An Exact Computer-Assisted Proof for Ten Points in Heilbronn's Triangle Problem

For ten points in the unit square, let Delta_10 be the largest possible minimum area among their 120 triangles. This preprint gives an exact computer-assisted proof that Delta_10 = A0, where A0 is the unique real root of 3456 A0^3 - 1764 A0^2 + 268 A0 - 9 = 0, approximately 0.04653741958254177256. The attaining arrangement is the known Comellas-Yebra construction. A written affine normalization reduces the unrestricted problem to three compact fourteen-parameter families. Complete exact certificate replays establish M4, M5 <= 46537/1000000 < A0 and M6 <= A0. The three continuous outer covers have 240, 838 and 2607 closed leaves, respectively, and no unresolved leaves. The M6 argument uses a fully stated analytic theorem for sixteen signed areas and verified projection and positive linear-combination rules. The verification uses only standard-library Python exact arithmetic and imports the published eight-point theorem and a weakened seven-point bound. The result establishes the optimum value; it does not classify every maximizer. The deposit includes the article, editable source, all three certificate datasets, exact verification code, archive checksums and fresh full replay receipts. The three certificate ZIPs are necessary components of the computer-assisted proof. No private conversation or account-specific link is required for verification. Conceptual origin and AI assistance. The initial idea and conceptual starting point of this investigation are based on the framework of the original geometric paper by Christos Panorias and Kostandina Theodhori, Canonical Parametric Representation of Plane Euclidean Geometry [3]. This attribution concerns the conceptual origin and methodological organization of the research; every mathematical premise required by the present proofs is stated explicitly here. OpenAI ChatGPT/Codex assisted verification of calculations, editorial polishing and selected, delimited arguments during proof development, including substantive mathematical reasoning in those arguments. Christos Panorias is the sole named author and retains responsibility for reviewing, adopting and submitting the mathematical claims and final text. Copyright 2026 Christos Panorias. All rights reserved. Preprint; no independent peer-review completion is claimed. References [1] F. Comellas and J. L. A. Yebra (2002), New lower bounds for Heilbronn numbers, Electronic Journal of Combinatorics 9(1), R6. https://doi.org/10.37236/1623 [2] N. Sudermann-Merx (2026), From Global Optimization to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Nonlinear Optimization, arXiv:2603.11107v3, 27 October 2026. https://arxiv.org/abs/2603.11107v3 [3] C. Panorias and K. Theodhori (2026), Canonical Parametric Representation of Plane Euclidean Geometry. Zenodo. https://doi.org/10.5281/zenodo.23837735 [4] G. Rote (2019), The Largest Contained Quadrilateral and the Smallest Enclosing Parallelogram of a Convex Polygon, author's manuscript, 4 July 2019. [5] L. Dehbi and Z. Zeng (2022), Heilbronn's problem of eight points in the square, Journal of Systems Science and Complexity 35, 2452-2480. https://doi.org/10.1007/s11424-022-1220-7 [6] Z. Zeng and L. Chen (2011), On the Heilbronn optimal configuration of seven points in the square, LNCS 6301, 196-224. https://doi.org/10.1007/978-3-642-21046-4_11