\section{Interpretation and prior-art boundary}\label{sec:interpretation} The common invariant of the paper is the filtration \[ \calL_{\le d}(S)\subset K(S). \] It has three readings that should not be conflated. Its quotient-group completion is the supportwise separating-atom invariant of Proposition~\ref{prop:dictionary}; its lattice points are the first dilations of the zero-sum face (2.5); and its complex character group is the phase ambiguity modulo translations in Proposition~\ref{prop:ratio}. All three readings now give the same closing threshold on every stable $C_{2pq}$ support: Section~\ref{sec:galois} realizes every positive-order failure below closure by rational and integer signals, while the depth-one support is direct. A complete parametrization of all surviving rational ratios is asserted only on the exact unit support. The two algebraic readings may be compressed as \[ \begin{array}{ccl} \calL_{\le d}(S)=K(S) &\Longleftrightarrow&\text{all complex invisible ratios are translations},\\ &\Longleftrightarrow&\text{all separating atoms over }S\text{ have length } \le d. \end{array} \tag{12.1} \] For rational signals, equality gives sufficiency immediately, but failure initially produces only a complex character obstruction. Realizing it by two rational Fourier vectors requires Galois covariance. Section~\ref{sec:galois} constructs such a lower-bound ratio on every stable $C_{2pq}$ support of depth greater than one, while Section~\ref{sec:signals} gives the stronger full ambiguity classification on $\units n$. The historical higher-autocorrelation literature established translation invariance, support-generation and ratio-character uniqueness mechanisms, and global finite-cyclic bounds \cite{AdlerKonheim1962,McLaughlinRaviv1968,ChazanWeiss1970, KakaralaIverson1993,GrunbaumMoore1995}. Konyagin--Lev supply the universal Freiman ambient group used in Section~\ref{sec:units} \cite{KonyaginLev2000}; the modern separating-invariant works supply the zero-sum generation criterion, non-closed-field Galois-stable criterion, quotient-group and degree language \cite{Domokos2017,Domokos2023,Domokos2026, ScheflerZhaoZhong2026,Huang2026}; and the recent autocorrelation papers supply the exact Fourier and rationality criteria and the support-specific questions \cite{AgulnickBusickWarner2026, CasperOrozco2026}. The zero-sum face reformulation connects the result to the polyhedral counting framework of Han--Wang--Zhang--Zhang \cite{HanWangZhangZhang2026}. For subsets, Pebody supplies the finite-abelian reconstruction framework, the cyclic upper bound six, the multiset classification and the even-order value-four conjecture \cite{Pebody2004,Pebody2007}. Together with Keleti--Kolountzakis's even-order three-deck obstruction, that classification already gives the exact infinite families $C_{2^a}$ for $a\ge4$ and $C_{2\ell^k}$ for primes $\ell>3$ whenever $2\ell^k>10$ \cite{KeletiKolountzakis2006}. The family $C_{2pq}$ treated here is disjoint from both. Kiss--{\L}aba--Marshall--Somlai study general mask-polynomial lower bounds \cite{KissLabaMarshallSomlai2026}. For four of the deep support patterns used here, the needed obstruction instead follows directly by evaluating the three prime-power cyclotomic divisors at $1$; the remaining rows still require the uniform binary argument. The lower bound in Theorem~\ref{thm:set-reconstruction} is the Keleti--Kolountzakis pair; the translation/reflection lower bound is a specialization of Simon's deck construction and equivalence tests \cite{Simon2017}. The binary increment here is the uniform exclusion of all deep $2pq$ supports, not ownership of these general mechanisms. The overlap with Domokos's 2026 criterion is architectural and direct: Theorem~5.4 there decides separation by testing integral generation on every Galois-stable character subset of a regular representation. Here a support $S$ is fixed first, and the object computed is the least degree at which its block-monoid group completion reaches $K(S)$. The general criterion therefore owns the field and support framework; it does not supply the local CRT presentations, the profiles $(1,2,154,80,4,14)$ and $(1,2,154,92,4,2)$, or the rational norm-one classification on the unit row. Within that boundary, the contribution assembled here is one theorem chain. For unit supports of all even cyclic groups, the fourth-order quotient has a sharp dichotomy, the exceptional branch closes at six, and its entire rational phase family is the norm-one torus over $\Q(\zeta_6)$ modulo residual translations. For $C_{2pq}$, the local presentations, Cubic Skeleton, Quartic Skeleton, bridges, and critical families classify every one of the 255 stable supports; the nine canonical Galois lifts, two bridge signs, and depth-six phases realize every nontrivial first failure over both $\Q$ and $\Z$. Their literal overlap is the unit box, where the two independent derivations agree. The regime change has a local explanation. If $3\nmid2pq$, every parity-one punctured line has split by degree four into a parity constant and an odd-primary slope; only the diagonal parity on $P1\cup Q1$ survives later. When $p=3$, the $P1$ and $G$ boxes retain integral sign lifts, and seven masks preserve one through degree five. The difference between the two profiles is therefore propagated from one local change in Table~\ref{tab:local} through the same connector system. \begin{table}[ht] \centering \small \begin{tabularx}{\textwidth}{@{}YY@{}} \toprule Established here & Not asserted here\\ \midrule All-even unit quotient and exact degree-six closure & Arbitrary non-stable support classification\\ Complete rational ambiguity family on the exact unit support & Rational ambiguity-torus classification on every row of the 255-support atlas\\ Exact rational and integer thresholds on all 255 stable supports, with explicit lower-bound pairs whenever the depth exceeds one & Signals on non-stable supports, positivity, or fixed alphabets\\ Exact local separating-degree profile on stable $C_{2pq}$ supports & Ordinary separating Noether number or connector minimality\\ Uniform symbolic Cubic and Quartic Skeletons & Noise stability, positivity, or finite-alphabet reconstruction\\ Exact binary set-reconstruction number four on $C_{2pq}$ & Pebody's conjecture for all even cyclic orders\\ Oriented-deck value four modulo translations/reflections & A reflection-symmetrized deck or a weighted analogue\\ \bottomrule \end{tabularx} \caption{The theorem boundary of the unified paper.}\label{tab:boundary} \end{table} Several stronger-sounding statements are deliberately absent. We do not claim absolute priority, a complete ambiguity-torus parametrization on every atlas row, a classification for arbitrary non-stable supports, or a filtration through all cyclotomic subfields. We also do not claim the full even cyclic set-reconstruction conjecture. Nor do the exact-arithmetic replays in Appendix~\ref{app:replay} replace any uniform proof. They audit the theorem architecture and its finite routing only. The computations do have a reproducibility role. They check Boolean exhaustiveness, connector images, endpoint branches, and independent Smith presentations. Mutation controls reject missing X10B sign glue, altered failure sets, and corrupted atlases. The direction of dependence remains unchanged: the written lemmas imply the theorem, while the receipts test their implementation and transcription. The maximum orbit-stable local separating degree on $C_{2pq}$ is therefore six in both arithmetic regimes, but the locus attaining it changes sharply: one generic parity bridge becomes seven prime-three sign families. That change is already visible in the unit-support dichotomy and is explained locally by the free sign directions in Table~\ref{tab:local}. This is the structural link between the two halves of the paper. Several separate questions remain natural: a complete rational ambiguity parametrization on every atlas row, analogous Skeletons for higher-rank groups, and robust reconstruction from approximate data. They require, respectively, a finer cocycle/descent analysis, a larger orbit geometry, and quantitative conditioning estimates. The exact cyclic result here is a base case for those directions, not a claim to solve them.