A Kohlrausch or Weibull Exponent Below One Admits a Classical Mixture: Pollard's Criterion as a Licence Condition, Two Predictions Refutable Against Existing Literature, and a Self-Audit of the Framework That Produced Them
Consolidated edition. This version merges the three previous papers of the programme into a single document, reordered by logical dependency rather than chronology so that the surviving core is separable from the speculation built on it. 63 pages, 26 sections in 7 parts, 8 appendices. Eloy René Becerra Daly What is established, in decreasing order of confidence [EXACT] Identities. The Λ-hierarchy reduces to τ = ℏ/E and ℓ = the reduced Compton wavelength. The accessible information satisfies IQ = (γ/τ)β exactly. A distinguished point at D/L = 1/2 carries four coincident properties — exactly one bit, the entanglement sudden-death point for C0 = 1, the maximum of D·C̄ at L²/4, and a Bures arc of π/3 — all invariant in β, τ and L. [OBS] Verification. D + C = 1 holds to machine precision across 650 circuits on two IBM backends and ~5.5×106 shots. [THM] Limits. A no-go theorem shows the temporal exponent of dephasing is independent of the pulse-sequence filter, which removes the derivation previously claimed for Axiom 3 and relabels it as an empirical postulate. [THM] The transferable result. Pollard's criterion used as a licence condition: β ≤ 1 certifies that the curve admits a positive rate density, β > 1 proves that none exists. In reliability engineering an infant-mortality exponent licenses reading the batch as heterogeneous, while a wear-out exponent forbids it under any partition. Six transfers are registered, each with the observation that would retire it. The inference this work does not license A fitted β < 1 is not evidence of quantum behaviour. The theorem establishes representability of the curve, not identification of the mechanism — the map from mechanism to curve is many-to-one — so what follows is weaker and sufficient: the data cannot be adduced as evidence of coherence, because a classical account of them exists. The decisive test, which is not performed here On a single sample, the anomalous-diffusion exponent and the stretched-relaxation exponent must satisfy β = (1 + αdiff)/2, with β ≥ 1/2 as a corollary. Both quantities are already tabulated in the NMR relaxometry and diffusion-imaging literature for overlapping sample sets. No new experiment is required to attempt this refutation. Errata to previous versions The critical qubit number is n* = 1.848, not 2.6. The LIGO ringdown significance, with the internal error scaled by √(χ²/ν) = 1.669 as the catalogue's own goodness of fit requires, is 12.5σ; we record that our first correction of it, 20.9σ, was itself wrong for neglecting that scaling, and that the originally published 13.8σ was closer to the defensible value than our correction. A reported 18 kHz emergence mode lies below the Rayleigh limit of its own data. The published comoving density of remnants corresponds to Ω = 0.063, not the 0.12 it is stated to reproduce. Endpoint, labelled CONJ throughout The programme closes with a dark-matter candidate: Planck-scale black-hole remnants at Mmin = mP/(2√π) = 6.140 µg, an exact identity rather than the approximation 0.28 mP under which it was published. Evaluating it shows the usual dismissal to be right for the wrong reason: the remnant carries 148.6 J at galactic velocity and one transit is expected per 0.86 km²·yr, so rarity is not the obstacle; but the mean free path in rock is 1023 light years and the probability of one interaction while crossing the Earth is 10-32. The candidate is cross-section-limited, not exposure-limited, and the two demand opposite experiments. Four falsifiers are given, two requiring no new instrument. Reproducibility Every numerical claim is recomputed from the equations by a dependency-free implementation: 354 regression tests and 37 published-number checks, over a registry of 138 equations each carrying source, evidence level and declared domain. Where a published number could not be reproduced it is recorded as an erratum rather than adjusted; where a correction of ours proved wrong it is recorded as such. The register documents 27 inconsistencies, of which 11 remain open, each with the observation that would close it.
