The second polarization: the covariant polar graviton tower of a black-hole universe, and the constant that closes it

The one-loop weight of the no-boundary state of the black-hole universe (S¹×S² topology, complex Schwarzschild–de Sitter caps) was computed in Paper II of this series for every propagating sector except one: the second (polar, "even") graviton polarization away from the Nariai slice, where the tower of Zerilli master operators was shown to have no ζ-function — its rows carry a λ⁻²ln λ² term, the sum over ℓ has a pole, and the determinant is an O(1) unknown in any master-variable scheme. This paper builds the instrument that computes the sector without master variables: the scalar-type Kodama–Ishibashi reduction of the gauge-fixed (de Donder) graviton operator itself, seven metric and three ghost components per ℓ, as a coupled two-dimensional system whose ζ-regularized determinant is a matrix Gel'fand–Yaglom mode sum on a conformally flat disc. The instrument is gated end to end (exact Einstein-space identities, the Volkov–Wipf S²×S² spectrum, the exact decomposition into the known spin rows on the slice, all at 10⁻¹⁰ or better), and three things come out. First, the covariant polar tower is log-free: at (u,v) = (3, 0.8) its rows fit the local large-λ series to 2.6×10⁻⁹ with the structural K = 3 coefficient reproduced to 4×10⁻⁵, and a Zerilli-sized logarithm is excluded by a factor of forty — the second polarization has a ζ-function. Second, the logarithm sits, with the predicted coefficient −A and the opposite (ln L)² growth, in the Jacobian between the covariant and the master-variable descriptions, which is a determinant ratio and not a measure factor; the pole is a property of the bounded cap, where the two parities are not isospectral. Third, the two standard mixed boundary-condition sets change the rows by a local series whose K ≤ 3 coefficients are the mixed-boundary heat-kernel constants of the string frame, including two structures forced by the splitting of helicity pairs by component projectors — Vassilevich's χ_{:a}χ_{:a} term and the Branson–Gilkey–Kirsten–Vassilevich cross-set E² rule (w₁ = −180, w₂ = 180) — each confirmed exactly on the flat disc. Along the way the mode-sum scheme's last empirical input, the per-spin scheme constant, acquired a closed law, C(s) = s²∫₀¹(c² − 1) d ln ρ with c = a′/N: the Born integral of the spin-curvature potential the scheme omits, equivalently a Liouville functional of the flat-form conformal factor, confirmed on twelve caps to 10⁻⁶. The paper is written as a standalone, step-by-step account for a wider readership, with plain-language summaries at the end of each section; every detour — a falsified conjecture, a "noise" that was an exact tail, a per-row residual that was three omitted heat-kernel structures — is retained (fifteen corrections, Appendix A), and every row is tabulated (Appendix B). Scripts, caches, dated logs and figures are in version 5 of the working record (below). A prior-art check (memo in the record) found no previous computation of the polar graviton tower on a bounded Schwarzschild–de Sitter cap, no previous statement of the Zerilli tower's pole, and no previous numerical test of the BGKV interaction terms; the author would be glad to be corrected.