Fundamental Foundations and Derivation of Model Parameters: Tensioned Vacuum, Coherent Volume and Y-Junction Mechanism

We define the initial state and microscopic foundation of the T^3 vacuum elasticity model and derive its effective parameters from noncommutative geometry, Loop Quantum Cosmology (LQC) and fractal network theory. The zero state is a tensioned empty vacuum: a single 1D filament of zero thickness and zero energy that acquires finite core radius r_S = 1 fm. Unit norm |T|=1 follows from vortex topology. Bare tension ∼1e38 GeV^2 is renormalized by dressing factor Z_T3 ∼1e-39 to 0.18 GeV^2. Fermion coupling appears as Berry phase. From the LQC Euclidean bounce we obtain S_inst = 50.38, chi_LQG = exp(-S_inst) = 1.31e-22. Bare coherence length l_coh^bare = 0.122 pm. Spectral dimension d_s = 11/3 from rigorous heat-kernel fit ln K_int = A - d_int/2 ln t, d_int = 0.666 +-0.008, gives renormalized grain l_coh^eff = 0.1136 um, V_coh = (l_coh^eff)^3, filling factors f_bare = 2.09e-4, f_eff = 2.43e-16. Quantization: N_cross = L_eff / d = 56,800,000, N_length = 931145 = 5 * 186229 from APS eta-invariant eta_bare = -2/9, eta_full in [1.59, 2.366], Delta L in [221.09, 315.76] fm. Theorem 1: orbifold base S^2(2,3,6) with chi_orb = 0 cancels 1/sqrt(t) divergence in Tr exp(-t Delta_M), yielding regularized zeta_M and self-adjoint Delta_M with APS boundary. Ramanujan bound |lambda_1(p)| <= 2 sqrt(p) for Gamma(N)\T_p. Adelic completion M_A = M x prod_p (Gamma(N)\T_p) with cuspidal filter and length spectrum l(gamma_p) = ln p leads to conjecture Spec_cusp(M_A) = Spec(xi), equivalent to Riemann Hypothesis. Complementary series lambda < 1/4 corresponds to Cheeger bottlenecks, super-voids and w_DE(z). Physical normalization 1/V_coh yields 0.027 rho_crit for harmonic n=3 and 0.26 rho_crit for n∼9. Stability E_tot(n) = c n^2 - ln|_q|, c = 0.912 GeV, selects only n <=3 as elementary. 1e102 grains are derived via Y-junction cross-link instability: single infinite filament with winding N∼1e102 unfolds when curvature exceeds threshold, 3.48 < E_Y < 7.78 GeV, Gamma t ∼235, giving 78 volume e-folds to L = 50.4-75 Gpc. M_Z = 91.2 GeV, sin^2 theta_W = 0.23121 from = 0.88, b = 2.047, k_bare = 85, f_APS^-1 = 0.4225. The scale l_coh^eff = 0.1136 um gives falsifiable signatures: deviation from 1/r^2 below micron, Casimir correction, PdH_x NMR shift, GW dispersion, and T^3 topology. Same Vol_coh = (0.1136 um)^3 yields G_pred = 6.99e-39 GeV^-2 vs G_N = 6.7088e-39 GeV^-2 (4.2%), M_Z,phys = 91.2 GeV (0%) and M_W,phys = 80.0 GeV vs 80.379 GeV (0.5%) with no free parameters.