Consolidated edition. This version merges the three previous papers of the programme into a single document, reordered by logical dependency rather than chronology so that the surviving core is separable from the speculation built on it. 63 pages, 26 sections in 7 parts, 8 appendices. Eloy René Becerra Daly What is established, in decreasing order of confidence [EXACT] Identities. The Λ-hierarchy reduces to τ = ℏ/E and ℓ = the reduced Compton wavelength. The accessible information satisfies IQ = (γ/τ)β exactly. A distinguished point at D/L = 1/2 carries four coincident properties — exactly one bit, the entanglement sudden-death point for C0 = 1, the maximum of D·C̄ at L²/4, and a Bures arc of π/3 — all invariant in β, τ and L. [OBS] Verification. D + C = 1 holds to machine precision across 650 circuits on two IBM backends and ~5.5×106 shots. [THM] Limits. A no-go theorem shows the temporal exponent of dephasing is independent of the pulse-sequence filter, which removes the derivation previously claimed for Axiom 3 and relabels it as an empirical postulate. [THM] The transferable result. Pollard's criterion used as a licence condition: β ≤ 1 certifies that the curve admits a positive rate density, β > 1 proves that none exists. In reliability engineering an infant-mortality exponent licenses reading the batch as heterogeneous, while a wear-out exponent forbids it under any partition. Six transfers are registered, each with the observation that would retire it. The inference this work does not license A fitted β < 1 is not evidence of quantum behaviour. The theorem establishes representability of the curve, not identification of the mechanism — the map from mechanism to curve is many-to-one — so what follows is weaker and sufficient: the data cannot be adduced as evidence of coherence, because a classical account of them exists. The decisive test, which is not performed here On a single sample, the anomalous-diffusion exponent and the stretched-relaxation exponent must satisfy β = (1 + αdiff)/2, with β ≥ 1/2 as a corollary. Both quantities are already tabulated in the NMR relaxometry and diffusion-imaging literature for overlapping sample sets. No new experiment is required to attempt this refutation. Errata to previous versions The critical qubit number is n* = 1.848, not 2.6. The LIGO ringdown significance, with the internal error scaled by √(χ²/ν) = 1.669 as the catalogue's own goodness of fit requires, is 12.5σ; we record that our first correction of it, 20.9σ, was itself wrong for neglecting that scaling, and that the originally published 13.8σ was closer to the defensible value than our correction. A reported 18 kHz emergence mode lies below the Rayleigh limit of its own data. The published comoving density of remnants corresponds to Ω = 0.063, not the 0.12 it is stated to reproduce. Endpoint, labelled CONJ throughout The programme closes with a dark-matter candidate: Planck-scale black-hole remnants at Mmin = mP/(2√π) = 6.140 µg, an exact identity rather than the approximation 0.28 mP under which it was published. Evaluating it shows the usual dismissal to be right for the wrong reason: the remnant carries 148.6 J at galactic velocity and one transit is expected per 0.86 km²·yr, so rarity is not the obstacle; but the mean free path in rock is 1023 light years and the probability of one interaction while crossing the Earth is 10-32. The candidate is cross-section-limited, not exposure-limited, and the two demand opposite experiments. Four falsifiers are given, two requiring no new instrument. Reproducibility Every numerical claim is recomputed from the equations by a dependency-free implementation: 354 regression tests and 37 published-number checks, over a registry of 138 equations each carrying source, evidence level and declared domain. Where a published number could not be reproduced it is recorded as an erratum rather than adjusted; where a correction of ours proved wrong it is recorded as such. The register documents 27 inconsistencies, of which 11 remain open, each with the observation that would close it.
