A Sharp Linear Computability Gap for Direct Sums of Binary Cubics
Alper Ferudun, Mercury Software GmbH

Over an algebraically closed field of characteristic zero, the sum F_m
of m disjoint binary cubics x_i y_i^2 has Waring rank 3m, but the largest
linear-computability colon bound is 2m+1. An exact inverse-system
calculation covers every linear ideal and identifies the unique
maximizer. In particular, xy^2+zw^2 is a complete counterexample to
closure under disjoint sums of linearly computable forms, answering
the question on page 503 of Oberwolfach Report 9/2015.

The rank formula is known and credited. The paper includes its elementary
proof and a standard-library exact checker. The latter passes 1264
quotient-length calculations on 629 distinct sampled subspaces and
scope 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-verification
claim. OpenAI Codex assisted with research and preparation.
