Licensing: paper, figures and data are under CC BY 4.0; the source code is under the Apache License, Version 2.0. The paper text names the code licence "Apache License 2.0" — this is the same licence. Mutual information between distant regions of a topologically ordered state can witness long-range nonstabilizerness; for anyon states it takes an operational form: the state is certified when the Shannon entropy of the anyon-type distribution is not an integer in some frame generated by the modular matrices S and T. For SU(2)_k these frames are Clifford operations on a single qudit of dimension 2(k+2), a discrete Fourier transform and a quadratic phase, for which Talbot-type optics has been proposed, and whose Fourier stage has been used on time bins. We show that the entropy of the time-bin distribution equals the label entropy plus one bit in every frame, so the witness is read from bin intensities. In a pre-registered calculation for k=2 and 4, a naive certification rule falsely certifies blind labels in more than half of the runs at 10^4 photons per frame; it does so in 45 to 50 % of the runs even without phase noise, and phase noise of 0.05 rad adds 12.5 to 23.2 percentage points. In the many-photon limit a protected rule keeps its margin for phase errors up to 0.14 rad (k=2) and over the whole tested range up to 0.40 rad (k=4, beyond the predicted range, so this prediction is refuted), and for uniform background counts up to 7 % and 3.5 %. The frame S, which separates blind from non-blind labels for k=2 and 4, needs no phase stage; the phase thresholds apply to rules that also evaluate frames with a phase stage. An independently written second implementation reaches the same pre-registered verdicts. This is a proposal and a calculation, not an experiment: a single qudit can be emulated with classical light, so the scheme reproduces the frame structure of the witness and does not certify magic. This record contains the manuscript, the scripts, the result file of the pre-registered run (run 1) and the console outputs of the check scripts. The README describes the files and how to re-run them. 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, 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.
Licensing: paper, figures and data are under CC BY 4.0; the source code is under the Apache License, Version 2.0. The paper text names the code licence "Apache License 2.0" — this is the same licence. Mutual information between distant regions of a topologically ordered state can witness long-range nonstabilizerness; for anyon states it takes an operational form: the state is certified when the Shannon entropy of the anyon-type distribution is not an integer in some frame generated by the modular matrices S and T. For SU(2)_k these frames are Clifford operations on a single qudit of dimension 2(k+2), a discrete Fourier transform and a quadratic phase, for which Talbot-type optics has been proposed, and whose Fourier stage has been used on time bins. We show that the entropy of the time-bin distribution equals the label entropy plus one bit in every frame, so the witness is read from bin intensities. In a pre-registered calculation for k=2 and 4, a naive certification rule falsely certifies blind labels in more than half of the runs at 10^4 photons per frame; it does so in 45 to 50 % of the runs even without phase noise, and phase noise of 0.05 rad adds 12.5 to 23.2 percentage points. In the many-photon limit a protected rule keeps its margin for phase errors up to 0.14 rad (k=2) and over the whole tested range up to 0.40 rad (k=4, beyond the predicted range, so this prediction is refuted), and for uniform background counts up to 7 % and 3.5 %. The frame S, which separates blind from non-blind labels for k=2 and 4, needs no phase stage; the phase thresholds apply to rules that also evaluate frames with a phase stage. An independently written second implementation reaches the same pre-registered verdicts. This is a proposal and a calculation, not an experiment: a single qudit can be emulated with classical light, so the scheme reproduces the frame structure of the witness and does not certify magic. This record contains the manuscript, the scripts, the result file of the pre-registered run (run 1) and the console outputs of the check scripts. The README describes the files and how to re-run them. 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, 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.
