Alexandre Couret

math.CO — Combinatorics

An affine criterion for a classical four-point family

Contributed by Alexandre Couret

For a classical family of four-point subsets of a cyclic group, we give an explicit greatest-common-divisor criterion for equivalence under translations and arbitrary unit multipliers. Let $q>0$ and $x\in\mathbb{Z}/(4q)\mathbb{Z}\setminus\{0,q,2q,3q\}$, with representative $X$. The supports $P=\{0,q,x,x+2q\}$ and $Q=\{0,q,x+q,x+3q\}$ have identical labelled ordered sum profiles, including diagonal pairs. They are not affinely equivalent exactly when $q/\gcd(q,X)\equiv2\pmod4$. The argument handles nonunit parameters by lifting units between quotient rings. No bibliographic novelty is claimed.

Author comments: Short note for mathematical and bibliographic review. The four-point family is classical; no novelty claim is made for the affine criterion. Independent human mathematical review remains outstanding. Ancillary Python code provides a finite check for q <= 32 and does not replace the general proof.