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Topological Trace Theorem and Virtual Anyon Tunneling in SU(2)_2 String-Net Transfer Matrices

Adam Bazih

Published Apr 21, 2026• 0 citations
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Abstract

[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] Abstract. We prove an exact algebraic theorem for the Levin-Wen string-net transfer matrix of the SU(2)_2 (Ising) modular tensor category on a cylinder of circumference L_x. The transfer matrix T commutes with the fermion (ψ) Wilson loop operator W_ψ = ⊗_x diag(1,−1,1), yielding two topological sectors T_+ and T_−. We prove: Trace Splitting Theorem: Tr(T_+) − Tr(T_−) = Tr(F^{L_x}) where F is a 3×3 matrix with exact algebraic eigenvalues {D, 2^{5/4}(2^{1/4}−1), −2(2^{1/4}−1)}, and D = 2 is the total quantum dimension of SU(2)_2. Therefore Tr(T_+) − Tr(T_−) → D^{L_x} as L_x → ∞. Proof method: The ψ-signed trace decomposes as Tr(W_ψ T) = Tr(F^{L_x}) where F = W[1,1,:,:] − W[σ,σ,:,:] + W[ψ,ψ,:,:] is the "ψ-signed diagonal channel" of the MPO column tensor. We compute F exactly: W[1,1] = W[ψ,ψ] = diag(1, √2, 1) — the vacuum/fermion channel W[σ,σ] = [[2^{1/4}, 0, 2^{1/4}], [0, 2^{5/4}, 0], [2^{1/4}, 0, 2^{1/4}]] — the σ channel F eigenvalues: 2 (= D, exact), 2^{5/4}(2^{1/4}−1) ≈ 0.450, −2(2^{1/4}−1) ≈ −0.378 All entries of the column tensor W[a,b,v',v] are powers of 2: {1, 2^{1/4}, 2^{1/2}, 2^{3/4}, 2^{5/4}}, reflecting the algebraic structure of SU(2)_2 fusion rules. Ground state splitting (numerical): We compute the exact transfer matrix eigenvalues for L_x = 2 through 9 (sector dimensions up to 9842). The signed splitting s(L_x) = λ_+(top) − λ_−(top) is decomposed via 4-term Prony analysis into: s(L_x) = A_1 ρ_1^{L_x} + Re[A_2 ρ_2^{L_x}] + A_4 ρ_4^{L_x} with four exponential rates (machine-precision fit to all 8 data points): ρ_1 = −0.5916 (real negative): dominant for L_x ≥ 8, rate α_1 = ln(C/|ρ_1|) = 2.092 ≈ ln(8) ρ_{2,3} = −0.368 ± 0.108i (complex pair): dominant for L_x ≤ 7, rate α_2 = ln(C/|ρ_2|) = 2.527 ≈ ln(12) ρ_4 = +0.130 (real positive): fastest-decaying term, α_4 = 3.607 where C = 4.79290 is the per-site growth rate. The complex pair with α_2 ≈ ln(D_total) = ln(12) dominates at moderate system sizes (explaining earlier numerical coincidences with ln(12)), while the real term with α_1 ≈ ln(D^3) = ln(8) = 3ln(2) dominates asymptotically. Key structural results: The Z_2 topological symmetry is the ψ Wilson loop: W_ψ = ⊗_x diag(1,−1,1) = (−1)^{n_σ} Both sectors share the same per-site growth rate C to 10+ significant digits The winning sector (+ or −) follows a complex pattern due to multiple interfering tunneling channels The ground state is NOT in the vacuum virtual channel — it is dominated by σ-string fluctuations The mean-field optimal state has ψ_opt ≈ (0.504, 0.702, 0.504), close to but not exactly √d_a-weighted Physical interpretation: The trace splitting D^{L_x} represents the topological amplification of sector differences — the transfer matrix enhances the trivial dimension difference (dim_+ − dim_− = 1) to a topologically protected D^{L_x} signal. The ground state splitting involves virtual σ-anyon tunneling across the cylinder, with multiple interfering channels (real and complex exponentials) reflecting the non-commutative structure of the MPO virtual space. The σ anyon tunneling amplitude is complex because the 3×3 virtual-space matrices at different sites do not commute, leading to quantum interference in the tunneling paths. Data table (L_x = 2..9): L_x λ_+ λ_− Winner δ_rel 2 22.9758 22.9680 + 3.38×10^{−4} 3 110.102 110.103 − 1.93×10^{−6} 4 527.711 527.710 + 2.05×10^{−6} 5 2529.26 2529.26 − 1.73×10^{−7} 6 12122.5 12122.5 + 1.42×10^{−8} 7 58102.0 58102.0 − 7.00×10^{−10} 8 278477 278477 − 1.58×10^{−11} 9 1334711 1334711 + 1.13×10^{−11} Relation to previous work: This paper provides the analytical foundation for the numerical hierarchy D^{L_x} observed in our earlier work (DOI: 10.5281/zenodo.19673053). The Trace Splitting Theorem proves that D appears exactly in the sector-resolved partition function, while the ground state splitting involves a richer multi-exponential structure with rates connected to D^3 and D_total = D_{SU(3)_3} × D_{SU(2)_2} = 12. Connection to AME(4,6) framework: The SU(2)_2 Ising MTC is one factor of the SU(3)_3 ⊗ SU(2)_2 product theory that emerges from the AME(4,6) quantum error-correcting code. The trace theorem generalizes: for the product theory, Tr(T_+) − Tr(T_−) = D_total^{L_x} = 12^{L_x}.