Licensing: paper, figures and data are under CC BY 4.0; the source code is under the Apache License, Version 2.0. The paper text names the code licence "Apache License 2.0" — this is the same licence. Mutual information between distant regions of a topologically ordered state can witness long-range nonstabilizerness ("long-range magic"). For anyon states on a torus it takes an operational form: the Shannon entropy of the flux (anyon-type) statistics along a loop, evaluated in every frame generated by the modular transformations S and T. The witness is one-sided: a non-integer entropy certifies magic, an integer entropy certifies nothing. We ask whether it can certify the states that were actually prepared in the trapped-ion demonstration of non-Abelian D(D_4) topological order. In exact arithmetic, under a model of the prepared states (below), we find that it cannot. Modeling the 22 ground states of that experiment as the joint eigenbasis of the abelian Wilson loops, we find all of them blind to the witness in every frame, because S and T permute them up to phases and each is uniform over two or eight anyon types, so the entropy is 1 or 3 bits throughout. We then give a protocol that leaves the blind set: one logical string operator, controlled by an ancilla and applied at a rotation angle t≠π/4. In simulation, and in an exploratory robustness analysis, at least 42.5 % of all angles give a distance of at least 0.25 bit from the nearest integer; the budget is 200 to 300 flux samples (about 300 with a conservative decision rule). Because the target states are blind, even 1 % depolarizing noise produces false certificates at a few thousand samples unless the decision rule is protected; in our model the protected rule tolerates up to 6 % in the best case. One pre-registered prediction was refuted and one proved ill-posed; both are reported as such. A short chapter derives requirements for a photonic realization with qudits of dimension 8: announced state-independent loss is harmless, and state-dependent loss shifts the tested state only at second order. Scripts, result files, and the pre-registration are deposited with this paper. This record contains the manuscript, the 13 scripts (verification, analysis, and the figure), their result files, and the pre-registration with its deviations. The README explains how to re-run the scripts and which result files are reproduced bit for bit after normalizing line endings to LF (Python on Windows writes CRLF). 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, 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.
Licensing: paper, figures and data are under CC BY 4.0; the source code is under the Apache License, Version 2.0. The paper text names the code licence "Apache License 2.0" — this is the same licence. Korbany, Ellison, Stephen and Piroli showed that a non-integer value, in some modular frame, of the Shannon entropy of the topological-sector weights of a topologically ordered ground state, which equals the mutual information between separated regions, certifies long-range nonstabilizerness. They treated the toric code and doubled Fibonacci. We ask where this witness is blind on doubled SU(2)_k string nets. For minimal-entropy states whose modular-data columns are flat, we prove that the witness is integer in every modular frame if and only if the column supports are powers of two. The proof embeds the modular representation into the Clifford/Weil representation of a qudit of dimension 2(k+2). We map all minimal-entropy states for k ≤ 32 (the flat-column characterization was registered before the computations for 17 ≤ k ≤ 32), show that the doubled-Ising state σσ̄ is bit-blind in exact arithmetic although the ground-space dimension criterion of Wei and Liu certifies strong long-range magic in a qubit embedding, and show that the blind spots depend on the encoding of the local degrees of freedom. The modular-invariant diagonal state is blind exactly when the number of anyons is a power of the local prime. We argue that it is the ground state obtained from the empty string-net configuration, and confirm this numerically on the doubled-Fibonacci honeycomb torus. This record contains the manuscript, the 16 verification scripts and their result files, the pre-registration documents (one of them redacted, see README) and the material of the cold reads described below. The README explains how to re-run the scripts and states which hash cannot be checked against the deposited copy.Notice (2026-10-__). Two sentences of the introduction will be revised in the next version; no computed value and no result changes.(1) The value H = 1.700979 of Ref. [Sayim, 5a] is described as "one of two long-range-nonstabilizerness certificates reported there". Following the correction notice of that work (2026-10-10), it is the long-range certificate reported there; the other item shows only that sampled ground-space states are not stabilizer states.(2) The sentence "the witness and its theorem are due to Ref. [Korbany et al., arXiv:2605.22424]" will also state that the idea of certifying long-range nonstabilizerness by non-integer mutual information goes back to D. A. Korbany, M. J. Gullans and L. Piroli, Phys. Rev. Lett. 135, 160404 (2025), and N. Parham, arXiv:2504.19966, as Sec. 3.1 of arXiv:2605.22424 states. Both works are already cited in the paper. 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 and, for part of the work, in the Claude AI web interface). 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, and all references are checked against the original sources. Before release, the statements and proofs of the main theorem were cold-read in one run, and two of its lemmas in a second run, each in a new Claude AI chat (Opus 5.5, high reasoning effort) given only extracted text: the first reader computed the list of blind states for k <= 16 itself, conjectured the rule of Theorem 1 and tested it at four larger k, and the readers reported imprecisions in the statements of two lemmas, which I corrected; the corrected text was not read again by a fresh reader. 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: 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.21134454 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.Correction notice (2026-10-10). In the summary above and in the section on the certificates of the paper, item (2) is described as a second long-range certificate: “a gauge-invariant minimum ground-space stabilizer Rényi entropy, ≥6.5”. This description is too strong. The values reported for item (2) are Monte Carlo estimates of the total stabilizer Rényi entropy of individual sampled ground-space vectors on small lattices; they are neither a minimum over the ground space nor a lower bound. They show that these states are not stabilizer states. A value of the total magic at one fixed system size does not certify long-range magic, which is a property of families of states of growing size, defined by robustness under shallow local circuits (F. Wei and Z.-W. Liu, arXiv:2503.04566; D. A. Korbany et al., arXiv:2605.22424; Y. Zhang et al., arXiv:2605.15150). The long-range certificate of this work is item (1); that non-Abelian string-net ground states carry long-range magic is shown by Zhang et al. In the same section, the fraction of Pauli expectation values in {0, ±1}, measured for one ground-space state, is a nonstabilizerness indicator only; it is not a diagnostic proposed by Wei and Liu. No computed value changes. The paper text will be corrected in the next version. 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.
