"This record is a duplicate. The updated and complete version available at https://doi.org/10.5281/zenodo.19842968" Within the Relational-Informational Model (RIM), where gauge fields emerge from the Uhlmann curvature of the observer’s reduced density matrix ρObs, I show that the number of fermion generations is determined by the topology of the degeneracy locus—the set of spacetime points where eigenvalues of ρObs coincide. By the von Neumann–Wigner non-crossing theorem, pairwise eigenvalue de- generacy in a Hermitian family requires three independent real conditions (codi- mension 3). In four-dimensional spacetime, each degeneracy condition therefore defines a one-dimensional submanifold—a degeneracy string. For the SU(3) sector of RIM, the spectrum of ρObs contains three eigenvalues {p1,p2,p3}, admitting ex- actly 3 2 = 3 independent pairwise degeneracies. Consequently, three topologically distinct degeneracy strings exist, each carrying a distinct sector of the Uhlmann holonomy. I identify these three sectors with the three fermion generations.An explicit qutrit toy model demonstrates that a closed loop in parameter space encircling all three degeneracy points produces a Z3 monodromy—a cyclic permuta- tion of eigenvalues—whose braid structure provides an information-geometric origin for the CKM mixing matrix and CP violation. The SU(2) sector contributes no additional generations (since 2 2 = 1) but instead generates the isospin doublet structure within each generation. This result constitutes a sharp, falsifiable prediction: the number of fermion generations is exactly three, as a topological consequence of pairwise degeneracies in a three-level quantum system embedded in four-dimensional spacetime. The prediction is consistent with precision electroweak data (Nν = 2.984 ±0.008 from LEP) and current LHC bounds on fourth-generation quarks.
"This record is a duplicate. The updated and complete version available at https://doi.org/10.5281/zenodo.19842968" Within the Relational-Informational Model (RIM), where gauge fields emerge from the Uhlmann curvature of the observer’s reduced density matrix ρObs, I show that the number of fermion generations is determined by the topology of the degeneracy locus—the set of spacetime points where eigenvalues of ρObs coincide. By the von Neumann–Wigner non-crossing theorem, pairwise eigenvalue de- generacy in a Hermitian family requires three independent real conditions (codi- mension 3). In four-dimensional spacetime, each degeneracy condition therefore defines a one-dimensional submanifold—a degeneracy string. For the SU(3) sector of RIM, the spectrum of ρObs contains three eigenvalues {p1,p2,p3}, admitting ex- actly 3 2 = 3 independent pairwise degeneracies. Consequently, three topologically distinct degeneracy strings exist, each carrying a distinct sector of the Uhlmann holonomy. I identify these three sectors with the three fermion generations.An explicit qutrit toy model demonstrates that a closed loop in parameter space encircling all three degeneracy points produces a Z3 monodromy—a cyclic permuta- tion of eigenvalues—whose braid structure provides an information-geometric origin for the CKM mixing matrix and CP violation. The SU(2) sector contributes no additional generations (since 2 2 = 1) but instead generates the isospin doublet structure within each generation. This result constitutes a sharp, falsifiable prediction: the number of fermion generations is exactly three, as a topological consequence of pairwise degeneracies in a three-level quantum system embedded in four-dimensional spacetime. The prediction is consistent with precision electroweak data (Nν = 2.984 ±0.008 from LEP) and current LHC bounds on fourth-generation quarks.
