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Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports, 2pq Orbit Profiles, and Binary Reconstruction

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics

Contributed by Oleksiy Babanskyy ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

For a finite cyclic group and an exact Fourier support SS, we study the lattice L≤d(S)\mathcal{L}_{\le d}(S) generated by zero-sum relations of length at most dd inside the full relation lattice K(S)K(S). Its closing degree is both a local separating degree and the order at which every invisible phase ratio becomes a translation character. For the exact unit support of every even cyclic group, we compute $$ \mathbb{Z}^{U_{n}}/\mathcal{L}_{\le4}(U_{n})\cong \begin{cases} \mathbb{Z}/n\mathbb{Z},&3\nmid n,\\ \mathbb{Z}\oplus\mathbb{Z}/(n/6)\mathbb{Z},&6\mid n. \end{cases} $$ Moreover L≤5(Un)=L≤4(Un)\mathcal{L}_{\le5}(U_{n})=\mathcal{L}_{\le4}(U_{n}). Thus fourth- and fifth-order rational ambiguity coincide: fourth order reconstructs up to translation when 3∤n3\nmid n, while for n=6mn=6m the complete surviving family has normalized ratios parameterized by norm-one elements of Q(ζ6)\mathbb{Q}(\zeta_6), modulo the residual translation subgroup μ6\mu_6; order six closes it. For n=2pqn=2pq with distinct odd primes, we classify every nonempty Aut⁡(Cn)\operatorname{Aut}(C_n)-stable support. The distributions of these 255 supports by closing degree are $$ (1,2,154,80,4,14)\quad(p=3),\qquad (1,2,154,92,4,2)\quad(p,q\ge5). $$ The proof uses seven local CRT presentations, cubic and quartic Skeletons, and explicit depth-five and depth-six obstructions. Galois-equivariant lower-bound ratios show that the lattice, rational-signal and integer-signal closing degrees coincide on every stable support. Finally, for binary indicators, a supportwise crosswalk and a uniform exclusion of all deep supports give rset(C2pq)=4r_{\rm set}(C_{2pq})=4 under translations. The value is also four for the same oriented analytical deck when outputs are identified up to translation or reflected translation.

Author comments: Mathematics unchanged from aiXiv 2609.03274 v1. This version adds an explicit AI-assistance statement and a CC BY 4.0 notice. Companion and reproducibility instructions: https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/tree/74a839355ba9ffd96a73fea8aadceabea3f9e57b

Provenance statement

The author selected the research programme, directed the work, and retains responsibility for the claims, attributions and publication decisions. The author reports substantial mathematical, computational and writing contributions from GPT-5.6 Sol via OpenAI Codex, consistent with the earlier aiXiv disclosure. Individual proof steps are not attributed to a specific model session. The paper computes the unit-support phase quotient for even cyclic groups; classifies closing degrees for the 255 nonempty automorphism-stable supports in C_(2pq), with distinct odd primes p and q; constructs rational and integer lower-bound witnesses; and treats binary reconstruction on C_(2pq). The complete parametrization of rational ambiguities is stated for exact unit support, not for every one of the 255 supports. The binary theorem is restricted to C_(2pq), not the full all-even Pebody conjecture. Prior Fourier, zero-sum and invariant-theoretic frameworks are explicitly credited. The companion contains exact-arithmetic, symbolic and finite checks, source-boundary records and reproducibility instructions. These checks and AI-assisted adversarial reviews are assurance evidence; they are not independent human peer review or an end-to-end formal proof. For this submission, the author reports that no part of the contribution has been checked and understood by a human. No new comprehensive mathematical review or priority certification is claimed for this deposit. This deposit preserves the mathematics of aiXiv 2609.03274 v1 and public commit 74a839355ba9ffd96a73fea8aadceabea3f9e57b. The only manuscript changes are the Hexagon-specific CC BY 4.0 license notice and an explicit AI-assistance and verification-status statement. Prior deposit: https://aixiv.online/abs/2609.03274v1 Pinned source, companion and claim boundaries: https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/tree/74a839355ba9ffd96a73fea8aadceabea3f9e57b

Tools used

OpenAI
CodexVersion GPT-5.6 Sol
Python Software Foundation
Python
SymPy
SymPy

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Version history

  1. v1 · 49 pagesSubmitted by Oleksiy BabanskyyMathematics unchanged from aiXiv 2609.03274 v1. This version adds an explicit AI-assistance statement and a CC BY 4.0 notice. Companion and reproducibility instructions: https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/tree/74a839355ba9ffd96a73fea8aadceabea3f9e57bCurrentNov 09, 2026