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.19600752 We quantify the mutual information I(T;S) between topological sector labels T and CHSH values S in two-dimensional lattice models. In the Z_2 lattice gauge theory on a 16×16 torus at inverse temperature β = 2.0, we measure I = 0.015 ± 0.011 bits, significant at >2σ in 9 out of 10 independent Monte Carlo runs (corroborated by gauge-invariant plaquette-strip observables, I = 0.018 ± 0.011 bits, >3σ in 21/21 runs); the crossover decomposition attributes this β = 2.0 signal to the anyonic (local-defect) component of the sector label rather than to the Wilson-loop component. A finite-size scaling analysis indicates that this sector–CHSH mutual information decays toward zero with system size. A crossover near β ~ 2.0 shifts the dominant information source from local defect structure to global Wilson-loop sectors. In the quantum toric code via exact diagonalization, abelian Z_2 anyons produce no local CHSH correlations at system sizes L ≥ 3, consistent with the inaccessibility of abelian topological entanglement to local Pauli operators. We systematically test and rule out three classical mechanisms for Bell violation, consistent with Fine's theorem. Most significantly, within an idealized TQFT model we find that braiding of Fibonacci anyons in the fusion space produces CHSH-form violation with |S| = 2.811 (99.4% of the Tsirelson bound), and that this violation is strongly sector-dependent: with fixed measurement settings (optimized at the winner sector), |S| ranges from near zero to 2.72 (CHSH violation in only 14% of sectors), while sector-adapted settings achieve CHSH violation in 99.5% of sectors. This is consistent with topology neither creating nor destroying entanglement, but determining the measurement basis required to observe it. Keywords: CHSH inequality, topological order, mutual information, lattice gauge theory, toric code, Fibonacci anyons, Ising anyons, Bell correlations, non-Abelian braiding Version notes (v2.3, following a comprehensive internal review of the full series):• Title changed from “Topological Information in Bell Correlations: Mutual Information Between Topological Sectors and CHSH Values in 2D Lattice Models” to “Sector-Dependent CHSH Violation in Fibonacci Anyons, and Finite-Size Topological–CHSH Mutual Information in 2D Lattice Models”. (Added on 2026-10-09: the title change was made in v2.3 but was not listed in these notes.) • The gauge-invariant plaquette-strip confirmation is re-anchored: the previously documented preliminary value I = 0.035 ± 0.016 bits (5 seeds) is replaced by I = 0.018 ± 0.011 bits with σ = 14.2 ± 11.4 and 21/21 seeds > 3σ, obtained from the expanded 21-seed recompute and confirmed by a second, independently written route (both scripts and result JSONs deposited). • The information decomposition of Table II is completely recomputed: every entry now comes from a deposited validation script (crossover_decomposition_validator.py) as a seed-set mean over five seeds with per-run optimized CHSH settings, replacing values that were previously hard-coded in the figure script from an analysis that was not part of the record; the significance column now reports how many of the five seeds exceed 2σ instead of a single-run σ, and the table gains a β = 1.5 row. Comparing every entry against the previous version: the mean-anyon column moves at β = 1.0 (17.8 → 17.9), β = 2.0 (2.5 → 2.7) and β = 2.5 (0.0 → 0.3); I_full moves at β = 1.0 (0.001 → 0.006), β = 2.5 (0.057 → 0.017), and β = 3.0 (0.320 → 0.000); I_Wilson moves at β = 1.0 (0.000 → 0.001), β = 2.5 (0.055 → 0.011), and β = 3.0 (0.316 → 0.000); I_anyons moves at β = 1.0 (0.001 → 0.002) and β = 2.0 (0.004 → 0.003). The β = 2.0 value of I_full (0.011) and of I_Wilson (0.001) are unchanged in the third decimal. The most consequential changes are at β = 2.5 and β = 3.0, where the earlier single-run values are replaced by seed-set values. The β = 3.0 row is zero because at that run length the sampled CHSH value is exactly constant in all five seeds (a single distinct value with zero variance in the deposited characterization), so any mutual information with it vanishes identically; sector transitions are not absent there (see the frozen-regime entry below). A second, independently implemented Monte Carlo engine confirms the β = 2.0 structure of the decomposition. • A characterization of the frozen regime is deposited (frozen_regime_characterization.py), covering both deposited engines at β = 2.5 and β = 3.0 on 40,000-measurement runs and, for the anchor engine at β = 3.0, additionally the 4000-measurement scale of Table II. Sector transitions occur on both of these scales at β = 3.0 (5–19 per 4000-measurement run in the anchor engine, 124–133 per 40,000 in the same engine, and 214–237 per 40,000 in the second engine), and in the anchor