually stated, false. This paper gives it a sharp form. On the context stack of maximal abelian subalgebras the canonical degree-2 class has order exactly n, and its cocycle holonomy along closed context loops detects EXACTLY the all-versus-nothing sector of contextuality for closed-triple Weyl families at even local dimension. Both containments are strict, each by an explicit certificate: at d=3 the class is nontrivial of order 3 while the achievable state-independent AvN value is 0, and KCBS exhibits state-dependent contextuality where the class shadow vanishes. The Wigner-friend loop generated by H and S splits by level: its projective (Pancharatnam) holonomy is +1 while its determinant-section holonomy is exactly -i, so any interpretational claim about "the" holonomy of that loop must state which one it means. ATTRIBUTION. The classification of state-independent values for qudit Weyl families, the even/odd dichotomy and the fixed value d/2, on which the state-independent side of the equivalence rests, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024). Attainment at general even d is the Raussendorf-Okay-Zurel-Feldmann construction. This paper uses both and rediscovers neither. A RETRACTION IS CARRIED IN THIS PAPER, NOT BURIED. An earlier draft presented the tier-two facets of the d=4 census on a (1/(2*sqrt2))Z grid with a single coefficient magnitude 3/(2*sqrt2), and read the resulting unit-coefficient bounds 4*sqrt2 and 14*sqrt2/3 as placing Tsirelson's sqrt2 inside the tier-two geometry. The whole presentation was a normalization artifact and is withdrawn (Retraction 1): the stored lift has coefficients in {0,+-1} with integer bounds 6 and 7, every facet functional in the census is rational, and Q(sqrt2) is not among the eighteen canonical arithmetic fields. Certificate: d4_arithmetic_profile.py. WHERE THE EVIDENCE IS THIN, STATED IN THE PAPER. Two sweeps quoted here drew no positive instances at all: the d=4 half of the 2,100-family equivalence test, and the 6,500-trial false-certificate sweep, which draws no contextual family at any of d=2,4,8 and therefore establishes the absence of false certificates only vacuously. Both are disclosed where they appear. Theorem 2 is labelled machine-validated rather than proved: the closed form is established end to end for the deletion families treated, while the general closed-triple d=2 statement in compressed coordinates remains open and is recorded as open in the verification ledger. HARDWARE CAVEAT, STATED UP FRONT. The quoted significances, about 56 sigma for the ray-level Wigner-friend holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented witness model and conditional on compatibility and nondisturbance assumptions. They are not systematics-aware and neither measurement is loophole-free; a roughly 3.5 sigma preparation offset on one strand is itself evidence of unmodelled systematics. We also record that the pinned value S = 4.6125 cannot be recomputed from the stored counts (a naive recomputation gives 4.5662, a gap of about 2.7 standard errors); the archived value 4.558 +- 0.018 is the one an independent reader can reproduce from the deposited data. Read these as consistency checks on real hardware, not as precision measurements or certifications. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. It was held back from the first publication batch and released only after three adversarial review passes; what each pass found, and how each item was closed, is recorded in verification/INDEX.md rather than quietly fixed. One defect found there is NOT closed and belongs to the companion Paper B, which cites a false-certificate artifact that is not in fact in the repository; it is recorded rather than left for a reader to discover. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Note: https://doi.org/10.5281/zenodo.21630739 . Code archive: https://doi.org/10.5281/zenodo.21498498 AI-ASSISTANCE DISCLOSURE. Large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance by the scripts named in the text, and the author takes full responsibility for the results, including any errors. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Quantum measurements need not reveal preexisting values: the outcome assigned to an observable can depend on which compatible observables are measured with it. This contextuality underlies the Peres-Mermin proof of the Kochen-Specker theorem and, in quantum computation with magic states, tracks the even/odd parity of the local dimension separating the classically simulable from the genuinely quantum. ATTRIBUTION: the even/odd value classification of state-independent qudit contextuality (the parity dichotomy and the fixed obstruction value d/2) is due to Abramsky, Cercelescu and Constantin, FSCD 2024; this paper supplies its geometric and cohomological ACCOUNT, not a rediscovery of it. Organizing all measurement contexts of M_n(C) into the stack [X/PU(n)] of maximal abelian subalgebras, we show the lifting class of U(1) -> U(n) -> PU(n) is vacuous on the action groupoid U(n) x X but, on PU(n) x X, is a canonical degree-2 class of order exactly n. Its real shadow, through the signed-permutation group B_n, is a Z_2 extension splitting if and only if n is odd, with "-I is a rotation square iff n is even" as the mechanism; at n=4 it restricts to the real two-qubit Pauli class, whose pairing with the Peres-Mermin word is -1, exactly the obstruction to a noncontextual value assignment. We compute H^2 = Z/n (+) Z_2 for n >= 4 and identify the canonical class as a generator of the Z/n summand. STATUS AND SCOPE. Every step is tagged Complete, Sketch (reduced to a named standard theorem whose hypotheses are verified), or Gap, and a status table closes the paper. The finite and algebraic core is machine-checked by scripts in the public suite. The geometric layer rests on standard but non-elementary facts about differentiable-stack cohomology and the Lyndon-Hochschild-Serre spectral sequence; that bookkeeping is reduced to named standard theorems and AWAITS SPECIALIST REVIEW. This is a preprint, not a peer-reviewed article, and should be read at that strength. The finite Weyl value spectrum is developed in a companion paper. Code, data and verification suite: https://github.com/manuflog/contextuality-obstructions Note on methods: large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance, and the author takes full responsibility for the results, including any errors.
Preprint. Fourth paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), and a note on local validity. This paper makes "observer" a mathematical object: an isometric enlargement of the system together with a measurement context on the enlarged algebra, organized into a category whose invertible switches form a subgroupoid and which carries the U(1) cocycle of the companion papers. Four results follow. (1) Relative facts. Every observer admits an exact local classical section, while no global value assignment exists for the Peres-Mermin family, nor for any enlarged family containing its image under a Wigner enlargement. (2) A gauge-robust Z2 invariant of switch loops: +1 on all loops of Weyl switches, -1 on the oriented mutually-unbiased triangle. Nontriviality requires complementary switching; the converse fails, and the counterexample is pinned. The invariant counts switching parity at the tau level, the same object as the Wigner-friend -i. (3) The movable cut. The comparison map between cut placements is fixed only up to a choice of section: its ray-level content is invariant, the section-level bookkeeping shifts by the loop holonomy, and the update rule available to each observer is forced. (4) The Frauchiger-Renner chain is localized. Its observer cycle has trivial class and holonomy exactly +1, which refutes the natural conjecture that such paradoxes measure holonomy. Its contradiction is instead state-sector, with contextual fraction exactly 1/6 and the CHSH inequality at value 7/3 as the unique separating facet of the codeword polytope. ATTRIBUTION. The classification of value assignments on the Peres-Mermin square that this program builds on is due to Abramsky, Cercelescu and Constantin (FSCD 2024). Their priority is acknowledged in the text and here. HARDWARE CAVEAT, STATED UP FRONT. One result is anchored by data taken on ibm_fez. The quoted significance of roughly 56 sigma is shot-noise statistical only, computed under the implemented interferometric model. It is not a systematics-aware figure, and the experiment is not loophole-free. Read it as a consistency check of the ray-level prediction on real hardware, not as a precision measurement. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. Results are tagged in the text as Complete, Sketch, or Gap. The categorical and cohomological layer awaits specialist review. The paper presents structural characterizations within its stated axioms; it is not an interpretation-free derivation of Copenhagen, and it does not claim to resolve the measurement problem. AI-ASSISTANCE DISCLOSURE. This work was carried out with substantial AI assistance. Every load-bearing claim is machine-verified by scripts that ship with the paper, and the author takes full responsibility for the contents. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
