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

Adam Bazih

Published Apr 21, 2026• 0 citations
Publisher

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 In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951.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.] 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.

DOI
10.5281/zenodo.19673052

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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 In particular the "zero free parameters" framing and several numerical predictions are withdrawn; see the Status Note at DOI 10.5281/zenodo.20618848 and the maintained mathematics at DOI 10.5281/zenodo.20128951.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 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.
[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 perform a seven-part holographic analysis of the SU(3)_3 ⊗ SU(2)_2 string-net model associated with the AME(4,6) quantum error-correcting code. The transfer matrix is reinterpreted as a discrete renormalization group (RG) operator, and its iterated action is studied as a proxy for the MERA tensor network that defines an emergent holographic bulk. Key results: (1) Perfect conformal self-similarity — gap ratios preserved to 15 significant digits through 5 RG steps; (2) WZW tower matching — entanglement spectrum matches (0,0)⊗σ primary at h=3/16 exactly (Δh=0.0000); (3) Entanglement temperature β_ent = 2.380; (4) N-layer stacking shows exponential RG convergence (gapped system); (5) Topological hierarchy: ξ_topo/ξ = 5171 at Lx=2, scaling as exp(4·Lx), giving ~10^17 at Lx=10 — matching the Planck-electroweak ratio M_Pl/v; (6) Brown-Henneaux G₃ = 0.156 lattice units from c=11/2, L_AdS=ξ=0.572; (7) Stress tensor candidates at h≈1.81 (9.5% below h=2, finite-size at Lx=2). The topological hierarchy mechanism is the most provocative result: an O(1) topological constant (α≈4) combined with modest system size (Lx≈10) generates the 16 orders of magnitude separating v from M_Pl, without fine-tuning.
Nós calculamos a entropia de emaranhamento topológica para um grande conjunto de modelos em dimensão d. Sabe-se que muitos sistemas quânticos podem ser construídos a partir de teorias de gauge na rede. Em dimensões maiores a 2 existem generalizações além das teorias de gauge. Chamadas higher gauge theories, estas são baseadas em generalizações de ordem superior do conceito de grupo. O nosso objeto de estudo é um conjunto grande de modelos d-dimensionais, que são obtidos a partir de teorias Abelianas de higher gauge. Neste trabalho, calculamos a entropia de emaranhamento para dito conjunto de modelos. O nosso formalismo permite fazer a maior parte do cálculo para dimensão arbitrária d. Mostramos que a entropia de emaran- hamento S_A , em uma sub-região A do sistema, é proporcional à log(GSD_Ã ), onde GSD_Ã é a degenerescência do estado fundamental de uma restrição particular do modelo na região A. Quando A tem a topologia de uma bola de dimensão d, a quantidade GSD_Ã conta o número de estados de borda. Neste caso, S_A escala com a área da borda (d 1)-dimensional de A. O resultado exato da entropia que obtemos está em concordância com os resultados conhecidos na literatura.
[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 conduct a four-pronged investigation of the gravitational sector of the SU(3)₃ ⊗ SU(2)₂ string-net model associated with the AME(4,6) quantum error-correcting code. Phase 1: perturbation study sweeping string tension λ ∈ [0, 2] reveals that the degeneracy structure {1, 12, 18, 18, 1} is topologically protected with no phase transition. Phase 2: transfer matrix eigenvalue analysis gives physical correlation length ξ = 0.57 (Lx=2), confirming the system is gapped. Phase 3: factored column-operator computation at Lx=3 (27000-dimensional product space) gives ξ = 0.38. Phase 4: Crane-Yetter state sum on CP² yields phase exp(2πi × 11/16) = 247.5°, the gravitational anomaly indicator. Five rigorous results: (1) stress tensor T(z) at h=2 exists (2+1D graviton); (2) gravitational anomaly on CP²; (3) k_grav = 11/48; (4) γ = ln(12), ΔS_BH = -ln(144); (5) thermal Hall κ_xy/T = (11/2)π²k²_B/(3h). We are explicit that the theory cannot derive G_N, explain M_Pl/v, or produce dynamical 3+1D gravity from topological data alone.
[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.