From the flat band to the Dirac point: one mean-field gap ratio on the strip −1 < a < 2, the poles of two Mellin transforms, and the place of BCS (Paper P21)

For a density of states N(ξ) ∝ |ξ|^a near the Fermi level, the mean-field ratio Δ₀/T_c of a constant-gap superconductor at the weakest coupling that still pairs is a single analytic function r(a) = R(a)1/a, R(a) = 2ᵃ⁺¹ η(a) Γ(1 + a/2)/Γ(1 − a/2), on the whole strip −1 < a < 2. On the side 0 < a < 2 it is the threshold ratio of an earlier paper of this series; on the side −1 < a < 0, where pairing has no threshold, it is the weak-coupling ratio obtained in Γ–ζ form for van Hove singularities of order n, a = 1/n − 1, by López, Sánchez and Wang (2026). We give a Mellin proof that joins the two sides across the BCS point, r(0) = πe−γ. One expansion of the two gap equations holds on the whole strip; its two leading terms collide exactly at the three distinguished points: at a = −1 (both Mellin moments have a pole, the flat band, r = 2), at a = 0 (the logarithm of BCS), and at a = 2 (the gap moment has a pole, r → 0). We prove exact cutoff relations with error bounds, show that the ratio 2 at the flat edge holds for every fixed coupling, and prove with interval arithmetic that r decreases strictly from 2 to 0. Hence a measured mean-field ratio fixes the exponent a uniquely. Two final sections place the strip within the series, first classically (the Fermi factor 1 − 2¹⁻ᵃ, the Bose–Einstein pole and the BCS constants are the objects that appeared in thermal radiation, condensation and BCS theory) and then in a labelled framework-internal reading, in which the special constants of the strip arise at collisions or a cancellation in one expansion; a table sorts the constants of the series in the same way. All results are classical mathematics and physics; no result depends on the reading, and none contradicts the literature. The paper is written for physicists and mathematicians alike: each section opens with the classical reading, then states what the mathematics adds, and every result carries a status label (proved, proved computer-assisted, classical, classical in substance, numerical, negative, mean-field prediction, framework-internal, open). The companion code (Python 3 with mpmath and python-flint) contains the preregistered interval-arithmetic proof checks in two runs at 64 and 128 bit, the checks of the unified expansion, the poles and residues, the comparison with López, Sánchez and Wang, the constants table and the figure, together with the reference logs, the preregistration and a README that maps each result to its script and log line. The LaTeX source compiles in Overleaf as is (pdfLaTeX + BibTeX).