engine they are frequent at β = 2.5 on the long runs; the conditional variance of the CHSH value given the sector label remains close to its unconditional value wherever that value is nonzero. At the 4000-measurement scale the β = 3.0 decomposition row vanishes because the sampled CHSH value is exactly constant there, not because the sector label freezes. The earlier statements that the CHSH value becomes a near-deterministic function of the frozen sector label, and that sector transitions are rare throughout β ≥ 2.5, are corrected accordingly. Two undocumented transition counts are removed from the β ≥ 2.5 caveat: the β = 3.0 count is replaced by the measured rates (the earlier figure of about 7 per run lies inside the measured 5–19 range — it was undocumented, not wrong), and the β = 2.0 comparison count is removed without replacement, as no corresponding measurement is deposited. • The β = 3.0 row of the simulation-history table is marked as a single early frozen-regime run that is not reproduced by the deposited systematic seed set, and the corresponding bar in the overview figure carries the same annotation. • Carried over, not new in this release: the finite-size scaling section (I(L) at β = 2.0 for L = 8–64, two independent Monte Carlo engines, BIC model comparison — the Z_2 mutual-information signal decays toward zero with system size and is reported as a finite-size effect) and the Related Work and Prior Art section were introduced in v2.1 and are retained in this version. (Corrected on 2026-10-09: an earlier wording of this note listed them as new since v2.2.) • Wording and framing corrections from the review pass: the Fine-bound passage is corrected (Fine's theorem requires a setting-independent hidden-variable weighting; the tested ρ(n|a,b) family is setting-dependent, so the empirical S = 2.000 cap is reported as a property of this particular family, not a general re-derivation of the Fine bound), and the abstract claim is aligned with the body ("confirming Fine's theorem empirically" → "consistent with Fine's theorem"). The uncertainty of the exponential-fit asymptote is corrected by a factor of ten (SEM reporting). Numerical values that changed in this release are listed explicitly in the entries above. • Companion-work title updated to match Paper 2's ratified retitling. Non-Abelian capitalization corrected to the series-wide form throughout. A hardcoded table reference is corrected to the actual table carrying the cited value. • Self-cite hardening: the three load-bearing companion citations point to the frozen Version-DOI of the exact cited version (10.5281/zenodo.21282566, 10.5281/zenodo.21327673, 10.5281/zenodo.21327799), and the companion Bell-algebra paper is now cited with its final title and Concept-DOI (10.5281/zenodo.21134322). • Deposit hardening from the series review: a deposited second-route script is repaired (one-line indentation error) and its comparison target updated to the v2.3 values; auxiliary script documentation is standardized to English; one deposited script and its result file are renamed, so their file names differ from v2.2. All headline values re-verified by rerunning the deposited reproduction scripts (bit-identical / statistically identical); the frozen-regime characterization script was extended in this release and is covered by its own entry above. In a final usability pass the deposit was exercised as a reader would: one script's output path is now anchored on its own location instead of the working directory, outdated header comments in two scripts are corrected (an obsolete paper title and version, a reference to a file that is not part of the record, a stale table number, and a byte-identity claim about a file outside the record), and the README now lists the third-party packages and the interpreter versions the release was checked with. No computed value is affected. • Clarifications added in this release: the detector reference angles θ_A, θ_B of the classical measurement model are now defined in the text; the sector-shuffle-corrected I(T;S) is defined explicitly (permutation-null-subtracted signal); and a caveat marks the Fibonacci sweep parameter δ as a representation-deformation parameter rather than a physical anyon parameter (verified in the companion landscape analysis, Appendix C). The simulation-parameter appendix now states the thermalization and measurement counts per deposited code path, replacing a single figure that combined two of them. • Three rows of the Fibonacci table (L = 4, 6, 7) are aligned with the cited companion version: the companion re-anchored its sector-phase optimisation from a 25-point to a 50-point δ grid, which raises those optima in the third decimal (|S| 2.224 → 2.225, 2.394 → 2.398, 2.535 → 2.536, with the corresponding concurrence and %
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.
