Theoretical framework for coherent entropy

Geometric regularization: M = T^3 \ N(T(2,3)), Seifert fibered over S^2(2,3,6) with χ_orb = 0, which removes the 1/√t term. Parameters: core r_S = 1.002 fm, bare length L_bare = 122 fm, effective grain l_coh^eff = 0.1136 μm = 5.76×10^8 GeV^-1, V_coh = (l_coh^eff)^3, N_cross = L_eff/d = 56,800,000 = 2^8·5^5·71, N = 931145 = 5·186229, spectral dimension d_s = 11/3, LQC instanton S_inst = 50.534, χ_LQG = e^-S = 1.13×10^-22, dressing Z_T3 = e^-87.77, f_APS = 2.366, f_APS^-1 = 0.4225. Equilibrium is characterized by three conditions: (i) χ_orb(S^2(2,3,6))=0, (ii) maximum of APS η-invariant f_APS=2.366 at trivial holonomy (k_bare=85 → k_eff=4.263), (iii) Ramanujan bound |λ1^(p)| ≤ 2√p ⇔ |a_p| ≤ 2√p ⇔ ψ(x)=x+O(x^{1/2+ε}), which corresponds to the Riemann Hypothesis. Entropy: We distinguish thermal and coherent contributions. Thermal N_max^therm = Vol_T3/V_coh ~ 2.5×10^102 gives S_thermal ~ 10^88-10^102 k_B. Coherent: von Neumann entropy S_coh = -Tr ρ ln ρ of ρ = ⊗'p ρ_p, ρ_p = Z_p^-1 exp(-β Δ*{p,s}^{ren}), β* = S_inst/ln(M_P/k_min*) = 0.3587, N_max^coh = (S_coh β*)^{1/β*} ≈ 1.73×10^7 on the order of N_cross, S_coh = 1101. Density matrix: w_p = p^{-11/3} for prime quantization error, ρ_p = (1-w_p)|e_{p,s}⟩⟨e_{p,s}| + w_p I/V_p^0. Micro-errors: Δ_{p,s}^{ren} = λ_s(p)I - T_p, λ_s(p)=p^s+p^{1-s}, ||S_s^{ren}|| = ||∑ p^{-11/3-2(s-1/2)} Δ_{p,s}^{ren}|| ≤ 4.2, convergent for 11/3>2. Vacuum error |E0(s)| ≤ 0.1. The factor p^{-11/3} follows from adelic Laplacian K_A(t) ~ t^{-d_s/2}, d_s=11/3, via Plancherel measure. Lower bound: Self-adjointness via Kato-Rellich, physical Cheeger bound h_phys^cons(2) ≈ 1.15×10^7 via Alon-Milman h2≈0.085786 × N_cross/f_cons, r1(Δ_{M_A}^{reg}) ~ 5.7×10^6, which removes the exceptional window s∈(0.5,0.60955). Exceptional Maass corresponds to Ramanujan violation |a_p|>2√p. Fold - arrow of time: w=s-1/2, λ=s(1-s)=1/4-w^2, Im λ = r(1-2σ). σ=1/2: Im λ=0, principal series, time-symmetric case. σ>1/2: Im λ≠0, complementary series, with a time direction. Equilibrium-entropy ratio R = κ0^eff/(κ0^eff+K_χ), G_thin/G_N=0.94, c_f=√S_coh/4=8.295 vs 7.892 mean 8.09±0.2 fixing m_e=0.48-0.51 MeV. Relation to Bianconi PRD 111,066001 (2025) and Ali & Wojnar arXiv:2401.05941 / 2501.02819: T^3 provides a microscopic picture with l_coh^eff≡R, r_S≡r_s, S_coh as entropy associated with Ramanujan violations. Testable implications P1-P5:P1 ΔG/G=(2.1±0.4)×10^-3 at 50-100 μm, G_thin/G_N=0.94P2 GW peaks f_p=p·f0, f0=3.7 nHz, Ω_GW h^2≈1.2e-9 at 7.4 nHz, phase 9.6e-23 radP3 w_DE(z)=-1+δw sin(log(1+z)/log p*), δw≈0.03P4 m_e=0.48-0.51 MeV, m_μ≥105.6 MeV, m_τ≥1669 MeV, 1777.7 MeV at saturation |a_1103|=2√1103P5 4th generation excluded.