ABSTRACT We propose a framework in which physical reality is fundamentally a two-dimensional topological surface M², and quantum correlations—observed in Bell-type experiments as CHSH values above the classical bound—arise from the non-factorizability of anyonic fusion spaces built on M² rather than from nonlocal causal correlations. The structural label is the topological triple λ = (n, g, φ_T): winding number, genus, and topological phase. We show (i) that the d1b Fibonacci fusion space does not admit a tensor-product decomposition into spatially separated Alice/Bob Hilbert spaces; (ii) that on the n=8 Fibonacci fusion space of dimension D=F_7=13, using a sequential measurement protocol whose second setting contains the cross-bipartition generator σ_4, CHSH values up to |S|=3.406 are attained; and (iii) that λ carries independent operational signatures at the level of reduced channel dynamics and winding-number quantization. (A third claimed signature via surface topology is retracted in v1.6 — see Changelog below.) The framework yields a specific correction to the quantum prediction, C_TCH(a,b) = −cos θ + ε·f(n,g,θ). The dimensional hierarchy 0D → 1D → 2D → 3D_projected is derived from three premises: existence, finitude, distinguishability. TCH is categorically distinct from superdeterminism under the timeless TQFT ontology adopted throughout. VERSION 1.6 CHANGELOG Round 1 — Retraction of Predictions 1 and 6 following empirical tests (22 April 2026): Prediction 1 (detection-order autocorrelation ACF(1) ∈ [0.24, 0.59] in randomized-setting Bell experiments) was tested by reanalysis of Hensen et al. 2015 (N=245, MDE ≈ 0.18 below the predicted lower bound). Measured ACF(1) = −0.017 ± 0.062, p=0.87. Null hypothesis not rejected. Prediction retracted. Prediction 6 (genus-dependent Bell-fraction plateau as topology-specific signature) was tested on Fibonacci braiding simulations across S², T², T²\p, Σ_2, Σ_3. The Σ_2/Σ_3 plateau at 93.75% ± 0.52pp is exactly reproduced by a non-topological random-diagonal-phase control, and a Haar-random U(5) baseline yields 100% Bell-fraction. Topology-specific interpretation retracted. Coupled passages retracted: black-hole-handle enhancement bridge, ACF-as-Bohm-discriminator, "Autocorrelation as a theorem" paragraph, Abstract ACF-range sentence and Claim (iii) third signature (BH-α v2). Round 2 — Integration of Track-A (derivation check) and Track-B (structural audit) findings (25 April 2026): §III.C has been substantially sharpened without altering the numerical result |S|=3.406: Observable-edge labels corrected to (x_3, x_{n-2}=x_6) to match the calculation. For n=8 with total charge VAC, Bob's observable M_B = parity(x_6) = −I identically on all 13 basis states. The CHSH functional reduces to a one-party quantity with algebraic maximum 4; |S|=3.406 is approximately 85% of this maximum. Not a genuine Tsirelson violation in the EPR-bipartite sense. Haar-random U(13) probes without any braid structure yield |S| ∈ [3.16, 3.75] over 200 samples; the phenomenon is generic to non-factorized D=13 observables, not Fibonacci-MTC-exclusive. What remains specific is the analytical handle: only σ_4 fails to commute with M_A. Mechanism paragraph corrected: σ_4 appears in three of the four settings (A, A', B'), not only B'. §VIII.A Limitations table extended with three entries (Haar-U(13) universality; Fourier R²=0.38 at L=12; Z_2-MI universality baseline open). New scope-boundary paragraph identifies framing elements (TCH label, dimensional-emergence narrative, neutral-monistic positioning, Van-Raamsdonk references) as not entering the §III.C derivation; their status is deferred to Paper 3 (in preparation). §IV.A qualifier on Z_2-MI (no Haar-baseline yet). §IV.C context note on 91.8% Bell-fraction. §VI.C genus component of harmonic overtones removed. §I.C positive autocorrelation qualified as fixed-setting. §VIII.C Prediction 3 flagged as structurally coupled to Prediction 1. §VIII.D Protocol-dependence retitled "interpretive, not evidential". §VIII.F Conclusion list-sentence: "quantitative support" replaced by "first numerical construction". NOT AFFECTED: mathematical core §III (d1b non-factorisation, D=13 CHSH calculation, bipartition control ‖B‖=2.000 in 8000/8000), dimensional hierarchy §II, measurement barrier §VI (apart from §VI.C genus term), response to Conway-Kochen §VIII.B, Predictions 2, 2b, 2c, 4, 5, 7, 8. Retracted passages marked "[Retracted in v1.6]" in place; original v1.5 text preserved inline for transparency.
