GPT 6 Astra Ultra attempt on Math-719

Erdős Problem #213: independent continuation

Date: 23 September 2026. Source versions read in full: 213.tex in gpt_pro_5.4.

1) OBJECTIVE AND SOURCE AUDIT

The target is an arbitrarily large finite planar set with integral pairwise distances, no three collinear and no four concyclic. I read the supplied characteristic proof and seven-point coordinates. Those are established background, not a new construction. I independently executed all of its finite checks, then derived an exact bounded extension search.

2) EXACT VERIFICATION

Represent each source coordinate as $(x_i,t_i\sqrt{2002})$, with rational $x_i,t_i$. Exact rational arithmetic gives the 21 distances, in lexicographic pair order, \[ \begin{split} &22270,22098,16637,9248,8908,8636,21488,11397,15138,20698,13746,\\ &10795,14450,13430,20066,7395,11135,11049,5780,5916,10744. \end{split} \] All 35 determinants with rows $(x_i,t_i,1)$ are nonzero. All 35 determinants with rows $(x_i,t_i,x_i^2+2002t_i^2,1)$ are nonzero. Multiplying the second column by $\sqrt{2002}$ recovers the geometric determinants, so these exact checks certify both general-position conditions. The companion script verification/root_checks.py contains the rational calculation.

3) WORK: COMPLETE FINITE EXTENSION PARAMETRIZATION

More generally, normalize three old points to $O=(0,0)$, $B=(d,0)$, $C=(u,v\sqrt k)$ with $d>0$, $v\ne0$, and $d,u,v\in\mathbb Q$. Any new point $P=(x,t\sqrt k)$ at integer distances $a,b,c$ from them must satisfy \[ x=\frac{a^2-b^2+d^2}{2d},\qquad t=\frac{a^2+u^2+kv^2-c^2-2ux}{2kv}. \tag{1} \] These follow by subtracting the three squared-distance equations. Conversely, a triple $(a,b,c)$ of positive integers and $1$ gives a valid point at those three distances precisely when $x^2+kt^2=a^2$. Distances to the remaining old points are then tested by checking whether their rational squared distances are integer squares. The two determinant tests above enforce general position after adjoining $P$.

Thus all extensions with $a,b,c\le D$ are covered by at most $D^3$ exact tests. This is completeness for a specified bounded search, not a heuristic coordinate grid. It also avoids assuming the new point has rational Cartesian coordinates when the characteristic is nonsquare.

4) ADVERSARIAL VERIFICATION

The old seven-point determinant checks were run; an exhaustive $D^3$ extension search was not run and no eight-point example is claimed. Arbitrarily large configurations need not extend this particular heptagon, so even proving it unextendible would not decide the problem.

7) FINAL AND REFERENCE

UNRESOLVED. The old certificate is now independently reproducible, and $1$ provides an exact finite search specification. The first unsupplied construction remains eight points. The primary problem record checked via search is https://www.erdosproblems.com/213; it credits the heptagon to Kreisel--Kurz. No literature novelty is claimed.

8) COMPLETION ESTIMATE

COMPLETION: 10% This is a subjective estimate of progress toward the entire stated problem, not a probability that the conjecture or an unverified proof is correct.