CORRECTION NOTICE (v5, 2026-09-06). The central result of v1-v4 is WITHDRAWN. This record has not been peer reviewed. Issued 2026-09-06 by the author (v5 of this record). This notice was added to the archived record; the manuscript itself is unaltered. This record has not been peer reviewed. What is wrong The force exponent reported here as alpha = 3.92 +/- 0.16 is an artifact of the fitting procedure, not a property of the data. The fit was unweighted least squares in linear space applied to a steeply falling profile, which places 95.8 percent of its objective on the single point at the shortest retained separation. Re-fitting the same data with nothing discarded, changing only the error model, gives 2.82 (relative-error weighting) or 2.89 (log-space regression). Across fifteen combinations of standard estimator and fit window the exponent spans 2.53 to 3.92, and the value published here is the largest of them. What the data show instead, and how firmly The retraction is certain. The replacement is not. On the open 24-site chain the same data give alpha = 2.6 +/- 0.2, where the interval is the spread of a plateau in the local logarithmic slope over six consecutive separations, not a statistical error (the bootstrap error on the published fit to the same data is 0.63); the plateau value was reproduced at 2.5921 by a separately written implementation run on the same recorded state (SIM-2026-073). This is a single geometry at a single size. On a 24-site periodic ring, analysed with chord distance as pre-registered, two probes that are both energy-density-like (Delta = 1) give 1.83 and 2.37 against a CFT two-point value of 2 (SIM-2026-074): one size, not converged, disagreeing with each other by 29 percent, and not attributable to a single operator channel. No corrected exponent is established. The exchange channel is an open question. Treat 2.6 +/- 0.2 as the provisional open-chain value, not as a result. Also withdrawn or voided alpha = 3.92 +/- 0.16 (24 sites) and alpha = 3.97 +/- 0.15 (16 sites, exact) from stress-tensor exchange. Was: headline result and its first corroboration. Now: withdrawn. Why: the same estimator artifact: both move in lockstep under it (3.92 to 2.56, 3.97 to 2.75 on dropping the contact point), because they share one observable, one fit window and one fitting routine. scalar exchange (alpha = 2) excluded. Was: stated as excluded at about 12 sigma. Now: withdrawn. Why: the 12 sigma used the covariance error 0.16; at the bootstrap error 0.63 computed before release the exclusion is about 2.2 sigma, and under any window starting at separation 3 the point estimate itself falls to about 2.56. alpha = 4 is universal; alpha = 4.00 at Ising, tricritical Ising and Potts. Was: confirmed at three classes. Now: withdrawn; the operator towers at all three classes stand. Why: the exponent was measured only at Ising and inherited by the other two; the towers are spectra, not forces. third independent evaluation alpha = 4.07 (chi = 3 MERA), and with it the statement that this was the first holographic force measurement from an optimized tensor network state. Was: supporting evidence and a priority claim. Now: both withdrawn. Why: the profile changes sign inside its own fit window and cannot be a fixed-sign power law; the priority claim rests on that evaluation alone and was never gated against prior work. the discriminating test: stress tensor (Z_j Z_j+1) versus spin primary (X_j). Was: registered killer test. Now: void. Why: at the critical point these are one operator, related by exact Kramers-Wannier self-duality. Measured on the periodic ring: identical expectation values to 1e-15 and identical force profiles to 1e-13 at 14 sites, 2.4e-9 at 24 sites (SIM-2026-074). Open boundaries break the duality, which is why the open-chain run of this test (SIM-2026-073) showed a spurious difference between the two arms. The test compared the energy density to itself. Neither label was correct: Z_j Z_j+1 is the energy density, not the stress tensor; X_j is its dual, not the spin primary. transport time T = (pi/2) l_AdS to 1.2 percent as a geometric prediction. Was: confirmed. Now: measurement stands (T = 2.239 at N = 16); geometric reading not established. Why: l_AdS cancels in the cited derivation, which also gives pi rather than the pi/2 the manuscript compares