Mathematical companion to the T3 topological model: Seifert fibration, Goldman bracket and identification of u(1), su(2), su(3)

This is a rigorous, self-contained mathematical paper on the flat 3-torus T³ = R³ / L Z³ and the trefoil complement. It provides the exact mathematical building blocks of the model, without reference to physical parameter closure. 1. Free Laplacian on T³. Spectral degeneracy g(m) = r3(m) = number of representations of m as sum of three squares. By Legendre's Three-Squares Theorem (1798), r3(m)=0 iff m=4^a(8b+7). Hence exact spectral gaps at 7,15,23,28,31,... 2. Topology of M = T³ \\ N(T(2,3)). M is Seifert fibered over orbifold S²(2,3,6) with exceptional fibers (2,1),(3,1) and regular fiber h. Proof of H1(M)=Z⁴ via Mayer-Vietoris, vs H1=Z for S³ complement. Character variety Hom(pi1(M),SU(N))/Ad. For N=3 abelian part is T⁸/W, dimension 8 (8 gluons). Tangent T_ρM = H¹(M; ad_ρ). 3. Non-Abelian holonomies and q-integers. _q = sin(n theta/2)/sin(theta/2) with effective level k_eff(M_Z)=4.263, q=0.537+0.843i, _q=1.7536=2cos(theta_W), sin²theta_W=0.23121. 4. Dedekind sums and APS eta-invariant. s(1,2)=0, s(1,3)=1/18, bare eta=-2/9, full eta in [1.59,2.366] maximal at trivial flat connection. 5. Goldman bracket. f_gamma(ρ)=Tr ρ(gamma), {f_gamma,f_gamma'} = sum_p epsilon(p) f_{gamma_p∘gamma'_p}. Skew-symmetry from epsilon, Jacobi from dω_ABG=0 for Atiyah-Bott-Goldman form. 6. Total Hamiltonian and gauge algebra emergence. E_tot(n)=c n² - ln|*q|, c=0.912 GeV. E(2)=3.087, E(3)=7.478, E(4)=13.959 > 2E(2)=6.174, so n>=4 decays 4→2+2. Stable n≤3 have dimensions 1,3,8 via End(V_j): g0=End(V0)^ah=u(1) dim1, g*{1/2}=su(2) dim3, g1=su(3) dim8. Cartan-Weyl closure from Goldman and SU(3) trace identity.