A Sharp Linear Computability Gap for Direct Sums of Binary Cubics
Manuscript 21 October 2026 · Online 23 October 2026
Abstract
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.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Counterexample to additivity of linear computability
- Categories
- math.AC · math.AG
- Manuscript
- 21 October 2026
- Online release
- 23 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “A Sharp Linear Computability Gap for Direct Sums of Binary Cubics,” EulerSolve Research Papers, OWR-13494-006, 2026. https://doi.org/10.5281/zenodo.23586458.
BibTeX
@misc{ferudun2026owr13494006,
author = {Ferudun, Alper},
title = {A Sharp Linear Computability Gap for Direct Sums of Binary Cubics},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-13494-006/},
doi = {10.5281/zenodo.23586458},
note = {OWR-13494-006; unrefereed preprint}
}