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.

Let \(c=(63+8\sqrt{62})^{1/3}\), \(z=3/4-c/12-1/(12c)\), \(x=z/2\), and \(y=(1-z)(1-2z)\). The ten points are

\[(x,0),(1-y,0),(0,x),(1,y),(1-z,z),(z,1-z),(0,1-y),(1,1-x),(y,1),(1-x,1).\]

The parameter \(z\) is the unique real root of \(12z^3-27z^2+20z-4\), with \(z\approx0.3156111376086111\). Exact determinant reduction modulo that cubic checks all \(\binom{10}{3}=120\) triangle areas. Their minimum is

\[A=\frac{5z^2}{8}-\frac{z^3}{2}\approx0.04653741958254177256.\]

Sixteen triples attain \(A\). The remaining 104 are strictly larger. The next area is approximately 0.047806572540416697, attained by triples 134 and 568. This supplies the rigorous lower bound \(\Delta_{10}\ge A\).

Consider the ten points

\[(x,0),(1-y,0),(0,x),(1,y),(1-z,z),(z,1-z),(0,1-y),(1,1-x),(y,1),(1-x,1)\]

with \(0\le x\le y\le z\le 1/2\). This ansatz has diagonal-reflection and half-turn symmetry. It is a proper subclass of all ten-point configurations.

Three doubled triangle areas are

\[P=x(1-x-y),\qquad Q=y(1-2z),\qquad R=z(1-x+y)-y.\]

They come from triples 012, 145, and 058. The ordering makes all three nonnegative. Let \(t=\min(P,Q,R)\), let \(z_0\) be the unique real root of \(f(z)=12z^3-27z^2+20z-4\), and set \(t_0=(5z_0^2-4z_0^3)/4=2A\).

Assume \(t\ge t_0\). Then \(d=1-2z>0\), and \(Q\ge t\) gives \(L=t/d\le y\). Substituting \(L\) for \(y\) can only increase \(P\) and \(R\). The inequality \(R(x,L,z)\ge t\) gives

\[x\le U=1-\frac{t+(1-z)L}{z}.\]

The vertex of \(P(x,L)=x(1-x-L)\) is \((1-L)/2\), and

\[\frac{1-L}{2}-U=\frac{N}{2z(1-2z)},\qquad N=t(4-7z)+2z^2-z.\]

Since \(4-7z>0\), it is enough to check \(t=t_0\). The resulting quadratic in \(z\) has discriminant \(49t_0^2-18t_0+1<0\), using the exact brackets \(31/100<z_0<8/25\) and \(9/100<t_0<1/10\). Thus \(U\) lies at or before the vertex. From \(P(x,L)\ge t\) and \(x\le U\), we obtain \(P(U,L)\ge t\). Direct simplification gives

\[P(U,L)-t=\frac{tH}{z^2(1-2z)},\qquad H=(6z-4)t+2z^3-5z^2+2z.\]

Therefore

\[t\le h(z)=\frac{2z^3-5z^2+2z}{4-6z}.\]

Finally, \(h'(z)=-f(z)/(2(3z-2)^2)\). The cubic has one real root, so \(h\) reaches its unique maximum on \([0,1/2]\) at \(z_0\), and \(h(z_0)=t_0\). Hence every configuration in this family has a triangle of area at most \(A\). The parameters \(x=z_0/2\) and \(y=(1-z_0)(1-2z_0)\) attain \(A\), proving the family result.

The live canonical record was oriented before experimentation. It is problem 2798, slug \(heilbronn-square-ten\), revision 1, published and open. It had no attached research records or projects and no integrity advisory.

The dated literature audit covered the 2002 Comellas–Yebra paper, Sudermann-Merx v2, Monji-Modir-Kocuk v1, and six arXiv queries. Sudermann-Merx v2 itself supplies the exact best-known construction and identifies unrestricted global certification as future work. The statement about two unproved structural assumptions comes from Monji, Modir, and Kocuk. Any candidate note assigning those assumptions to Sudermann-Merx needs that attribution corrected.

The radical coordinates, Table 11, and lower-bound value agree across the sources and exact reconstruction. On p. 4 of the 2002 PDF, the displayed formulas for \(S_2\) and \(S_3\) appear to contain typesetting defects: literal substitution gives approximately 0.03479196 and -0.07965626 instead of the stated common area. Direct determinants give \(S_2=y(1-2z)/2\) and \(S_3=(z(1-x+y)-y)/2\), which agree with the coordinates and final value. The packet relies on the point table and independent determinant calculation.

A deterministic depth-first branch-and-bound used exact rational interval bounds and five witness triangles. Thresholds were tightened through 0.046539, 0.0465385, 0.046538, 0.04653775, and 0.0465376. The final completed run at 0.0465376 visited 649,645 nodes and closed every ordered box. This is 0.000000180417458... above the exact construction value.

At threshold 0.04653755, only 0.000000130417458... above the exact value, a 650,000-node cap left 35 boxes open at depth 64. The capped run is a budget-exhaustion trace. It supplies no counterexample. The coordinate ansatz also covers only a symmetric three-parameter family, so this route cannot certify the canonical unrestricted problem without a separate reduction theorem. The analytic argument in `heilbronn10-claim-symmetric-family-exact` settles the family exactly.

Use companion repository commit \(4e725664b35a6e5640876e56b1cd0ec3aeeef6e5\) and set \(n=10\) in \(optimization_models/heilbronn_final.py\). Seed the exact Comellas–Yebra coordinates and value as the incumbent. Retain the proved five-boundary-point canonicalization from Proposition 2. Run the full \(P_\Delta^*\) model over arbitrary ten-point configurations. Leave out the Monji-Modir-Kocuk two-points-per-edge conjecture and \(y_5\le1/2\) restriction.

A first budget is 24 wall-clock hours with at most 16 threads and 64 GiB RAM. Record Python, Gurobi, operating system, CPU, model parameters, deterministic seed, primal and dual bounds, node counts, logs, checkpoints, and the final solver status. Export each incumbent as exact decimal strings and save every certificate or box description needed for independent interval or rational checking.

A universal upper bound of 0.04654 is a useful intermediate result. A timeout or resource limit should become an explicit attempt record with its last certified upper bound. Keep the problem open until a universal exact or rigorous interval upper bound matches the exact construction value.

The script defines \(z\) as the isolated real root of the cubic, constructs the ten exact points, and checks coordinate containment and distinctness. It computes every triangle determinant, reduces it in \(\mathbb Q[z]/(f)\), proves all 120 areas are at least \(A\), and prints every exact row. It also checks five symbolic identities or inequalities used in the family upper-bound argument. The artifact checks the algebra behind the proposed family theorem. A human proof review remains appropriate.

The script forms all 120 signed determinant polynomials for the ten affine points. For each dyadic box in \([0,1/2]^3\), it computes exact termwise rational bounds. A box closes once one of five fixed witness determinants lies throughout \([-2T,2T]\), which proves that every configuration in that box has a triangle of area at most \(T\). Ordered-infeasible boxes are pruned exactly. At \(T=14543/312500=0.0465376\), the run closes the family after 649,645 nodes. The analytic family theorem is sharper; this artifact records an independent computational route and its scaling boundary.