ABSTRACT We propose a framework in which physical reality is fundamentally a two-dimensional topological surface M², and quantum correlations—observed in Bell-type experiments as CHSH values above the classical bound—arise from the non-factorizability of anyonic fusion spaces built on M² rather than from nonlocal causal correlations. The structural label is the topological triple λ = (n, g, φ_T): winding number, genus, and topological phase. We show (i) that the d1b Fibonacci fusion space does not admit a tensor-product decomposition into spatially separated Alice/Bob Hilbert spaces; (ii) that on the n=8 Fibonacci fusion space of dimension D=F_7=13, using a sequential measurement protocol whose second setting contains the cross-bipartition generator σ_4, CHSH values up to |S|=3.406 are attained; and (iii) that λ carries independent operational signatures at the level of reduced channel dynamics and winding-number quantization. (A third claimed signature via surface topology is retracted in v1.6 — see Changelog below.) The framework yields a specific correction to the quantum prediction, C_TCH(a,b) = −cos θ + ε·f(n,g,θ). The dimensional hierarchy 0D → 1D → 2D → 3D_projected is derived from three premises: existence, finitude, distinguishability. TCH is categorically distinct from superdeterminism under the timeless TQFT ontology adopted throughout. VERSION 1.6 CHANGELOG Round 1 — Retraction of Predictions 1 and 6 following empirical tests (22 April 2026): Prediction 1 (detection-order autocorrelation ACF(1) ∈ [0.24, 0.59] in randomized-setting Bell experiments) was tested by reanalysis of Hensen et al. 2015 (N=245, MDE ≈ 0.18 below the predicted lower bound). Measured ACF(1) = −0.017 ± 0.062, p=0.87. Null hypothesis not rejected. Prediction retracted. Prediction 6 (genus-dependent Bell-fraction plateau as topology-specific signature) was tested on Fibonacci braiding simulations across S², T², T²\p, Σ_2, Σ_3. The Σ_2/Σ_3 plateau at 93.75% ± 0.52pp is exactly reproduced by a non-topological random-diagonal-phase control, and a Haar-random U(5) baseline yields 100% Bell-fraction. Topology-specific interpretation retracted. Coupled passages retracted: black-hole-handle enhancement bridge, ACF-as-Bohm-discriminator, "Autocorrelation as a theorem" paragraph, Abstract ACF-range sentence and Claim (iii) third signature (BH-α v2). Round 2 — Integration of Track-A (derivation check) and Track-B (structural audit) findings (25 April 2026): §III.C has been substantially sharpened without altering the numerical result |S|=3.406: Observable-edge labels corrected to (x_3, x_{n-2}=x_6) to match the calculation. For n=8 with total charge VAC, Bob's observable M_B = parity(x_6) = −I identically on all 13 basis states. The CHSH functional reduces to a one-party quantity with algebraic maximum 4; |S|=3.406 is approximately 85% of this maximum. Not a genuine Tsirelson violation in the EPR-bipartite sense. Haar-random U(13) probes without any braid structure yield |S| ∈ [3.16, 3.75] over 200 samples; the phenomenon is generic to non-factorized D=13 observables, not Fibonacci-MTC-exclusive. What remains specific is the analytical handle: only σ_4 fails to commute with M_A. Mechanism paragraph corrected: σ_4 appears in three of the four settings (A, A', B'), not only B'. §VIII.A Limitations table extended with three entries (Haar-U(13) universality; Fourier R²=0.38 at L=12; Z_2-MI universality baseline open). New scope-boundary paragraph identifies framing elements (TCH label, dimensional-emergence narrative, neutral-monistic positioning, Van-Raamsdonk references) as not entering the §III.C derivation; their status is deferred to Paper 3 (in preparation). §IV.A qualifier on Z_2-MI (no Haar-baseline yet). §IV.C context note on 91.8% Bell-fraction. §VI.C genus component of harmonic overtones removed. §I.C positive autocorrelation qualified as fixed-setting. §VIII.C Prediction 3 flagged as structurally coupled to Prediction 1. §VIII.D Protocol-dependence retitled "interpretive, not evidential". §VIII.F Conclusion list-sentence: "quantitative support" replaced by "first numerical construction". NOT AFFECTED: mathematical core §III (d1b non-factorisation, D=13 CHSH calculation, bipartition control ‖B‖=2.000 in 8000/8000), dimensional hierarchy §II, measurement barrier §VI (apart from §VI.C genus term), response to Conway-Kochen §VIII.B, Predictions 2, 2b, 2c, 4, 5, 7, 8. Retracted passages marked "[Retracted in v1.6]" in place; original v1.5 text preserved inline for transparency.
