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Heilbronn's triangle problem: a power-saving lower bound n^(-2+eta), refuting the almost n^-2 upper bound

Theorem 1.1: absolute constants $\eta,c_1>0$ and $n_0$ with $\Delta(n)\ge c_1n^{-2+\eta}$ for every $n\ge n_0$ . Hence the almost $n^{-2}$ upper-bound formulation is false. The construction samples integer columns in a box with finite-field norm congruence conditions and deletes the few small triangles; $\eta$ is explicit but extremely small. It does not determine the order of $\Delta(n)$ : the gap to the upper bound $n^{-7/6+o(1)}$ remains, and no conjecture for the true exponent is made.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Heilbronn's triangle problem: a power-saving lower bound n^(-2+eta), refuting the almost n^-2 upper bound · Heilbronn triangle power lower bound

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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VibeMathed record: Heilbronn's triangle problem: a power-saving lower bound n^(-2+eta), refuting the almost n^-2 upper bound

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Heilbronn's problem and the almost $n^{-2}$ formulation (1.1); it gives $\Delta(n)\ge c_1n^{-2+\eta}$ for all large $n$ , with an explicit but tiny $\eta$ . The proof was not refereed. The challenge lean/ComparatorChallenges/HeilbronnTriangle.json (solution module OAI.Geometry.HeilbronnTriangle.Main, present at the pinned commit) is not in the formalization catalogue; its statement HeilbronnTriangle.lean was read here. heilbronn_power_lower_bound gives an explicit $\eta=1/(10^5K)>0$ and point sets of sizes $n_j\to\infty$ in $[0,1]^2$ with every triangle of area at least $n_j^{-2+\eta}$ ; almost_n_minus_two_refuted states that $\Delta(n)\le Cn^{-2+\eta/2}$ fails for all large $n$ . This states the refutation; the 'every sufficiently large $n$ ' form is broader than the Lean. Not rebuilt here. The manuscript notes two earlier preprints (Ellmann, Agama) claiming stronger power bounds, with gaps it identifies.