This paper is a duplicated. Updated and complete version available at https://doi.org/10.5281/zenodo.21591128" I determine the behaviour, under a change of internal quantum reference frame (QRF), of the geometry and gauge structure obtained from the mixed-state quantum geometric tensor (msQGT) of a reduced density matrix ρObs. The setting is the Relational–Informational Model (RIM), in which the emergent metric and gauge field are the real and imaginary sectors of the msQGT of an observer’s state, defined relationally within a Wheeler–DeWitt universe; I take that identification as given and ask only how the resulting structure transforms between internal observers. Since ρObs is a subsystem state conditioned on a QRF, it inherits the frame relativity of subsystems established by the perspective-neutral QRF programme, which I make precise. In the ideal regime, where the frame-change map is a unitary, the observer’s spectral degeneracy pattern—which fixes the emergent gauge group—and the topological charges of the Uhlmann curvature are frame-invariant, while the metric and gauge potentials are frame-covariant; I verify the topological invariance numerically for spin-1/2 and spin-1 monopoles, and show that it holds only for smooth, single-valued frame changes (singular, Dirac-string changes inject charge). In the physical regime, where the relevant constraint generator is bounded below, the factorisation condition of Ali Ahmad et al. fails in some frames, so that whether ρObs—and hence the emergent gauge content—is even definable becomes frame-relative. The new element relative to existing QRF results is that the msQGT endows the QRF-relative reduced state with an information-geometric (Uhlmann/Yang–Mills) structure whose integer-valued topological charges constitute a new class of QRF invariant, distinct from—and complementary to—the known continuous resource invariant (the conserved entanglement-plus-coherence sum), which carries no gauge structure. The upshot is a sharp division: the objective content of the emergent structure is topological, its metric/gauge representation is perspectival, and in the physical regime even the observer’s existence as a subsystem is frame-relative.
I analyse the geometry of the Uhlmann connection for a smooth family of finitedimensional density matrices in a neighbourhood of a pairwise eigenvalue-degeneracy locus,which by the von Neumann–Wigner theorem has codimension three in parameter space.First, I derive from the parallel-transport condition that the holonomy-generating weightattached to an eigenvalue pair (pa,pb) is wab = (√pa − √pb)2/(pa + pb), and verify thedefining operator identity numerically to 10−10; the frequently quoted heuristic weight√papb/(pa + pb) fails the same identity at order unity. Second, I establish an exact blockdecomposition of the Uhlmann curvature in the moving eigenframe: restricted to the twonearly degenerate bands, the curvature equals the Yang–Mills field strength of the reducedu(2)-valued connection plus a spectator residual ∆⊥ mediated by the remaining bands. Fortwo-level systems ∆⊥ vanishes identically; for three-level systems it is generically nonzeroand, together with every other Uhlmann-layer quantity, remains finite on the locus, because wab vanishes quadratically in the gap and cancels the eigenvector divergence—thepair connection itself vanishes linearly with the distance ϵ to the locus. The quantisedmonopole content (integer eigenbundle Chern numbers, verified by the Fukui–HatsugaiSuzuki method for spin-1/2 and spin-1 models) is carried by the eigenframe, not by thefinite Uhlmann curvature. Third, for the canonical depolarised-monopole family I obtainthe closed form ΦU = argcos(πwab) for the Uhlmann holonomy phase around a great circle,exhibiting a Z2 transition exactly at wab = 12, i.e. at Bloch radius rc = √3/2, depolarisationλc = 1−√3/2≈0.1340, purity Trρ2 = 7/8; the heuristic weight would instead forbid anytransition at finite mixedness. The results give a self-contained local field-theoretic characterisation of mixed-state gauge structure at level crossings and correct several statementsin the literature that rest on the heuristic weight, including the “half-Berry” limit of theUhlmann connection near degeneracy.
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 %
Important status notice: This version has been superseded by v2.0. Subsequent work shows that the single-domain three-diagonal construction used here produces an experimentally excluded common-mode photon-speed anisotropy. Readers should consult the revised version and 10.5281/zenodo.21464971. We propose that conservation of probability — the single axiom A=1 — is sufficient to derive the principal results of quantum mechanics without additional postulates. On a bipartite three-dimensional causal lattice T³_diamond, each particle is represented as a causal session: an independent phase oscillator that propagates amplitude across the lattice subject to |ψ_R|² + |ψ_L|² = 1 at every tick. No Hamiltonian and no measurement postulate are assumed; a Hilbert-space structure with unitary evolution emerges from the U(1) oscillator and the per-session amplitude constraint rather than being postulated a priori. From this single constraint, the paper derives: (i) the Dirac