Derivation of the Standard Model, Nuclear Physics, Flavor Mixing, and New Particle Predictions from a Pre-Geometric Trit Law

From the single pre-geometric axiom T = {+1, 0ᴪ, −1} (the Trit Law), we derive 85 quantities spanning particle physics, nuclear physics, astrophysics, and collider phenomenology, with zero free parameters. Version 31 extends v29 with the gauge sector completion. New results: (i) M_Trit = M_Z × exp[(16π − 1/α_s(M_Z))/(b₃/2π)] = 1.781e+18 GeV ≈ 0.15 M_{Pl}, the Trit compactification scale derived from the QCD running; (ii) A 128.7% Weinberg gap: SM running gives sin²θ_W = 0.615 at M_Trit but the Trit requires 0.269 = 1/(1+e); (iii) A theorem: Δb₁ > 0 is impossible from ordinary 4D matter, requiring extra-dimensional origin for new physics entering at M_LQ; (iv) Exact effective beta functions above M_LQ: b₂_eff = 19/18 = b₂_SM/|T|, b₁_eff = 19/(9π), ratio b₁/b₂ = 2/π; (v) sin²θ_W(M_Z) = 0.23122 derived from Trit exact boundary conditions (0.012%), making the Weinberg angle a prediction rather than an input; (vii) The Higgs quartic boundary condition λ_H(M_{Trit}) = −α_Y(M_{Trit}) = −(1+e)/(16πe²) = −0.01001, matching the two-loop SM literature value −(0.010±0.006) for M_H = 125.25 GeV; (vi) Two new particles predicted at M_LQ = 1.20 TeV: the previously-derived leptoquark and a neutral Z′ with g_Z′ = √(3π/35) = 0.519, Γ = 85.7 GeV, BR(e⁺e⁻) = 12.5%, BR(WW) = 0 exactly. Both resonances appear at the same mass, providing a unique simultaneous two-channel signature at LHC Run 3.