Gravity and the Cosmological Constant from the AME(4,6) String-Net: Five Directions toward Emergent Spacetime

Paper 50 in the AME(4,6) series. Five independent approaches to extracting gravitational physics from the AME(4,6) quantum error-correcting code (SU(3)_3 ⊗ SU(2)_2 modular tensor category). REVISION 3 (April 2026) — TEE-corrected cosmological constant formula: The corrected CC prediction includes the topological entanglement entropy (TEE) normalization factor 1/D = e^{-γ}, where γ = ln D is the TEE from Turaev-Viro: ρ_Λ/ρ_Pl = (1/D) × (D/C)^{Lx} = D^{Lx-1}/C^{Lx} With D_total = 12, C_total = 367.38 (fusion-multiplicity-corrected product transfer matrix), and Lx = 82 = 2D²_{SU(3)} + n_{SU(3)}: log₁₀(ρ_Λ/ρ_Pl) = -122.926 vs observed -122.940 (Planck 2018) Gap = 0.014 orders of magnitude (3.3% linear ratio) This is the most precise derivation: the 1/D factor closes 1.09 orders of the previous gap, arising from the Turaev-Viro partition function Z = D^{-1} · Tr(T^{Lx}). Previous values (C_total ≈ 1833, Lx ≈ 56) used the binary transfer matrix without fusion multiplicities. Thread 3 of Paper 51 identified this: the single multiplicity-2 channel in SU(3)_3 shifts C from 382→76.7 and Lx from 56→82. The product C_total = C_{SU3}^{n_{SU2}} × C_{SU2}^{n_{SU3}} = 76.651³ × 4.793¹⁰ = 367.38. Direction 1: The per-site growth rate C = 4.7928992722626482 is transcendental. Direction 2: Topological hierarchy mechanism with TEE correction. Direction 3: Full product transfer matrix with fusion multiplicities gives C_total = 367.38, Lx = 82. Direction 4: Newton constant from entanglement entropy G_N = 1/(4α) with α = (13/6)ln2 + (3/2)ln3 ≈ π (0.26% match). Direction 5: Turaev-Viro partition function confirms static TQFT gives O(1) Λ — dynamical mechanism essential, but the 1/D normalization provides the crucial TEE prefactor.