equation in the continuum limit a→0, with the gamma matrices identified as geometric encodings of the bipartite RGB/CMY sublattice geometry; (ii) wave-particle duality and the Born rule as consequences of amplitude redistribution under A=1, with Gleason's theorem securing the |ψ|² uniqueness from the framework's emergent Hilbert space; (iii) quantization of atomic orbitals as Arnold tongue attractors of the two-body phase dynamics, recovering the hydrogen Bohr radius to four significant figures (k_min = 0.0970 vs k_Bohr = 0.0971) without fitting; (iv) gravity as a clock-density gradient rather than spacetime curvature — a refraction of causal paths through regions of higher session density, reproducing the 1/r² force law with an explicit weak-field metric tensor g_μν as a functional of the clock-density field; (v) photon emission as a structural consequence of A=1: the Coulomb interaction is a probability source that a single session cannot accommodate, mandating creation of a new photon session at orbit lock-in; (vi) the structural form of the induced gauge action (Sakharov / Zeldovich), whose leading-order term has the Maxwell form on O_h-averaged observables, with a closed-form gauge-sector birefringence sharing the same optical axis (1,1,−1) as the kinematic-sector prediction. Energy and momentum conservation are not independent inputs — they are consequences of A=1 applied to closed session sets. The numerically tractable results are verified or partially verified by a suite of more than fifteen experiments with no free parameters beyond the Planck-scale calibration of the lattice spacing; an audit table records the status of every claim. The paper conjectures that the Standard Model gauge group SU(3) × SU(2) × U(1), together with Lorentz SO(3,1), is the automorphism group of the framework's extended per-site amplitude — a precise 18-dimensional Lie-algebra equality. The existing framework is shown to carry SO(3,1) × U(1) (dim 7) on the per-site (ψ_R, ψ_L) amplitude; the remaining factors require structural extensions (per-site C² isospin and C³ colour indices) that are the subject of a companion paper. Six computational scaffolding scripts establish the conjecture's components. This is Paper I of the A=1 Discrete Causal Lattice series. Companion papers in preparation address the Standard Model gauge derivation, the Farey-sequence mass spectrum (and its Veneziano-amplitude limiting case), plasma and gradient-ionization phenomenology, the proton-interior structure, quantum measurement theory, and a Hilbert-6th-style combinatorial foundation for continuum physics. What changed in v1.0 (since v0.98-RC): * Three new audit-table rows: long-horizon escape mechanism (PART), lock-in resonance grid-independence across {65, 81, 97, 113}³ grids (PART), and the Standard Model gauge-group conjecture as a precise 18-dimensional Lie-algebra equality (STUB).* A Gleason-uniqueness subsubsection in the Born-rule discussion, addressing the necessity question for the |ψ|² measure.* An explicit metric-tensor formula in the gravity section, identifying g_μν as a functional of ρ_clock in the weak-field limit.* Refined automorphism conjecture with explicit decomposition into established and structural-extension factors.* New Acknowledgments section. Source code, paper sources, and data files: https://github.com/JackDMenendez/discrete-causal-latticeLicense: Paper text and figures CC BY 4.0; source code MIT.
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-k density matrices: minimal purifications make the fixed-rank stratum the base of a principal U(k)-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at k=1, at equal weights, and at k=N. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
Files of this version are restricted (18 October 2026) because it was released under earlier licence terms: the source code was under the MIT License. All current versions use one clear licensing: paper, figures and data under CC BY 4.0, source code under the Apache License, Version 2.0. Please use the latest version: https://doi.org/10.5281/zenodo.21362245 We report a real-space resolution of the full modular data of a doubled-Fibonacci Levin–Wen string-net, obtained by exact diagonalization on finite torus supercells (2×2, 2×3, 3×3), together with a sharply scoped negative result for the dynamical half of the same program — and we keep the classes of evidence strictly separate. (i) The vacuum row of the modular S-matrix, |S₀ₐ| = dₐ/D, is confirmed by two structurally distinct lattice routes — an operator route built from a flux basis, and a Verlinde–Casimir route that reads the quantum dimensions from the lattice loop spectrum without using the F or R symbols — the latter reproducing the analytic values at 1.1×10⁻¹⁶ and the two agreeing at <10⁻⁹, the precision at which the operator-route values are recorded. (ii) The ground-state degeneracy GSD=4 is a separate check, invariant across 2×2, 2×3 and 3×3; |S| is invariant under 24 O(4) gauge transformations of the ground-state manifold and size-robust to ≤2.5×10⁻¹¹. (iii) The Wilson-loop algebra alone does not fix the chiral sign, and we measure that limit rather than assert it: the loop algebra is ℂ⊕M₃(ℂ) (dim=10=1²+3²), its antisymmetric part vanishes identically on the