math.CO — Combinatorics

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

Contributed by Oleksiy Babanskyy

For a finite cyclic group and an exact Fourier support $S$, we study the lattice $\mathcal{L}_{\le d}(S)$ generated by zero-sum relations of length at most $d$ inside the full relation lattice $K(S)$. 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 $\mathcal{L}_{\le5}(U_{n})=\mathcal{L}_{\le4}(U_{n})$. Thus fourth- and fifth-order rational ambiguity coincide: fourth order reconstructs up to translation when $3\nmid n$, while for $n=6m$ the complete surviving family has normalized ratios parameterized by norm-one elements of $\mathbb{Q}(\zeta_6)$, modulo the residual translation subgroup $\mu_6$; order six closes it. For $n=2pq$ with distinct odd primes, we classify every nonempty $\operatorname{Aut}(C_n)$-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 $r_{\rm set}(C_{2pq})=4$ 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