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.19601352 Version 1.4 adds an independent comparison with Xu, Ye, and You (Phys. Rev. A 112, 062232, 2025) via a unified Bell-operator-norm diagnostic (new Sec. VII.E); a condensed verification note on the five-dimensional representation under sector-phase modulation (Appendix C); first-priority and prior-art hedging in the abstract, introduction, and conclusion; a corrected companion-paper attribution of the |S| = 3.406 value to the cross-bipartition construction; finite-size framing of the referenced Z2 mutual-information result; affiliation simplified to "Independent Research, Germany"; and new references. No primary numerical result is changed. Abstract: We report a complete sequence-by-sequence numerical landscape of Bell-inequality violation across Fibonacci anyon braiding words of length L = 3 to 12 in two complementary encodings of six anyons. Earlier work by Brennen et al. established the existence of Bell-test constructions for Fibonacci anyons, and Xu, Ye, and You have independently studied three-copy CHSH correlations with maximum |S| ~ 2.633; the present contribution is the complete enumeration over all braid words of a given length class. In the two-generator d1b encoding (native four-dimensional fusion space) the optimal sequence at L = 12 reaches |S| = 2.811 (99.4% of the Tsirelson bound), with concurrence C = 0.988; in the five-dimensional encoding, Tsirelson saturation occurs already at L = 8. Bell violation is generic: a single topological mechanism (the sigma_3 cross-bipartition generator) gates whether entanglement is accessible — blocking it collapses concurrence to zero exactly. Sequential braiding produces positive lag-1 autocorrelation, a falsifiable prediction absent in standard quantum mechanics with independent measurement settings.
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.
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.21134322 We construct four spatially separated Majorana zero modes (MZMs) per party on a finite Kitaev honeycomb lattice in its non-Abelian phase, bound to well-separated Ising vortices, and use them to realize a local, noncommuting CHSH measurement algebra. Two static fusion-parity observables per party, Z_A=iγ_{A,1}γ_{A,2} and X_A=iγ_{A,2}γ_{A,3} (and analogously for Bob), commute exactly between parties (P₂=0) and fail to commute within a party (P₁=‖[Z_A,X_A]‖=2.000000); on the maximally entangled fusion-parity Bell state the CHSH expectation reaches the algebraic Tsirelson value S=2√2=2.828427. This value survives fermion-parity superselection exactly, because the Bell state is a definite-parity eigenstate with unit weight in its own sector — a statement specific to the present four-MZM, measurement-only construction, not to braid-then-fuse protocols, which are a different experiment. A commuting-only control setting returns K=√2 ≤ 2, and the finite-size Majorana footprint leaking across the bipartition decays monotonically with system size (isotropic scaling 17.0% → 10.2% → 2.3%). We stress the critical scope-limiting fact, following Howard and Vala [Phys. Rev. A 85, 022304 (2012)]: a physical Bell-inequality violation using only topologically protected Ising-anyon operations does not follow from this construction, since Ising braiding generates only the Clifford group; that step requires a non-Clifford ("magic") resource, the subject of a companion paper. What we report here is the local realization of the noncommuting