# Verification report

**Manuscript:** Two Congruence Classes of Extremal Twenty-Point Sets.  
**Author:** Alper Ferudun, Mercury Software GmbH.  
**Source identifier:** EP-668.  
**Date:** 17 October 2026.

## Accepted scope

There are exactly two congruence classes of twenty-point planar sets maximizing the number of unit-distance pairs, including reflections in the equivalence relation. Explicit representatives are D + bV, where zeta=(1+i sqrt(3))/2, D={0,1,zeta,1+zeta}, V={0,1,zeta,zeta-1,1-zeta}, and b=zeta(5 +/- i sqrt(11))/6. Each has 54 unit pairs. The sum of fourth powers of all pair distances is (19070 +/- 160 sqrt(33))/9 and distinguishes the classes.

The value u(20)=54 and the uniqueness of the extremal abstract graph are external results of Alexeev, Mixon and Parshall (2025), not new results of this manuscript. The original asymptotic Erdős problem is unresolved.

## Written proof checks

The explicit graph6 correspondence identifies a 53-edge Cartesian product plus one extra edge. In any injective realization, the product squares satisfy the rhombus sum identity; connectivity propagates this identity to a global Minkowski decomposition. Equilateral triangles force both finite factors up to a unit multiplier and reflection. The extra edge leaves exactly two possible relative rotations. A quadratic-field argument excludes collisions. The fourth-moment identity is derived by expansion over ordered pairs, and its two exact values rule out every Euclidean congruence.

This argument covers all realizations. It does not infer global uniqueness from local rigidity, pictures or a finite numerical search. The source table's graph uniqueness is not confused with geometric uniqueness.

## Supplementary exact checks

The standard-library Python verifier performs 2,277 checks in each of normal and optimized execution. These cover the source hash, graph6 round trips, product graph correspondence, exact field arithmetic, all 190 pair distances in both representatives, all four normalized labelled branches and their reflection identifications, moments, diameters, relabellings and negative controls. A generic unit rotation gives only 53 unit pairs; the unrotated sum has collisions.

The separate C++17 implementation uses integer coordinate numerators with common denominator 12 and an explicit quadratic norm formula. It performs 783 checks in each of the normal and NDEBUG builds, checking every pair against the published graph and recovering the same two fourth moments. It does not import the Python field implementation.

The archive is extracted into a fresh directory, the manuscript is rebuilt and its six pages are compared with the original PDF text and content streams. The replay driver reruns both implementations in both modes. All six rendered PDF pages were directly visually inspected.

These are reproducible arithmetic checks and originating-researcher self-audit, not independent human or model review and not proof-assistant certification. They do not repeat AMP's graph enumeration. No mathematical gap was identified within the finite scope. AI assistance is disclosed in the article; no absolute priority is asserted.