against; the chi = 3 estimate of l_AdS gives a 57 percent miss; the transport driver never applies a MERA. The true spin primary Z_j was never in the design, and the direct replacement test is blocked: Z_j is odd under the Z2 symmetry, so the energy-difference observable used throughout this work subtracts across symmetry sectors and is undefined for it. A replacement test using two Z2-even probes has been pre-registered (PREREGISTRATION-T.md, committed before its driver) and run; its result is deliberately not part of this correction and will appear in a later version. Not previously reported The published fit has a reduced chi-squared of 6.008 on 9 degrees of freedom, and a bootstrap standard deviation of 0.63 rather than the quoted 0.16. Both were computed before release and omitted. What is unaffected Of the claims in this record: the two-phase optimization method, the chi = 3 criticality threshold, the exact-diagonalization gates, the measured OPE coefficient C_sigma-sigma-epsilon = 0.50 +/- 0.03 against the exact 1/2, and the conformal towers at all three classes. Not part of this record, mentioned only because they are sometimes read together with it: the later C_TTT stress-tensor self-coupling measurements (SIM-2026-067/068/069, dated after this record) and the matched off-critical ring control (SIM-2026-074) are unaffected by this correction and make no claim about the force exponent. How the error was found and what this record contains The error was found by the author's own internal adversarial audit, not by a reviewer; the journal submission it was found after is closed, and no review of this work has taken place. Files in this record: the full re-analysis (SIM-2026-072.json, driver, figure), the pre-registered chain test (SIM-2026-073.json) and ring test (SIM-2026-074.json) with both pre-registrations as committed before the runs, the runnable drivers with their shared module and the negative test suite for their validation gates, and the error register (ERRATA.md). Concept DOI 10.5281/zenodo.19660302 always resolves to the current version. The abstract of the original manuscript follows unchanged, for the record. Every claim in it about the force exponent should be read against the correction above. We report results from classical simulation of critical spin chains that establish quantitative physical content in the MERA-AdS correspondence. First, the interaction energy between boundary-primary excitations follows a power law F(dr) ~ dr^(-alpha) with three consistent evaluations yielding alpha = 3.92 +/- 0.16 (24q), 3.97 +/- 0.15 (16q), and 4.07 +/- 0.18 (16q chi=3 MERA), all consistent with stress-tensor exchange (alpha = 2*Delta_T = 4) and inconsistent with scalar exchange (alpha = 2); the exponent is explained by the Ising CFT fusion rule eps x eps = [1]. Because Delta_T = 2 is fixed by conformal symmetry in every unitary 2D CFT, alpha = 4 is universal: we confirm this at two further fixed points - the tricritical Ising model (c = 7/10), where a scale-invariant MERA recovers the full M(4,5) primary spectrum (Delta_sigma = 0.092, Delta_eps = 0.204, Delta_T = 1.999), and the three-state Potts model (c = 4/5), where a Z3-symmetric MERA at chi = 12 recovers Delta_sigma = 0.134, Delta_eps = 0.805, Delta_T = 2.001 - giving alpha = 4.00 at all three universality classes. Second, the holographic transport time matches the AdS_3 null geodesic prediction T = (pi/2) l_AdS to 1.2% at N = 16 sites on a time-grid-converged measurement, with the residual shrinking from 2.8% at N = 12. Third, we introduce a two-phase optimization method (ALM + Z2 projection) for Z2-symmetric MERA ground states at 24 qubits. Fourth, extending the MERA bond dimension from chi=2 to chi=3 overcomes the expressibility barrier, achieving 0.15% energy error at 16 qubits and enabling the first holographic force measurement from an optimized tensor network state. The OPE coefficient is also measured directly from the fixed-point tensors: C_sigma-sigma-eps = 0.50 +/- 0.03 against the exact Ising value 1/2. A hardware path is designed for Quantinuum Helios (57 CNOTs, 0.57% logical error). Version 4 changes (relative to v3): Transport: N=16 residual restated as the time-grid-converged value of -1.21% (dt = 1e-3), with -2.78% at N=12 showing the finite-size trend; the 0.72% figure in v3 was an artifact