A Sharp Linear Computability Gap for Direct Sums of Binary Cubics

A Sharp Linear Computability Gap for Direct Sums of Binary CubicsAlper Ferudun, Mercury Software GmbH Over an algebraically closed field of characteristic zero, the sum F_mof m disjoint binary cubics x_i y_i^2 has Waring rank 3m, but the largestlinear-computability colon bound is 2m+1. An exact inverse-systemcalculation covers every linear ideal and identifies the uniquemaximizer. In particular, xy^2+zw^2 is a complete counterexample toclosure under disjoint sums of linearly computable forms, answeringthe question on page 503 of Oberwolfach Report 9/2015. The rank formula is known and credited. The paper includes its elementaryproof and a standard-library exact checker. The latter passes 1264quotient-length calculations on 629 distinct sampled subspaces andscope controls; the universal statement is proved analytically.General linear elements and degree-one-generated ideals are essential. English unrefereed preprint, CC BY 4.0, 21 October 2026.No absolute priority, independent peer-review or formal-verificationclaim. OpenAI Codex assisted with research and preparation.