DOI
10.5281/zenodo.19673512

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Entanglement Spectrum of the SU(3)₃ ⊗ SU(2)₂ String-Net Model: A Baseline Computation for the AME(4,6) Theory

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[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We construct the Levin-Wen string-net model for the product modular tensor category SU(3)₃ ⊗ SU(2)₂ on a triangulated cylinder and compute its entanglement spectrum using a transfer matrix method. This product MTC, with 30 anyon types, total quantum dimension D=12, and central charge c=11/2, arises as the boundary theory of the Walker-Wang model associated with the AME(4,6) quantum error-correcting code. The entanglement spectrum at the exactly solvable point shows a degeneracy structure {1, 12, 17, ...} organized by quantum dimensions. We verify all modular data for both factors and confirm that no graviton mode appears at the topological point. This establishes the baseline for future investigations of emergent gravity.

•Apr 20, 2026•0 citations

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

Berkay Yüksel Sayim

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. 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[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We prove six theorems from the self-dual SU(3)_3 x SU(2)_2 modular tensor category.

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•May 20, 2026•0 citations

Topological Hierarchy from the AME(4,6) String-Net Model: Exponential Scale Separation at Rate α = ln D

Adam Bazih

[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We compute the topological ground-state splitting of the SU(2)₂ (Ising) sector of the SU(3)₃ ⊗ SU(2)₂ string-net model for cylinder widths Lx = 2, 3, 4, 5, 6 using a factored column-operator method. The Z₂ symmetry of the Ising topological order makes the two ground states nearly degenerate, with splitting δ(Lx) ~ exp(-α·Lx). We find the asymptotic protection rate converges to α ≈ ln D = ln 12 = 2.485, where D = 12 is the total quantum dimension. The fit 2α/ln(D²) = 1.001 holds to 0.1%. The hierarchy scales as D^Lx = 12^Lx. At Lx ≈ 15 the hierarchy reaches ~10^17, matching M_Pl/v. Key findings: (1) Five-point measurement confirming exponential suppression; (2) Non-monotonic anomaly at Lx=3-4 (lattice resonance); (3) Asymptotic α = ln(D) = γ (topological entanglement entropy); (4) Brown-Henneaux G₃ converges to 0.092 lattice units; (5) WZW primary matching exact at h=3/16 for all Lx; (6) Honest assessment of what is rigorous vs suggestive.

