Fundamental foundations and derivation of model parameters: tensioned vacuum, coherent volume and Y-junction mechanism

We define the microscopic foundation of the T^3 model and derive all effective parameters from first principles: noncommutative geometry, LQC bounce and fractal network theory. The zero state is a tensioned empty vacuum with a single 1D filament of zero thickness and zero energy, thickening into a tube r_S=1 fm. The filament carries unit norm $|T|=1$ from vortex topology, bare tension ~10³⁸ GeV^2 renormalized via ZT^3~10⁻³⁹ to $0.18$ GeV^2, and Berry phase coupling ψρ_Tln(G_t/G'_t)ψ to fermions. Qμν=1/3δμν from coarse-graining. The first grain is set by LQC instanton Sᵢₙₛₜ=50.38, χLQG=e^{-Sᵢₙₛₜ}=1.31×10⁻²² and spectral dimension d_s=11/3 from rigorous heat-kernel ln Kᵢₙₜ=A-dᵢₙₜ/2ln t, dᵢₙₜ=0.666±0.008, giving lcohᵇᵃʳᵉ=0.122 pm, lcohᵉᶠᶠ=0.1136\,μ m =113600000 fm, Vcoh=(lcohᵉᶠᶠ)^3 and filling fbare=2.09×10⁻⁴, feff=2.43×10⁻¹⁶. Quantization: Ncross=Leff/d=56800000, Nlength=931145=5·186229 from APS ηbare=-2/9, ηfull∈[1.59,2.366], Δ L∈[221.09,315.76] fm. Theorem 1: orbifold base S^2(2,3,6) has χorb=0 canceling 1/√t divergence, yielding regularized ζ_M and self-adjoint Δ_M. Ramanujan |λ_1⁽ᵖ⁾|≤2√p for Γ(N)_p, c_∞=π-s/2Γ(s/2). Adelic MA=M×∏_p(Γ(N)_p) with cuspidal filter and l(γ_p)=ln p gives Conjecture Speccusp(MA)=Spec(ξ) equivalent to RH, with complementary series λ<1/4 as Cheeger bottlenecks (super-voids, wDE(z)). Physical normalization 1/Vcoh yields 0.027ρcrit for harmonic $n=3$ and 0.26ρcrit for n≈9. Stability Eₜₒₜ(n)=c n^2-ln|[n]_q|, $c=0.912$ GeV gives only n≤3 elementary. We derive 10¹⁰² grains via Y-junction cross-link instability: single infinite filament with winding N~10¹⁰² unfolds when curvature exceeds threshold, 3.48<EY<7.78 GeV, Γ t~235, giving 78 volume e-folds to $L=50.4$-- $75$ Gpc. M_Z=91.2 GeV, sin^2θ_W=0.23121 from δσ^2=0.88, $b=2.047$, kbare=85, fAPS⁻¹=0.4225.