A Geometric Count of Fourth-Power Decompositions of Binary Octics

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. This preprint gives a geometric proof of the 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 is a ribbon on a rational quartic, with nilpotent line bundle of degree -4. The intersection correction is 112 - 64 + 28 = 76. The ordered affine map has degree 29184, and quotienting only by permutations gives 4864.

The complete theorem treats the perfect pair (k,d)=(4,2), not the full perfect-pair Problem E represented by AMR-014-0007 in UnsolvedMath. Real decompositions and closed-form construction of every summand are outside its scope. The numerical value and conic-restriction viewpoint are credited to prior work; no absolute priority is claimed.

The package contains the six-page English manuscript, source and a portable exact symbolic checker. AI assistance was used for literature navigation, calculations, proof development and drafting. The author takes responsibility for the claims. This is a self-audited, unrefereed preprint, not independently reviewed or proof-assistant formalized.