Here's what this paper means for actual humans: WHAT THIS IS ABOUT. When you build or use a quantum device, everything rests on knowing what its parts are: which degrees of freedom are "qubit 1," which are "qubit 2." The companion paper ("Forged Frames") showed that simple mathematical criteria cannot certify this; convincing fakes exist. This paper answers the question that result leaves standing: what is the MINIMUM experimental information that genuinely pins down a system's subsystem structure, and when can that information silently fail? WHAT WE FOUND, IN PLAIN TERMS. Four things. (1) EXACT LAWS INSTEAD OF MEASUREMENTS. Any wrong choice of parts must visibly distort the family of unentangled states, and we derive exactly how much, at every system size: closed-form constants (verified to seven decimal places), an explicit error bound, and a surprise. The two natural ways to measure "wrongness" (state distortion versus operator scrambling) turn out to be literally the same measurement, up to a fixed number, whenever the two parts have equal size. Wrongness cannot hide. (2) THE TWO-OPERATION THRESHOLD. You can pin down what your subsystems are by implementing just TWO generic operations and checking their statistics, while a THOUSAND mutually commuting operations tell you nothing at all. Diversity of control, not volume of control, is what fixes what your qubits are. The residual uncertainty is proportional to your measurement tolerance, with a constant we derive exactly and show how to compute in advance, so an experimenter can screen candidate operation pairs and pick a strong one (random pairs are typically several times worse, occasionally catastrophically so). (3) THE ALIASING LIMIT: CERTIFICATION CAN PASS WHILE TELLING YOU NOTHING. If you sample your certification data at special "resonant" times (set by the operations' energy scales), a maximally wrong assignment of subsystems passes the test perfectly. What your qubits are depends on when you looked. We characterize these blind spots exactly (for qubits they are universal and computable in advance; the danger windows shrink with the square root of your tolerance), and the fix costs almost nothing: sample at two different times, at least one away from the resonant set. With that, certification is restored; we prove it exactly and measure it. (4) REAL-LAB BUDGETS. We work out what certification costs in actual measurement shots (the error falls off unusually fast, inversely with the total number of shots rather than with its square root), how hardware noise floors the procedure, and how a simple reference measurement removes that floor to first order. A single seven-row table in the paper is effectively a lab protocol: it shows, condition by condition, which kinds of experimental access identify the subsystem structure (two generic operations: yes; one operation, commuting operations, resonant sampling: no). WHY IT MATTERS. For quantum hardware: this is a concrete recipe for certifying that the qubits you think you have are the qubits you physically have, with explicit tolerances, a sampling-time warning that is a real failure mode on calibrated hardware, and shot budgets. For foundations: the contested question "does physics determine its own subsystems?" gets a quantitative operational answer. Physics determines an operational subsystem CLASS (not one preferred coordinate representation of it), and that class is fixed by two operations sampled at two times, and by nothing less that we tested. WHAT IT OPENS UP. The seven-row identifiability table is a blueprint for a tabletop experiment on any small quantum platform; running it on hardware would move this from simulation to laboratory fact. On the theory side the paper leaves sharply posed open problems: the exact global constants when the two parts differ in size, the number-theoretic structure of long-time sampling (the failure times form a small-divisor hierarchy), and the extension from two parts to many. METHODS NOTE, BECAUSE IT MATTERS FOR TRUST. Every quantitative claim was pre-registered in the companion scripts before its run. The full ledger, including predictions that failed and one published-in-note constant that our own later theory overturned, is an appendix of the paper. All code, data, and working notes are in the deposit, seeded for exact reproduction. WHO AND HOW. A human-AI collaboration: research direction and standards of evidence by Ryan Cardwell (Crystalline Labs); design, code, derivations, and drafting by Claude (Anthropic), including autonomous sessions under the program's pre-registration rules, with adversarial reviews by other AI models incorporated and credited.