π as Physical Regulator Between Equilibrium and Entropy

We show that π emerges as a physical regulator between equilibrium (symmetry) and entropy (prime quantization errors) in the T³ elastic vacuum model. Space is primary: (2πθ)²Tr → Σ V_coh, <l>=π r_S/2 →1/8 from Santaló, heat kernel Vol/(4πt)^{3/2}. Equilibrium is defined by: χ_orb(S²(2,3,6))=0, maximization of APS η-invariant f_APS=2.366 (f_APS^{-1}=0.4225 fixed from M_Z=91.1876 GeV), Ramanujan |λ₁⁽ᵖ⁾| ≤2√p. Entropy is defined by: LQC instanton S_inst=50.534, β_*=S_inst/ln(M_P/k_min*)=0.3586943, S_coh=1101, N_max^coh=(S_coh β_*)^{1/β_*}=1.73×10⁷. Result 1: Π_reg = 8/(3 f_APS β_*) = 3.14216 = π within 0.018%, i.e. f_APS·β_*·π = 8/3. Result 2: N_max^coh = N_cross·R/π with N_cross=56.8×10⁶ gives R_exact=0.95849 vs observed G_thin/G_N=0.94 within 1.9%. Deviation measures physical imperfection K_χ and is not tuned. Failure at >5σ falsifies T³. This is Supplement 3 to:- Fundamental Foundations and Derivation of Model Parameters (Oct 2026)- Adelic Completion and Closure of the Window (0.5,0.60955) (Oct 2026)- Complete Entropy Model in T³ Elastic Vacuum (Nov 2026) All parameters: l_coh^eff=0.1136 μm, r_S=1.002 fm, L_bare=122 fm, N_cross=56.8M, N=931145=5·186229.