A Counterexample to a Linear-Junta Inverse Theorem for Spherical Triangles
Manuscript 12 October 2026 · Online 12 October 2026
Abstract
We construct smooth, real, mean-zero, 1-bounded functions on high-dimensional spheres whose average product over zero-sum unit triangles tends to 7/16384, while their correlation with every function of m linear coordinates is O(sqrt(m/d)) when m=o(d). The bound is uniform over all choices of coordinates and all test functions of L2 norm at most one. The construction is a centered cosine of a balanced, rescaled quadratic form. A conditional coupling proves the projection estimate, and a multivariate Gaussian limit gives the positive triangle average and a nonzero limiting variance. This supplies a complete counterexample to the dimension-free quantitative interpretation of Dmitrii Zakharov's spherical-triangle inverse question, AIM Problem 2.4 / UnsolvedMath AIM-COMBINATORICS-0007. It does not settle the related zero-sum-free hemisphere conjecture or the planar and finite-field analogues. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete quantitative counterexample
- Categories
- math.CO · math.PR · math.CA
- Manuscript
- 12 October 2026
- Online release
- 12 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 Counterexample to a Linear-Junta Inverse Theorem for Spherical Triangles,” EulerSolve Research Papers, AIM-COMBINATORICS-0007, 2026. https://doi.org/10.5281/zenodo.23314092.
BibTeX
@misc{ferudun2026sphericaltriangles,
author = {Ferudun, Alper},
title = {A Counterexample to a Linear-Junta Inverse Theorem for Spherical Triangles},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-combinatorics-0007/},
doi = {10.5281/zenodo.23314092},
note = {AIM-COMBINATORICS-0007; unrefereed preprint}
}