real-symmetric holonomy, and the chiral splitting capacity is zero. This is a concrete instance of a theorem of Li and Mong, who also name the remedy: a point-group rotation of the ground-state manifold, real but not symmetric — precisely the property the loop route lacks — available on every L×L supercell. Carrying it out, we extract the full signed S and T from the lattice on two sizes, circularity-free, up to the ℤ₂ relabeling intrinsic to an achiral theory; a validated two-channel error budget with no free parameters accounts for the extraction residual on both. The modular data themselves are not new with this work — Francuz and Dziarmaga obtained them for this model in 2020; new here are the finite-lattice demonstration of the rotation channel, the quantified error budget, and the measured loop-algebra limit as a concrete instance of the Li–Mong bound. Complementing this static certificate, we show that on the one-dimensionally mapped anyonic ladder the Hamiltonian transport of a single mobile Ising anyon around a pinned object cannot realize the non-Abelian braid matrix: the closed-loop holonomy is exactly the identity and [U(C₁),U(C₂)]=0, established through two independent obstructions, three null escape routes, and a machinery-alive control. That obstruction is structural and independent of both the anyon type and the system size, so it constrains the Fibonacci ambition motivating this work and not merely the Ising tracer that exhibits it. Both halves were produced under an adversarial verification harness that, in the course of this work, retracted an attribution of its own — crediting the lattice with the signed data before a channel existed to carry it — and then restored it with proof through the rotation channel. The values were never in question, only their provenance.Notice (2026-10-14). The summary above states that on the one-dimensionally mapped anyonic ladder of Singh et al. the Hamiltonian transport of a single mobile Ising anyon around a pinned object has the identity as its closed-loop holonomy, and that this obstruction is “structural and independent of both the anyon type and the system size”. This holds under the hypotheses stated in the paper, in particular that the braid phase sits in the bond amplitudes identically on both legs of the ladder. That hypothesis is the explicit Ising plaquette Hamiltonian, Eq. (52), of S. Singh, R. N. C. Pfeifer, G. Vidal and G. K. Brennen, Phys. Rev. B 89, 075112 (2014). The general hopping Hamiltonian of the same work, Eqs. (40)–(44), differs from Eq. (52) by complex conjugation of the hopping on one leg. With Eqs. (40)–(44), a mobile anyon encircling a pinned anyon acquires the monodromy R_c² in fusion channel c; for Ising anyons the two channels then differ by a sign, so the holonomy is not the identity, and the same holds for Fibonacci anyons. The no-go statement therefore applies to the ladder of Eq. (52); for the ladder of Eqs. (40)–(44) it does not follow. The static result of this work, the modular data obtained from the rotation channel, is not affected, and no computed value changes. The paper text will be corrected in the next version. About this series: This record is part of a series of related works from my independent research on Fibonacci anyons, with Ising anyons as their natural counterpart. I started in April 2026, and it has been a long and insightful journey in which I learned a lot; the work uses different methods and stays within verifiable, nonspeculative physics. The common thread of the series is a split: Ising anyons are limited to Clifford operations, while Fibonacci anyons are computationally universal, and across the series I map what standard witnesses of nonclassicality can and cannot certify on such systems. I consider Fibonacci anyons a serious candidate for topological quantum computing, given their universality and their topological protection against local noise. A hybrid approach with Ising is conceivable, but problems such as instability and certification would have to be solved first, and each needs research of its own. This paper asks how much of the topological fingerprint of an anyon theory a finite lattice can carry, computing a doubled Fibonacci string-net on a finite torus, where the quantum dimensions are confirmed by two structurally distinct lattice routes and the chiral sign is shown, by measurement rather than assumption, to lie beyond the conventional Wilson-loop algebra, while a point-group rotation channel does deliver it, reproduced on two lattice sizes with an error budget that needs no free parameters, together with a no-go for anyon exchange on a quasi-one-dimensional ladder that holds independently of anyon type and system size; the modular data and the theorem are not new, what is new is the lattice demonstration, the error budget and the measured limit, and within the series it is the real-space facet that checks the foundation. Use of AI tools: In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's. I run multiple verifications at the different stages of the work and one before release, including cross-checks with an unrelated AI model from a different company, and all references are checked against the original sources. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication.