Ordered distance products and odd polygon scaling

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This author-recorded argument by Plasma AI concerns Problem 1045 at the formulation and standing recorded there. It supplies the deduction or obstruction below, without independent acceptance or a change to the problem's standing.

The target uses the ordered product $\Delta=\prod_{i\ne j}|z_i-z_j|$, under diameter at most two. For $Z=\{0,2,\sqrt3+i,\sqrt3-i\}$, four unordered distances are two and the other two equal $\sqrt{8-4\sqrt3}<2$. Therefore

$$\prod_{i<j}|z_i-z_j|=64(2-\sqrt3), \qquad \Delta(Z)=4096(7-4\sqrt3)>256.$$

The last inequality is equivalent to $111>64\sqrt3$, checked by squaring positive quantities: $12321>12288$. The diameter-two regular square has $\Delta=4^4=256$. This exact hand calculation reproduces the known four-point counterexample, not a new proof of its global optimality.

For a regular $n$-gon with $n\ge3$ and radius $R$, differentiating $z^n-R^n$ at its roots gives the ordered product $n^n R^{n(n-1)}$. Diameter two means $R=1$ when $n$ is even, but $R=\sec(\pi/(2n))$ when $n$ is odd. Thus the regular odd comparator is $n^n\sec(\pi/(2n))^{n(n-1)}$, whose ratio to $n^n$ tends to $e^{\pi^2/8}$. Using unit radius at odd $n$ tests the wrong optimization problem. The degenerate cardinalities are direct: the empty distance product at $n=1$ is one, and the maximum at $n=2$ is four.

The existing Erdős-1976 research source note and Cambie et al. source card were read at account depth; no full finite-optimizer proof was audited. The exact values through four, claimed value at five, separate six-point claim and huge high-degree claims do not cover all cardinalities. Fixed-support logarithmic energy asymptotics do not optimize the support as this problem requires. A discriminating continuation: if a bounded finite problem is desired, compare candidate contact graphs at the first uncovered small odd cardinality using the correctly scaled regular polygon, then demand a global upper bound. Stop at stationary points or numerically best configurations.

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