The fine-structure constant as a topological invariant of the Poincaré homology sphere

We complete the companion paper series [1-5] on the SO(3,3) matrix model compactified on the Poincare homology sphere P3 = S3/2I* with six results that close the remaining open questions. (i) The spectral correction delta = log(2/3)/30, previously a Tier B numerical identification [2], receives a structural derivation from the orbifold one-loop structure: the denominator 30 = |C2| is the orbit-stabiliser ratio |G|/|Z(g)| = 120/4 for the order-4 conjugacy class, arising from the centraliser weight |Z(g)|/|G| in the orbifold partition function. The C2 class dominates because the identity sector is blocked (zeta_rho2(0) = 0), the triangular classes are invisible (chi_rho2 = 0), and the pentagonal classes contribute irrational terms whose net effect is suppressed by the Galois structure of Q(sqrt 5). This provides the structural explanation for the single open analytic step identified in [2]. (ii) The spectral zeta values zetaDelta,rho(1) for all five A5 irreps satisfy zeta_rho(1) = dim(rho) x c with universal constant c = 0.05953, a consequence of the homogeneity of P3 and the flatness of the rho-twisted bundles. The predictions zeta_rho3(1) = 0.17859 and zeta_rho4(1) = 0.23812 follow without computation. (iii) The chiral asymmetry of the Pati-Salam fermion content is derived from the character table of 2I*: the tensor product rho2 x rho6 = rho6 + rho8 (dim 6 = 2 + 4) splits into an SU(2)_L doublet and a Pati-Salam quartet, while rho2 x rho7 = rho9 (dim 6) remains irreducible. The lightest Kaluza-Klein level (k = 1) yields exactly one Pati-Salam family (4,2,1)_L in three generations. (iv) The one-loop vacuum energy, summed over all five irreps with the Lichnerowicz exponent -1/14 from [5], takes the closed form Lambda = [60/(14 pi^2)] log(pi^2/(4 phi)) / R_sigma^4. The Euler number e cancels in the Galois pair (det' rho2 x det' rho3 = 1/(2 phi)), and the sign follows from the inequality pi^2 > 4 phi (de Sitter). (v) The open coefficient c4 admits the structural identification c4 = dim(rho2) x (det' rho5)^2 = 3(pi/2)^2 = (3/4) pi^2, connecting the generation count, the rho5 determinant, and the Lichnerowicz curvature coupling R/4 = 3/2. This is consistent with the two-sector selection c4 containing pi from [5] and lies within CODATA 2022 uncertainty. We note that the pre-registered candidate c4 = e^2 from [2] is numerically closer to the CODATA central value by a factor of 60; the two candidates are distinguishable at 10⁻¹³ precision. (vi) Every ingredient of the alpha formula traces to a spectral-geometric invariant of P3: epsilon = 1/|pi_1(P3)|, phi = chi_rho2(C5), e = 1/(phi^2 det' rho2), pi = det' rho4. Since P3 admits a unique geometric structure (spherical, by the Thurston-Perelman geometrisation theorem), these spectral invariants are determined by the topology alone. The fine-structure constant is a topological invariant of the compactification manifold. All computations were performed in Python using mpmath at 50-digit precision.