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An affine criterion for a classical four-point family

Primarily AI-generated textHuman understanding: all partsmath.CO — Combinatorics

Contributed by Alexandre Couret ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

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>0q>0 and x∈Z/(4q)Z∖{0,q,2q,3q}x\in\mathbb{Z}/(4q)\mathbb{Z}\setminus\{0,q,2q,3q\}, with representative XX. The supports P={0,q,x,x+2q}P=\{0,q,x,x+2q\} and Q={0,q,x+q,x+3q}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)≡2(mod4)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.

Provenance statement

The mathematical development, calculations, Lean formalization and writing used substantial AI assistance. This note was prepared with OpenAI ChatGPT/Codex assistance from an existing project note and arithmetic argument. A private Lean development is reported to have passed compilation and axiom audits; no new Lean compilation was performed for this draft. The written proof is self-contained. Independent human mathematical review and bibliographic novelty assessment remain outstanding.

Tools used

OpenAI
Codex

References

  1. J. Rosenblatt, Phase retrieval , Communications in Mathematical Physics 95 (1984), 317--343. https://doi.org/10.1007/BF01212402. Consulted through later restatements for this draft.DOI
  2. F. Jedrzejewski and T. Johnson, The Structure of Z-Related Sets , Mathematics and Computation in Music, LNCS 7937 (2013), 128--137. https://arxiv.org/abs/1304.6608.link
  3. W. Q. Erickson and N. B. Jones, Homometric subsets of Zn\Z_n with cardinality 5: classification and enumeration , arXiv:2412.08997v4 (27 August 2026). https://arxiv.org/abs/2412.08997v4.arXiv

Version history

  1. v1 · 3 pagesSubmitted by Alexandre CouretShort 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.CurrentNov 07, 2026