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 (zetaᵣho2 (0) = 0), the triangular classes are invisible (chiᵣho2 = 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 zeta₃₄₋ₓ₀, ₑ₇₎ (1) for all five A5 irreps satisfy zetaᵣho (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 zetaᵣho3 (1) = 0. 17859 and zetaᵣho4 (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 pi²) log (pi²/ (4 phi) ) / Rₛigma⁴. The Euler number e cancels in the Galois pair (det' rho2 x det' rho3 = 1/ (2 phi) ), and the sign follows from the inequality pi² > 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) pi², 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 = e² 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/|pi₁ (P3) |, phi = chiᵣho2 (C5), e = 1/ (phi² 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.