Bell-operator algebra on a genuine microscopic lattice substrate, not a claim of Bell nonlocality. A loop-threading relative phase δ reproduces the analytic corollary S(δ)=2√2·cos²(δ/2) to a residual of 4.4×10⁻¹⁶, and an independent gauge-invariant flux-holonomy diagnostic confirms no adiabatic channel mixing at the quoted scale (‖[W,Z_A]‖_F=0.0132), with the sole open discrepancy being a 0.020-rad gap between the accumulated Berry phase at one full flux quantum and the target π, traced to genuine near-degeneracies along the loop rather than to a construction error.Correction notice (2026-10-23). The summary above states that the sole open discrepancy is a 0.020-rad gap between the Berry phase accumulated at one full flux quantum and the target π, “traced to genuine near-degeneracies along the loop rather than to a construction error”. This explanation is not correct. The single-mode phase summed along the loop, δ_full mod 2π = 3.1212, is not gauge invariant: the loop is closed on an eigenvector recomputed at Φ = 2π, whose phase the eigensolver fixes independently, so the quoted value is the gauge-invariant parallel-transport phase plus a closing phase that depends on the eigensolver’s start vector (here π). The gauge-invariant parallel-transport phases of the near-zero modes are −0.0204 and +0.0197 rad, all within 0.021 rad of zero; none is near π. The pre-registered single-mode closure criterion is therefore not met, and in this simulation the flux insertion does not generate δ dynamically. The “near-degeneracies” are particle–hole partner energies, equal by symmetry. The Bell-operator algebra, the gauge-invariant holonomy result and all other reported values are not affected. The paper text will be corrected in the next version. About this series: This record is part of a series of related works from my independent research on Fibonacci anyons, with Ising anyons as their natural counterpart. I started in April 2026, and it has been a long and insightful journey in which I learned a lot; the work uses different methods and stays within verifiable, nonspeculative physics. The common thread of the series is a split: Ising anyons are limited to Clifford operations, while Fibonacci anyons are computationally universal, and across the series I map what standard witnesses of nonclassicality can and cannot certify on such systems. I consider Fibonacci anyons a serious candidate for topological quantum computing, given their universality and their topological protection against local noise. A hybrid approach with Ising is conceivable, but problems such as instability and certification would have to be solved first, and each needs research of its own. This paper asks whether a real microscopic lattice can carry the Bell structure, and constructs four Majorana modes per party on a Kitaev honeycomb lattice in its non-Abelian phase, realizing the full CHSH measurement algebra locally, with observables that commute between the parties and fail to commute within a party; the algebra holds, reaching the algebraic Tsirelson value and surviving the fermion-parity superselection, but a physical Bell violation does not follow, because topologically protected Ising operations generate only Clifford gates and the missing step is a non-Clifford resource, which is the subject of a companion paper; within the series it is the substrate test, where algebra on real material is not yet a violation. 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.