•Apr 21, 2026•0 citations
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] Abstract. We prove an exact algebraic theorem for the Levin-Wen string-net transfer matrix of the SU(2)2 (Ising) modular tensor category on a cylinder of circumference L_x. The transfer matrix T commutes with the fermion (ψ) Wilson loop operator W_ψ = ⊗x diag(1,−1,1), yielding two topological sectors T+ and T−. We prove: Trace Splitting Theorem: Tr(T_+) − Tr(T_−) = Tr(F^{L_x}) where F is a 3×3 matrix with exact algebraic eigenvalues {D, 2^{5/4}(2^{1/4}−1), −2(2^{1/4}−1)}, and D = 2 is the total quantum dimension of SU(2)2. Therefore Tr(T+) − Tr(T_−) → D^{L_x} as L_x → ∞. Proof method: The ψ-signed trace decomposes as Tr(W_ψ T) = Tr(F^{L_x}) where F = W[1,1,:,:] − W[σ,σ,:,:] + W[ψ,ψ,:,:] is the "ψ-signed diagonal channel" of the MPO column tensor. We compute F exactly: W[1,1] = W[ψ,ψ] = diag(1, √2, 1) — the vacuum/fermion channel W[σ,σ] = [[2^{1/4}, 0, 2^{1/4}], [0, 2^{5/4}, 0], [2^{1/4}, 0, 2^{1/4}]] — the σ channel F eigenvalues: 2 (= D, exact), 2^{5/4}(2^{1/4}−1) ≈ 0.450, −2(2^{1/4}−1) ≈ −0.378 All entries of the column tensor W[a,b,v',v] are powers of 2: {1, 2^{1/4}, 2^{1/2}, 2^{3/4}, 2^{5/4}}, reflecting the algebraic structure of SU(2)2 fusion rules. Ground state splitting (numerical): We compute the exact transfer matrix eigenvalues for L_x = 2 through 9 (sector dimensions up to 9842). The signed splitting s(L_x) = λ+(top) − λ_−(top) is decomposed via 4-term Prony analysis into: s(L_x) = A_1 ρ_1^{L_x} + Re[A_2 ρ_2^{L_x}] + A_4 ρ_4^{L_x} with four exponential rates (machine-precision fit to all 8 data points): ρ_1 = −0.5916 (real negative): dominant for L_x ≥ 8, rate α_1 = ln(C/|ρ_1|) = 2.092 ≈ ln(8) ρ_{2,3} = −0.368 ± 0.108i (complex pair): dominant for L_x ≤ 7, rate α_2 = ln(C/|ρ_2|) = 2.527 ≈ ln(12) ρ_4 = +0.130 (real positive): fastest-decaying term, α_4 = 3.607 where C = 4.79290 is the per-site growth rate. The complex pair with α_2 ≈ ln(D_total) = ln(12) dominates at moderate system sizes (explaining earlier numerical coincidences with ln(12)), while the real term with α_1 ≈ ln(D^3) = ln(8) = 3ln(2) dominates asymptotically. Key structural results: The Z_2 topological symmetry is the ψ Wilson loop: W_ψ = ⊗x diag(1,−1,1) = (−1)^{n_σ} Both sectors share the same per-site growth rate C to 10+ significant digits The winning sector (+ or −) follows a complex pattern due to multiple interfering tunneling channels The ground state is NOT in the vacuum virtual channel — it is dominated by σ-string fluctuations The mean-field optimal state has ψ_opt ≈ (0.504, 0.702, 0.504), close to but not exactly √d_a-weighted Physical interpretation: The trace splitting D^{L_x} represents the topological amplification of sector differences — the transfer matrix enhances the trivial dimension difference (dim+ − dim_− = 1) to a topologically protected D^{L_x} signal. The ground state splitting involves virtual σ-anyon tunneling across the cylinder, with multiple interfering channels (real and complex exponentials) reflecting the non-commutative structure of the MPO virtual space. The σ anyon tunneling amplitude is complex because the 3×3 virtual-space matrices at different sites do not commute, leading to quantum interference in the tunneling paths. Data table (L_x = 2..9): L_x λ_+ λ_− Winner δ_rel 2 22.9758 22.9680 + 3.38×10^{−4} 3 110.102 110.103 − 1.93×10^{−6} 4 527.711 527.710 + 2.05×10^{−6} 5 2529.26 2529.26 − 1.73×10^{−7} 6 12122.5 12122.5 + 1.42×10^{−8} 7 58102.0 58102.0 − 7.00×10^{−10} 8 278477 278477 − 1.58×10^{−11} 9 1334711 1334711 + 1.13×10^{−11} Relation to previous work: This paper provides the analytical foundation for the numerical hierarchy D^{L_x} observed in our earlier work (DOI: 10.5281/zenodo.19673053). The Trace Splitting Theorem proves that D appears exactly in the sector-resolved partition function, while the ground state splitting involves a richer multi-exponential structure with rates connected to D^3 and D_total = D_{SU(3)3} × D{SU(2)_2} = 12. Connection to AME(4,6) framework: The SU(2)_2 Ising MTC is one factor of the SU(3)3 ⊗ SU(2)2 product theory that emerges from the AME(4,6) quantum error-correcting code. The trace theorem generalizes: for the product theory, Tr(T+) − Tr(T−) = D_total^{L_x} = 12^{L_x}.
