A general complex binary octic has exactly 76 decompositions as a sum of three fourth powers of quadratic forms, modulo permutations and independent fourth-root-of-unity rescalings. We give a geometric proof of this count, previously obtained numerically by Kowalczyk and Vill. Restriction to a conic realizes the problem as a projection of the degree-112 third secant variety of the quartic Veronese surface. Its scheme-theoretic base locus is a ribbon on a rational quartic, with nilpotent line bundle of degree -4. The resulting intersection correction is 112 - 64 + 28 = 76. The ordered affine Waring map consequently has degree 29184, and quotienting only by permutations gives degree 4864. This treats the perfect pair (k,d)=(4,2), not the general perfect-pair problem, real decompositions or closed-form construction of all summands. The package includes an English manuscript, source and exact symbolic checks. This is an AI-assisted, self-audited, unrefereed preprint; no independent review, proof-assistant formalization or absolute priority is claimed.