Abstract We present a unified analysis showing that the dominant numerical and qualitative structure of Bell–CHSH correlations—including the ubiquitous appearance of √2, sinusoidal correlation curves, and violations of the CHSH bound—arises generically from Euclidean projection plus measurement-induced dimensional reduction. The argument is developed in three interlocking layers. A geometry-first layer uses macroscopic analogues (tuned guitar strings with threshold detectors) to make the projection mechanism inspectable: a shared relational state, local measurement axes, and a recording map that collapses continuous amplitudes to binary outcomes. A threshold-scan (“dial”) test shows that conditioned CHSH values rise smoothly above 2 as the acceptance criterion tightens. A Fourier-analytic layer demonstrates that the sign function—the operator that converts continuous amplitudes to ±1—injects an infinite series of odd harmonics into a pure fundamental signal. The classical triangle-wave correlation E(θ) = 2θ/π − 1 is the fundamental plus harmonic distortion; the quantum correlation E(θ) = −cos θ shares the same fundamental mode but without higher-harmonic content and with unit amplitude. A detection threshold acts as a conditioning map that progressively strips the distortion harmonics. At threshold τ ≈ 0.23, total harmonic distortion reaches near zero, the correlation matches −cos θ, and |S| = 2√2. A combinatorial layer introduces the Q/R/x ontological framework—where Q is the object, R the definite binary relationship, and x the genuinely ambiguous state—and connects it to the formula (n2 − n)/2. Binary measurement (n = 2) supports 1 pairwise relationship and a linear correlation. Acknowledging x as a third state (n = 3) yields 3 pairs—exactly the minimum for planar angular geometry and the cosine. These three layers converge: the gap between |S| = 2 and |S| = 2√2 is the total harmonic distortion introduced by binary quantisation of a continuous angular signal—equivalently, the information-theoretic cost of projecting a ternary relational reality onto a binary measurement space. We show that the hard-threshold model produces CHSH violations only through post-selection on coincidences; when no-detection events are properly accounted for, the model respects all Bell-type inequalities, including the Eberhard/CH inequality used in modern “loophole-free” experiments. This honestly limits the model’s scope but sharpens the open question: whether the trial-definition process upstream of the detectors (heralding, phase-matching) constitutes a form of selection beyond the hard threshold that could affect 1 full-accounting statistics. We propose the “dial test”—sweeping acceptance criteria and plotting violation strength versus acceptance fraction—as a concrete, falsifiable protocol for probing ensemble neutrality in any Bell experiment. Nothing in this work challenges the empirical validity of quantum mechanics or the logical structure of Bell’s theorem. Rather, it challenges the necessity of interpreting Bell-type correlations as evidence for an ontological division of reality into “quantum” and “classical” domains. The analysis suggests that what distinguishes the so-called quantum regime is not a change in physical law, but a change in how much of the underlying relational structure survives measurement and recording. If this reading is correct, its most consequential implication is that the correlations which motivated the inference of “action at a distance” have a local geometric origin: they are properties of a shared state established in the past, not signals transmitted in the present, and their statistical character is determined by how severely the observation compresses what was already there.
We extend the density-feedback Faddeev–Niemi Hopfion framework of Papers I–VII to quantum Bell correlations. The suppression law S(θ) = sin4θ/φ6, derived from the Hopf map Jacobian in Paper I, together with measurement-induced density feedback 1/(1 + β(ρ + δρ)), generates a local deterministic model that reproduces quantum Bell correlations. The model avoids Bell’s theorem because the hidden variable is not a classical probability distribution over independent local properties but a single shared geometric flux line, a deterministic field structure whose angular correlations emerge from the suppression law. Structurally this is the Aharonov–Bohm situation: a single global, source-fixed connection fixes the correlation through its holonomy, with no local field or signal passing between the sites. In planar laboratory geometries, the configuration of every existing loophole-free Bell test, the model achieves CHSH ≈ 2.6 for density feedback β ≈ 0.452 & β ∈ [0.1, 1.0], within 7–9% of Tsirelson’s quantum bound 2√2 ≈ 2.828. The framework-fixed coupling β* ≈ 0.452 (determined from first principles by the Bogomolny structure of Papers I–VII) gives CHSHrescaled = 2.611, sitting smoothly within the scan range and confirming the result is not fine-tuned. The scaling 2.57–2.62 of the geometric rescale factor is a genuine geometric correction, not a fit parameter. Fully isotropic 3D sampling – where the hidden flux-line direction is distributed uniformly over the sphere rather than confined to the measurement plane – yields CHSH ≈ 1.7 even with standard detectors. Standard quantum mechanics predicts CHSH = 2.828 independent of geometry; this is the model’s primary falsifiable distinction. The model makes three testable predictions: (1) reduced Bell violations in multi-satellite quantum networks probing true 3D geometries; (2) continuous CHSH variation from ≈ 1.8 to ≈ 2.6 as detector back-action is varied from weak to projective measurements; and (3) φ-periodic corrections to the exact cos4 correlation form at the few-percent level. Monte Carlo simulations (N = 100,000) confirm robust violations in planar and biased geometries with smooth degradation toward the classical regime. Python simulation code is provided for independent verification.