Resolving the Full Modular Data of a Doubled-Fibonacci String-Net on a Finite Torus by Point-Group Rotation, and an Obstruction to Quasi-One-Dimensional Anyon Exchange

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. 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.---------------- Version notes (v1.0 → v1.1, following a comprehensive internal review of the full series): • No previously reported value changed, and no equation changed. The corrections below concern the justification of one result, the strength of several statements, and the completeness of the deposit. • Correction (justification, not result): the selection against the Galois-conjugate solution was attributed to the order-5 congruence. That congruence cannot do it — Fibonacci (h = +2/5) and Yang-Lee (h = -1/5) are roots of unity of the same order 5, so a congruence of level 5 is an invariant of the Galois orbit and does not separate them. What does separate them is the positivity of the extracted quantum dimensions (d_tau = phi > 0 rather than -1/phi), that is, unitarity. The extracted data and every reported value are unchanged; only the reason given for identifying them is. Three references are added for this (Freedman et al. for the case, Ng-Schauenburg and Dong-Lin-Ng for the congruence property, which had previously been cited to a work that conjectured rather than proved it). • Correction (printed multiset): the topological spins are listed as {0, 0, ±4π/5}. The published version listed {0, ±4π/5}; a rank-4 theory has two of them equal to zero. • Claim strengths reduced, in four places: the extraction is no longer called "circularity-free" but is described by what was withheld from the solver (no analytic doubled-Fibonacci S or T); the monodromy scalar is no longer "the heart of topological quantum computation" but a characteristic diagnostic of Fibonacci statistics, with universality attributed to the matrix-valued braid action of which it is one invariant; the gauge scan is described as over 24 sampled O(4) transformations rather than over 24 O(4) transformations; and the Verlinde-Casimir route now states explicitly what it is independent of. • Corrections to statements about the deposit itself: the driver that forms the reference error is released with this record, and the text said it was not; and the 2×2 basis is part of this record, and the text said it was not. • Center independence, previously expected but not verified, is now verified: the extracted T spectra agree across all three 2×2 centers at the level of the extraction residuals themselves, up to the intrinsic Z2 relabeling, and this is checkable from the deposited extraction file. • Added: a section identifying which unitary modular tensor category of rank 4 the data select. Exactly one is compatible; the quantum dimensions alone do not suffice, since two competitors share them exactly and are separated only by the topological spins; the residual Z2 is an automorphism of that single category rather than a competing candidate; and all excluded competitors differ by O(1), so the conclusion does not rest on the error budget. • The