ON THE VERSION NUMBER. "v3" is the version of the NOTE, not of this Zenodo deposit. Versions 1 (February 2026) and 2 were circulated privately and never deposited; v1 contained four errors of substance, which are stated with their repairs in the "What changed" section rather than quietly removed. This is the first Zenodo deposit, so if you see a deposit counter it will not match the document version. The document version is the one that matters for citation. Preprint, version 3 (complete rewrite). This note is the synthesis of the contextuality-obstructions program; the proofs live in the companion papers. The proposal is that the Copenhagen viewpoint, stripped of its rhetoric, is a set of structural statements about one geometric object: the transformation groupoid of measurement contexts (maximal abelian subalgebras) with its canonical U(1) 2-cocycle. In that formulation: local validity, the exact classicality of every single context, is a theorem with an explicit witness; complementarity is the nonvanishing holonomy of the cocycle along closed context loops, which detects exactly the all-versus-nothing sector of contextuality; collapse is pinned by statistics, repeatability and covariance to a one-parameter family per degenerate block whose unique efficient member is the Lueders rule, with a classical twin theorem identifying the same axioms' classical solution as Bayesian conditioning; observer-relativity of facts and the movable cut are theorems of an enlarged observer-context category; and the Frauchiger-Renner argument localizes to an ordinary state-dependent violation, with contextual fraction exactly 1/6, carried by a single CHSH facet, the loop holonomy of the observer switch being provably trivial. ATTRIBUTION. The classification of achievable state-independent values, the even/odd dichotomy and the fixed value d/2, is due to Abramsky, Cercelescu and Constantin (FSCD 2024). It is recast here in certificate language; the priority is theirs and is stated in the text. HARDWARE CAVEAT, STATED UP FRONT. Two signatures were measured on superconducting hardware. The quoted significances, about 56 sigma for the triangle holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented model. They are not systematics-aware, and neither experiment is loophole-free. Read them as consistency checks on real hardware, not as precision measurements. ERRATA. Version 1 of this note contained four errors of substance. Section "What changed" states each of them and its repair. They are recorded rather than quietly removed. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. Each tenet is either a theorem of the companion papers, a machine-verified certificate in the public suite, or is explicitly labeled a conjecture. The paper offers structural characterizations within its stated axioms; it is not an interpretation-free derivation of Copenhagen and it does not claim to resolve the measurement problem. AI-ASSISTANCE DISCLOSURE. This work was carried out with substantial AI assistance. Every load-bearing claim is machine-verified by scripts that ship with the papers, and the author takes full responsibility for the contents. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Paper C: https://doi.org/10.5281/zenodo.21633744 (posted 2026-07-28, after three adversarial review passes; it carries a labelled retraction of an earlier sqrt(2) reading). VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Files of this version are restricted (18 October 2026) because it was released under earlier licence terms: the source code was under the MIT License. All current versions use one clear licensing: paper, figures and data under CC BY 4.0, source code under the Apache License, Version 2.0. Please use the latest version: https://doi.org/10.5281/zenodo.21362245 We report a real-space resolution of the full modular data of a doubled-Fibonacci Levin–Wen string-net, obtained by exact diagonalization on finite torus supercells (2×2, 2×3, 3×3), together with a sharply scoped negative result for the dynamical half of the same program — and we keep the classes of evidence strictly separate. (i) The vacuum row of the modular S-matrix, |S₀ₐ| = dₐ/D, is confirmed by two structurally distinct lattice routes — an operator route built from a flux basis, and a Verlinde–Casimir route that reads the quantum dimensions from the lattice loop spectrum without using the F or R symbols — the latter reproducing the analytic values at 1.1×10⁻¹⁶ and the two agreeing at <10⁻⁹, the precision at which the operator-route values are recorded. (ii) The ground-state degeneracy GSD=4 is a separate check, invariant across 2×2, 2×3 and 3×3; |S| is invariant under 24 O(4) gauge transformations of the ground-state manifold and size-robust to ≤2.5×10⁻¹¹. (iii) The Wilson-loop algebra alone does not fix the chiral sign, and we measure that limit rather than assert it: the loop algebra is ℂ⊕M₃(ℂ) (dim=10=1²+3²), its antisymmetric part vanishes identically on the real-symmetric holonomy, and the chiral splitting capacity is zero. This is a concrete instance of a theorem of Li and Mong, who also name the remedy: a point-group rotation of the ground-state manifold, real but not symmetric — precisely the property the loop route lacks — available on every L×L supercell. Carrying it out, we extract the full signed S and T from the lattice on two sizes, circularity-free, up to the ℤ₂ relabeling intrinsic to an achiral theory; a validated two-channel error budget with no free parameters accounts for the extraction residual on both. The modular data themselves are not new with this work — Francuz and Dziarmaga obtained them for this model in 2020; new here are the finite-lattice demonstration of the rotation channel, the quantified error budget, and the measured loop-algebra limit as a concrete instance of the Li–Mong bound. Complementing this static certificate, we show that on the one-dimensionally mapped anyonic ladder the Hamiltonian transport of a single mobile Ising anyon around a pinned object cannot realize the non-Abelian braid matrix: the closed-loop holonomy is exactly the identity and [U(C₁),U(C₂)]=0, established through two independent obstructions, three null escape routes, and a machinery-alive control. That obstruction is structural and independent of both the anyon type and the system size, so it constrains the Fibonacci ambition motivating this work and not merely the Ising tracer that exhibits it. Both halves were produced under an adversarial verification harness that, in the course of this work, retracted an attribution of its own — crediting the lattice with the signed data before a channel existed to carry it — and then restored it with proof through the rotation channel. The values were never in question, only their provenance.Notice (2026-10-14). The summary above states that on the one-dimensionally mapped anyonic ladder of Singh et al. the Hamiltonian transport of a single mobile Ising anyon around a pinned object has the identity as its closed-loop holonomy, and that this obstruction is “structural and independent of both the anyon type and the system size”. This holds under the hypotheses stated in the paper, in particular that the braid phase sits in the bond amplitudes identically on both legs of the ladder. That hypothesis is the explicit Ising plaquette Hamiltonian, Eq. (52), of S. Singh, R. N. C. Pfeifer, G. Vidal and G. K. Brennen, Phys. Rev. B 89, 075112 (2014). The general hopping Hamiltonian of the same work, Eqs. (40)–(44), differs from Eq. (52) by complex conjugation of the hopping on one leg. With Eqs. (40)–(44), a mobile anyon encircling a pinned anyon acquires the monodromy R_c² in fusion channel c; for Ising anyons the two channels then differ by a sign, so the holonomy is not the identity, and the same holds for Fibonacci anyons. The no-go statement therefore applies to the ladder of Eq. (52); for the ladder of Eqs. (40)–(44) it does not follow. The static result of this work, the modular data obtained from the rotation channel, is not affected, and no computed value changes. The paper text will be corrected in the next version. 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 how much of the topological fingerprint of an anyon theory a finite lattice can carry, computing a doubled Fibonacci string-net on a finite torus, where the quantum dimensions are confirmed by two structurally distinct lattice routes and the chiral sign is shown, by measurement rather than assumption, to lie beyond the conventional Wilson-loop algebra, while a point-group rotation channel does deliver it, reproduced on two lattice sizes with an error budget that needs no free parameters, together with a no-go for anyon exchange on a quasi-one-dimensional ladder that holds independently of anyon type and system size; the modular data and the theorem are not new, what is new is the lattice demonstration, the error budget and the measured limit, and within the series it is the real-space facet that checks the foundation. 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.
Files of this version are restricted (18 October 2026) because it was released under earlier licence terms: all files, including the source code, were under CC BY 4.0. All current versions use one clear licensing: paper, figures and data under CC BY 4.0, source code under the Apache License, Version 2.0. Please use the latest version: https://doi.org/10.5281/zenodo.20531124 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.