The fine-structure constant as the spectral transfer function of the Poincaré homology sphere

We present a structural reading of the alpha formula established in the companion paper series on the SO(3,3) matrix model compactified on the Poincare homology sphere P3 = S3/2I*. The inverse fine-structure constant alpha⁻¹, expressed as a Laurent series in epsilon = 1/|2I*| = 1/120 with coefficients in Z[phi, e, 1/120], admits a natural decomposition into a ground term 360/phi^2 that is pure group theory (no spectral determinant, no analytic continuation) and a spectral correction that arises from the Mellin transforms of the rho_k-twisted heat kernels on P3. The ground term is the golden angle — the leading-order contribution from the character value phi = chirho_2(C_5) and the fundamental-group order |2I*| = 120, with no reference to the spectrum of the Laplacian. The spectral correction, contained in Blocks II and III of the formula, introduces the Euler number e (from the Galois-pair determinants) and pi (from the spherical-sector determinants) through the analytic operation zeta'_rho(0), which is the Mellin transform of the heat kernel evaluated at s = 0. In this reading, alpha⁻¹ is the spectral transfer function of P3: a single real number encoding how the local geometry (heat diffusion, eigenvalues) passes through the Mellin contour to become a global topological invariant. The Bromwich inversion, which translates the spectral data back into the time domain, provides the inverse direction. The two directions — Mellin (local to global) and Bromwich (global to local) — correspond to the two faces of the SO(3,3) framework: the compact face SO(6) (gauge symmetry, quantum mechanics) and the non-compact face SO(3,3) (curvature, gravity). The fine-structure constant sits at the interface. No new computations are performed; all numerical results are those of the companion papers.