of the dt = 0.05 integration grid and is now labeled as such. The N=24 transport point carries a new reliability caveat (double-lobe trace structure). The l_AdS comparison table is recomputed against the converged transport time (best fit 1.425, deviation 1.21%). Effective-mass derivation rewritten in a consistent AdS3 convention m^2 l^2 = Delta(Delta-2); m_eff = 0.693 = 1/l_AdS follows from Delta = 1 exactly. NEW: universality section extended to a third fixed point - a Z3-symmetric MERA at chi = 12 recovers the three-state Potts (c = 4/5) tower, giving alpha = 4.00 at all three universality classes (c = 1/2, 7/10, 4/5). NEW: the OPE coefficient C_sigma-sigma-eps = 0.50 +/- 0.03 measured directly from the fixed-point tensors, with C_sigma-sigma-sigma = 0 at machine precision as the Z2 structural check. Supplemental table row SIM-007 annotated (record lost; script preserved); data availability expanded; Zenodo DOI added to the front matter.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. NOTICE (2026-08-04). This record uses a magic measure that has been shown to be incorrectly signed, and it should not be cited for quantitative magic claims. The "total variation" quantity TV = sum |W(u)| over the discrete Wigner function was used with the condition TV < 1 characterising non-stabiliser states. The inequality runs the other way. For any quasi-probability distribution summing to 1, the l1 norm satisfies sum |W| >= 1 always, with equality exactly when W is non-negative. Direct computation: the stabiliser state |+> gives TV = 1.0000 and the magic state T|+> gives TV = 1.2071. The condition "TV < 1" therefore describes the empty set, and any tier defined by it has no members. Separately, TV as defined is exp(mana) -- a monotone reparameterisation of the established Wigner-negativity monotone of Veitch, Mari, Ferrie and Emerson -- and was never compared against it. Readers should consult Veitch et al. on mana and Wigner negativity, Howard and Campbell on robustness of magic, Bravyi et al. on stabiliser extent, and Leone, Oliviero and Hamma on stabiliser Renyi entropy. NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. We recast the three foundational quantum information protocols — quantum teleportation, superdense coding, and the CHSH Bell test — in the language of the Origami Instruction Set Architecture (ISA), and then lift each to its TriQ (3-qubit GHZ) and 6-731 (broken-Fano-line) extensions. In the ISA reading, teleportation is SPLIT;SPLAT with a classical wire, superdense coding is FLIP;FLOP on a Bell pair encoding log2∣G∣ bits via the group G=Z22, and the CHSH violation is a hybrid H¹/H² computation: the Bell pair is a stabiliser (H¹) resource but the Tsirelson-optimal measurement applies a T-gate rotation (H², TWIST=1). We prove that the Tsirelson bound cos2(π/8) equals the total variation of any dark-magic (Tier 1) state in the ISA magic-type classification — the Tsirelson bound is the Wigner mass of the dark-magic sector. Lifting to the TriQ register introduces the ORBIT opcode, raises superdense coding capacity to 3 bits per qubit (log2∣Z23∣), and replaces the CHSH game with the Fano Line Verification Game whose win conditions are the ±1 values of the 6j symbol of Z23. The 6-731 extension (one Fano line weakened to coupling η) makes teleportation directed, reduces superdense capacity to 3−H(η) bits per qubit, and interpolates the Fano game win probability continuously between classical and quantum bounds. Remarkably, the same asymmetry parameter η is independently measurable by three methods — thermodynamic Carnot efficiency (Paper 325), information-theoretic channel capacity, and game-theoretic win probability — whose agreement constitutes a Pentagon-identity self-verification test for the hardware. We close with the Hk Bell tower: for each cohomological depth k there is a Bell-type inequality whose maximum quantum violation requires an Hk resource, providing a game-theoretic ladder that extends beyond CHSH through the Fano game (H²) to a conjectured G2-correlation game (H³) accessible only to the 731-ISA SPIN opcode.Appendix B provides side-by-side quantum circuit and ISA spider diagrams for all three baseline protocols. Keywords quantum teleportation, superdense coding, CHSH inequality, Bell test, Tsirelson bound, Origami ISA, H^k cohomology, stabiliser states, magic states, Fano plane, GHZ state, TriQ register, 6-731 architecture, ORBIT opcode, TWIST opcode, topological heat engine, dark magic, Wigner mass, Bell hierarchy, Mermin inequality, Pentagon equation, quantum nonlocality, quantum channel capacity, G2 correlations
