# P2798: Exact ten-point Heilbronn number in the unit square

- ID: `P2798`
- Reference: `heilbronn-square-ten`
- Page: https://theoremdb.org/statements/P2798
- Record maturity: Reviewed problem with recorded work

## Problem

For ten distinct points \(P\subset[0,1]^2\), let \(a(P)\) be the smallest Euclidean area of a triangle spanned by three points of P. Determine \(\Delta_{10}=\max_{|P|=10}a(P)\).

### Problem setup

- **Remark.** Triangles are determined by unordered triples of distinct points, and collinear triples have area zero.
- **Remark.** The boundary of the unit square is allowed.
- **Definition.** A configuration is optimal when its smallest triangle area equals \(\Delta_{10}\).

### What counts as a solution

- Give an exact or rigorously interval-certified ten-point configuration attaining value A, and prove that every ten-point configuration in the unit square spans a triangle of area at most A.

## Status

The exact Comellas–Yebra construction proves Δ₁₀ ≥ 0.04653741958254177256.... No checked source proves a matching unrestricted upper bound, so the exact value of Δ₁₀ remains open. [1](#reference-1) [4](#reference-4) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (The best-known ten-point area is 0.0465374195825..., with global optimality open).** The exact Comellas–Yebra construction proves Δ₁₀ ≥ 0.04653741958254177256.... No checked source proves a matching unrestricted upper bound, so the exact value of Δ₁₀ remains open.

Sudermann-Merx, arXiv:2603.11107v2, gives globally certified solutions for \(n\le9\). Section 7 says that extending certification to \(n=10\) remains open and will require substantial computation. Appendix A, Table 11 gives the exact Comellas–Yebra ten-point construction and identifies it as best known.

Monji, Modir, and Kocuk, arXiv:2512.14505v1, also leave \(n=10\) open. Their reported ten-point run uses a restricted model with two unproved assumptions: exactly two points on each edge and \(y_5\le1/2\). The solver reached no matching upper bound within one day.

The sources therefore support an exact best-known construction and an open unrestricted upper-bound problem. The two structural assumptions belong to the Monji-Modir-Kocuk restricted computation.

### Background and intake notes

This is the first small unit-square instance beyond the current certified frontier. Local optima, boundary patterns, and branch-and-bound boxes can be reused independently.

- Original intake status: UNKNOWN: Current 2026 work certifies the unit-square problem through nine points; the ten-point best configuration still depends on unproved structural restrictions.
- 2026-07-27: An eight-point target was discarded after current papers were found to certify it. The March 2026 exact-coordinate paper proves global optimality through n=9 and states that its n=10 result uses two unproved structural conjectures.
- 2026-07-27: No duplicate ten-point Heilbronn target was found in TheoremDB candidate or live prospecting data.
- A useful record should separate coordinate lower bounds, interval-certified triangle areas, structural lemmas, and the remaining global branch boxes.

- Recorded example: The ten points \((i/9,(i/9)^2)\) for \(0\le i\le9\) lie in the square and have no collinear triple.

### Other known results

- **Computation 1** (reproduced): Exact algebra checks all 120 triangles in the ten-point construction. Sixteen attain A = 5z²/8 - z³/2, and the other 104 have strictly larger area. [1](#reference-1)
- **Claim 2** (reported): Within the ordered boundary family with parameters 0 ≤ x ≤ y ≤ z ≤ 1/2, exact algebra gives maximum minimum triangle area A = 5z₀²/8 - z₀³/2, where 12z₀³ - 27z₀² + 20z₀ - 4 = 0. The unrestricted ten-point upper bound remains open.

### Prior approaches

- **Route 1** (supported): The audit fixes the attribution boundary, confirms the exact construction, finds no unrestricted n = 10 certificate, and records one apparent display defect in the 2002 derivation. [4](#reference-4)
- **Route 2** (inconclusive): The exact cover succeeds 1.8 × 10⁻⁷ above the construction value. A run 1.3 × 10⁻⁷ above it leaves 35 boxes after 650,000 nodes, exposing rapid scaling near the optimum.