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We construct the Levin-Wen string-net model for the product modular tensor category SU(3)₃ ⊗ SU(2)₂ on a triangulated cylinder and compute its entanglement spectrum using a transfer matrix method. This product MTC, with 30 anyon types, total quantum dimension D=12, and central charge c=11/2, arises as the boundary theory of the Walker-Wang model associated with the AME(4,6) quantum error-correcting code. The entanglement spectrum at the exactly solvable point shows a degeneracy structure {1, 12, 17, ...} organized by quantum dimensions. We verify all modular data for both factors and confirm that no graviton mode appears at the topological point. This establishes the baseline for future investigations of emergent gravity.
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.
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We present the complete R-matrix braiding catalog for SU(3)_3 modular tensor category, comprising all 144 eigenvalues across 55 anyon pairs. Each eigenvalue takes the form exp(in*pi/18) where n is an integer, establishing that pi/18 serves as the universal phase quantum. The catalog is obtained through direct Sage computation (127 eigenvalues), charge conjugation symmetry (15 eigenvalues), and a novel Hopf link derivation for the multiplicity-2 channel (1,1) x (1,1) -> (1,1) (2 eigenvalues). We verify consistency through the Verlinde formula (144/144 channels confirmed) and charge conjugation symmetry (71/71 pairs, zero failures). Universal patterns discovered include: (i) the adjoint anyon (1,1) produces a universal elastic phase of -pi/2 for all non-simple-current anyons, (ii) adjoint self-braiding yields exclusively fourth roots of unity, and (iii) the simple current (3,0) generates phases that are exact multiples of pi/3. This catalog provides the foundational braiding data for the AME(4,6) theory of everything based on the Walker-Wang model.
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We prove six theorems from the self-dual SU(3)_3 x SU(2)_2 modular tensor category.
[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras, and whose condensed-phase identification did not resolve the sixteen-fold way (a D^2 = 4 condensed phase is determined only together with the central charge mod 8). Affected classifications should be re-checked against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We establish rigorous obstructions to fermionic anyon condensation in product modular tensor categories and analyze the lattice structure of condensable algebras. Our key results include: (1) a proof that fermionic condensation is impossible in SU(2)_k ⊗ SU(2)_k for all k ≤ 8, and provide the mechanism via conformal weight analysis; (2) computation of the full condensation lattice (partial order under inclusion) for SU(2) levels 1-8, revealing rich combinatorial structure; (3) identification of maximal condensable algebras as Lagrangian when they exist. These results contribute to the classification program for topological phases of matter and have implications for the construction of fermionic topological orders from bosonic ones.
[NOTE, June 2026: This paper is part of the AME(4,6)/TOE series whose headline claims were partially retracted. In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951. The text below is unchanged and remains as the record of the programme's history.] We compute the topological ground-state splitting of the SU(2)₂ (Ising) sector of the SU(3)₃ ⊗ SU(2)₂ string-net model for cylinder widths Lx = 2, 3, 4, 5, 6 using a factored column-operator method. The Z₂ symmetry of the Ising topological order makes the two ground states nearly degenerate, with splitting δ(Lx) ~ exp(-α·Lx). We find the asymptotic protection rate converges to α ≈ ln D = ln 12 = 2.485, where D = 12 is the total quantum dimension. The fit 2α/ln(D²) = 1.001 holds to 0.1%. The hierarchy scales as D^Lx = 12^Lx. At Lx ≈ 15 the hierarchy reaches ~10^17, matching M_Pl/v. Key findings: (1) Five-point measurement confirming exponential suppression; (2) Non-monotonic anomaly at Lx=3-4 (lattice resonance); (3) Asymptotic α = ln(D) = γ (topological entanglement entropy); (4) Brown-Henneaux G₃ converges to 0.092 lattice units; (5) WZW primary matching exact at h=3/16 for all Lx; (6) Honest assessment of what is rigorous vs suggestive.