Three recent experimental studies of D-Wave annealers each report an unexplained effective-temperature phenomenon: fitted inverse temperatures whose dependence on annealing time is left empirical, with sampling quality degrading at a critical frustration point [Pelofske, PRR 8, 033280 (2026)]; a coupling-independent temperature offset t-bar in [0.2,0.4] that violates the assumed T_eff ∝ 1/J scaling [Grattan et al., arXiv:2512.03162]; and hardware inverse temperatures a "consistent" factor of 5–7 above closed-system theory, attributed to uncontrollable hardware effects [Kim et al., PRE 113, 035302 (2026)]. We show that a single account, built from the diabatic effective-temperature relation of Gyhm et al. evaluated on the vendor's published fast-anneal schedules and one thermal bath, organizes all three. First, on real schedules the coherent inverse temperature is nearly linear in annealing time — not ∝ sqrt(t_a) as for linearly vanishing drivers — because the transverse field decays exponentially and dies by s ≈ 0.6; exact statevector simulation confirms the first-order integral to 0.4–4.2% in its validity window. Second, a units identity, beta_cyclic(t_a) = beta_correct(t_a/2pi), resolves the factor of 5–7: the published theoretical reference curves in this lane are quantitatively consistent with schedule data evaluated in cyclic rather than angular frequency units. The foundational paper's own schedule evaluations carry the rescale: its earlier single evaluation, beta ≈ 0.54 at 5 ns, equals our correct-convention value at t_a/2pi (0.539), and all 24 of its current tabulated beta and c_2 values that we can recompute from vendor schedule tables are reproduced at t_a/2pi to within 0.11%; a free-rescale fit to Kim et al.'s published simulation curve gives r = 6.40, with 2pi = 6.283 inside the fit's shallow region; what remains after the correction is consistent with open-system cooling. Third, the coupling-independent offset is a residual-excitation floor read through a domain-wall thermometer, t-bar ≈ 2/ln(1/rho): the implied kink densities (9e-5 to 4e-3) overlap published kink measurements, the excitation-floor form beats the constant-offset (noise) model on AIC for all four reported machines, and the logarithm compresses two and a half decades of rho into the narrow observed band, explaining its apparent universality. In the coherent nanosecond regime the closed-system law is an upper envelope on hardware inverse temperature; the shortfall is a candidate in-situ decoherence diagnostic, machine-resolved, and at the one matched cell where both machines are coherent the measured cross-machine ratio (2.63) agrees with the parameter-free prediction at the few-percent level. All hypotheses, tolerances, and kill criteria were registered before data extraction; failed and undecidable gates are reported alongside the successes. No quantum-advantage claim is made. AI use and validation: This work was produced in close collaboration with AI research assistants, directed by the author, who set the research direction, the standards of evidence, and all decisions about what is claimed and published. Anthropic's Claude (Fable 5) performed the analysis, simulation, extraction, and drafting; xAI's Grok (via Cursor) provided independent pre-publication review; and Anthropic's Claude Opus 5.5 performed a post-publication adversarial audit of the manuscript against its pinned evidence, with findings and dispositions logged in the project ledger and incorporated in subsequent versions. The registered-verdict protocol is designed so that no result rests on trusting a model's output: hypotheses, tolerances, and kill criteria were committed to the repository before any data was extracted; failed and undecidable gates are reported in the paper; every numeric claim traces to pinned, machine-readable extraction files and analysis code included in this deposit; figure data was extracted from the source PDFs' vector drawing operations rather than read by a model; and verbatim quotes were verified against raw page dumps.
