%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Autocorrelation Phase Lattices on Cyclic Groups
%% Standalone public-repository manuscript source.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt,a4paper]{article}
\ifdefined\pdfinfoomitdate
  \pdfinfoomitdate=1
\fi
\ifdefined\pdfsuppressptexinfo
  \pdfsuppressptexinfo=-1
\fi
\ifdefined\pdftrailerid
  \pdftrailerid{}
\fi

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{lmodern}
\usepackage[protrusion=true,expansion=false]{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{booktabs,array,longtable,tabularx}
\usepackage{geometry}
\geometry{margin=2.6cm}
\usepackage{xcolor}
\usepackage[hidelinks]{hyperref}
\Urlmuskip=0mu plus 2mu\relax
\urlstyle{tt}
\setlength{\emergencystretch}{3em}
\setlength{\parskip}{0pt}
\linespread{1.015}
\raggedbottom

\input{macros}

\hypersetup{
  hypertexnames=false,
  pdftitle={Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports, 2pq Orbit Profiles, and Binary Reconstruction},
  pdfauthor={Oleksiy Babanskyy},
  pdfsubject={higher-order autocorrelations, zero-sum lattices, Galois realization, local separating degree, and set reconstruction},
  pdfkeywords={autocorrelation, cyclic group, zero-sum lattice, Galois descent, rational signal, integer signal, separating invariant, phase retrieval, set reconstruction, analytical deck}
}

\title{Autocorrelation Phase Lattices on Cyclic Groups:\\
Unit Supports, $2pq$ Orbit Profiles, and Binary Reconstruction}
\author{Oleksiy Babanskyy}
\date{}

\pagestyle{plain}

\begin{document}
\maketitle

\begin{abstract}
\input{abstract}
\end{abstract}

\smallskip
\noindent\textbf{Keywords:} higher-order autocorrelation; finite cyclic group;
zero-sum sequence; separating invariant; phase lattice; Galois descent;
rational and integer signals; set reconstruction; analytical deck.

\smallskip
\noindent\textbf{MSC 2020:} 43A25; 11B75; 20K01; 13A50.

\smallskip
\noindent\footnotesize Copyright \textcopyright\ 2026 Oleksiy Babanskyy.
The repository's MIT license applies to the assurance software, not to this
manuscript.  This Hexagon version is distributed under the
Creative Commons Attribution 4.0 International license (CC BY 4.0).
\normalsize

\input{sections/01_introduction}
\input{sections/02_zero_sum_lattices}
\input{sections/03_phase_lattices}
\input{sections/04_unit_supports}
\input{sections/05_signal_classification}
\input{sections/06_crt_boxes}
\input{sections/07_cubic_layer}
\input{sections/08_quartic_layer}
\input{sections/09_profiles}
\input{sections/10_galois_realization}
\input{sections/11_binary_set_reconstruction}
\input{sections/12_interpretation}

\appendix
\input{appendices/A_local_connectors}
\input{appendices/B_boolean_routing}
\input{appendices/C_reproducibility}

\section*{Data and code availability}
The theorem proofs are contained in this manuscript.  The
\href{https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups}{companion repository}
is available at
\begin{center}
\small
\texttt{https://github.com/aconsciousfractal/FCIG-}\allowbreak
\texttt{Autocorrelation-Phase-Lattices-}\allowbreak
\texttt{on-Cyclic-Groups}.
\end{center}
It contains the exact-arithmetic checks, symbolic certificates,
source-boundary records and deterministic replay described in
Appendix~\ref{app:replay}.
Those artifacts are assurance layers rather than premises of the proofs.
Third-party full texts are not redistributed.

\section*{Acknowledgment of prior frameworks}
The Fourier monomial and Galois criteria used here are taken from
Casper--Orozco; the classical translation-invariant and global sixth-order
lines include Adler--Konheim and Gr\"unbaum--Moore.  The universal ambient
group belongs to the Konyagin--Lev framework.  Domokos supplies the
zero-sum generation criterion, including its non-closed-field,
Galois-stable-support form; the quotient-group/separating-atom language is
drawn from the separating-invariant literature.  Precise uses are identified
at the relevant statements.  The supportwise realization step here consists
of explicit Galois-equivariant lower-bound ratios and does not re-claim those
general criteria.

\section*{AI assistance and verification status}
This work involved substantial AI assistance using GPT-5.6 Sol via OpenAI
Codex, as reported by the author.  AI systems contributed to mathematical
exploration, candidate proof arguments, verification code, and manuscript
drafting and revision.  The author directed the research and is responsible
for the claims and source attributions.  The computational checks and AI
reviews are assurance evidence, not independent human peer review or an
end-to-end formal proof.  At the time of this Hexagon submission, the author
reports that no part of the contribution has been checked and understood
by a human.

\bibliographystyle{alpha}
\bibliography{refs}

\end{document}