### Open directions

- **Route 3** (reported): Pin the public model and exact incumbent, retain only proved symmetry reductions, and export enough solver state for independent interval or rational checking. [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (reproduced): A self-contained SymPy replay reconstructs the algebraic point set, reduces every triangle area modulo the cubic, and checks the identities and sign brackets used by the family argument.
- **Artifact 2** (reproduced): A standard-library branch-and-bound closes the whole ordered parameter cube at threshold 0.0465376 using five witness triangles and exact dyadic rational interval arithmetic.

### Computational notes

- Exact determinant algebra for the displayed parabolic configuration gives minimum triangle area 1/729, attained by consecutive parameter values. This is a baseline lower bound only.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `heilbronn-square-ten`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1 entry has incomplete source metadata. Each affected row names the fields that still need editorial review.

1. <a id="reference-1"></a>Nathan Sudermann-Merx, “From Computational Certification to Exact Coordinates: Heilbronn's Triangle Problem on the Unit Square Using Mixed-Integer Optimization”. arXiv:2603.11107 (2026). Sudermann-Merx, arXiv:2603.11107v2, Section 1.4 on p. 3, Section 7 on p. 17, and Appendix A/Table 11 on p. 19; Monji-Modir-Kocuk, arXiv:2512.14505v1, Conjecture 1 and final n=10 experiment https://arxiv.org/abs/2603.11107
   - Also cited at Exact reconstruction and 120-triangle certificate in heilbronn10-artifact-exact-certificate, executed 2026-07-28 UTC
   - preprint; reference source; arXiv source revision v2; checked 2026-07-28
   - Source use: citation_only
   - Certifies the Heilbronn frontier through nine points and reconstructs the ten-point incumbent without an unrestricted optimality proof.
   - For Exact ten-point Heilbronn number in the unit square: The exact Comellas–Yebra construction proves Δ₁₀ ≥ 0.04653741958254177256.... No checked source proves a matching unrestricted upper bound, so the exact value of Δ₁₀ remains open.
2. <a id="reference-2"></a>Amirhossein Monji, Amirali Modir, and Burak Kocuk, “Solving the Heilbronn Triangle Problem using Global Optimization Methods”. arXiv:2512.14505 (2025). Conjecture 1 and the n=10 results. https://arxiv.org/abs/2512.14505
   - preprint; reference source; arXiv:2512.14505v1; checked 2026-07-28
   - Source use: citation_only
   - Provides global-optimization evidence for the ten-point Heilbronn problem under stated structural restrictions.
3. <a id="reference-3"></a>Nathan Sudermann-Merx, heilbronn, companion optimization models for From Computational Certification to Exact Coordinates, GitHub commit 4e725664b35a6e5640876e56b1cd0ec3aeeef6e5 (2026). README, optimization model, and exact incumbent at commit 4e725664b35a6e5640876e56b1cd0ec3aeeef6e5 https://github.com/spiralulam/heilbronn/tree/4e725664b35a6e5640876e56b1cd0ec3aeeef6e5
   - Also cited at optimization_models/heilbronn_final.py and the n=10 configuration
   - software; software source; commit 4e725664b35a6e5640876e56b1cd0ec3aeeef6e5; checked 2026-07-28
   - Source use: citation_only
   - Reused material: README, optimization model, and exact incumbent at commit 4e725664b35a6e5640876e56b1cd0ec3aeeef6e5
   - Reuse basis: fair_use_reviewed; rights holder: Nathan Sudermann-Merx; checked 2026-08-01; by Philip Weiss, TheoremDB staff
   - Required attribution: Nathan Sudermann-Merx, Heilbronn’s Triangle Problem, spiralulam/heilbronn GitHub repository, commit 4e725664b35a6e5640876e56b1cd0ec3aeeef6e5, March 12, 2026.
   - Contains the pinned global-optimization model proposed for an unrestricted ten-point search.
   - Provides the pinned unrestricted optimization model proposed for the next global certificate run.
4. <a id="reference-4"></a>Francesc Comellas and J. Luis A. Yebra, New Lower Bounds for Heilbronn Numbers, Electronic Journal of Combinatorics 9(1) (2002), R6 https://doi.org/10.37236/1623
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source metadata incomplete: exact locator.
   - Source use: citation_only
   - Supplies the exact ten-point Heilbronn construction that gives the current checked lower bound.