Stabilisation exponentielle par rétroaction de systèmes quantiques ouverts soumis à des mesures en temps continu Dans cette thèse, nous nous intéressons à la stabilisation par rétroaction des systèmes quantiques ouverts soumis à des mesures imparfaites en temps continu. Tout d'abord, nous introduisons la théorie du filtrage quantique pour décrire l'évolution temporelle de l'opérateur de densité conditionnelle représentant un état quantique en interaction avec un environnement. Ceci est décrit par une équation différentielle stochastique à valeurs matricielles. Deuxièmement, nous étudions le comportement asymptotique des trajectoires quantiques associées à des systèmes de spin à N niveaux pour des états initiaux donnés, pour les cas avec et sans loi de rétroaction. Dans le cas sans loi de rétroaction, nous montrons la propriété de réduction de l'état quantique à vitesse exponentielle. Ensuite, nous fournissons des conditions suffisantes sur la loi de contrôle assurant une convergence presque sûre vers un état pur prédéterminé correspondant à un vecteur propre de l'opérateur de mesure. Troisièmement, nous étudions le comportement asymptotique des trajectoires de systèmes ouverts à plusieurs qubits pour des états initiaux donnés. Dans le cas sans loi de rétroaction, nous montrons la réduction exponentielle de l'état quantique pour les systèmes N-qubit avec deux canaux quantiques. Dans le cas particulier des systèmes à deux qubits, nous donnons des conditions suffisantes sur la loi de contrôle assurant la convergence asymptotique vers un état cible de Bell avec un canal quantique, et la convergence exponentielle presque sûre vers un état cible de Bell avec deux canaux quantiques. Ensuite, nous étudions le comportement asymptotique des trajectoires des systèmes quantiques ouverts de spin-1/2 avec les états initiaux inconnus soumis à des mesures imparfaites en temps continu, et nous fournissons des conditions suffisantes au contrôleur pour garantir la convergence de l'état estimé vers l'état quantique réel lorsque le temps tend vers l'infini. En conclusion, nous discutons de manière heuristique du problème de stabilisation exponentielle des systèmes de spin à N niveaux avec les états initiaux inconnus et nous proposons des lois de rétroaction candidates afin de stabiliser le système de manière exponentielle.
Préface de la première éditionLa naissance de la physique quantique date d'un siècle et cette description des phénomènes physiques, qui a transformé notre vision du monde, n'est toujours pas remise en cause, ce qui est exceptionnel pour une théorie scientifique.Ses prédictions ont toujours été vérifiées par l'expérience avec une précision impressionnante.Les concepts fondamentaux, comme les amplitudes de probabilité, les superpositions linéaires d'états, qui semblent si étranges pour notre intuition quand on les rencontre pour la première fois, restent toujours essentiels.Une évolution importante s'est toutefois manifestée au cours des dernières décennies.Les progrès spectaculaires des techniques d'observation, des méthodes de manipulation des atomes, permettent maintenant de réaliser des expériences si délicates qu'elle n'étaient considérées que comme des "expériences de pensée" par les pères fondateurs de la mécanique quantique.L'existence des corrélations quantiques "non séparables", qui est à la base du "paradoxe" de Einstein-Podolsky-Rosen et qui violent les fameuses inégalités de Bell, a pu être confirmée expérimentalement avec une grande précision.Les états "intriqués" de deux systèmes, qui manifestent de telles corrélations quantiques, sont mieux compris, et même utilisés pour des applications concrètes, comme la cryptographie quantique.L'intrication d'un appareil de mesure avec son environnement se révèle une piste intéressante pour une meilleure compréhension du processus de mesure.Parallèlement à ces progrès conceptuels, on assiste également à une invasion de notre monde quotidien par des dispositifs dont le principe de fonctionnement repose sur des phénomènes quantiques.Les sources laser qui sont utilisées pour la lecture des disques compacts, l'ophtalmologie ou les télécommunications optiques, sont basées sur l'amplification de la lumière par des systèmes atomiques dont les populations sont inversées.La résonance magnétique des noyaux des atomes est couramment utilisée dans les hôpitaux pour prendre des images de plus en plus précises des organes du corps humain.Des millions de transistors sont inclus dans les puces qui permettent à nos ordinateurs d'effectuer des opérations à des vitesses prodigieuses.
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 %