Blame for release/docs/sections/01_framework.tex

\section{The question and the mathematical framework}\label{sec:framework}
Graph incidence constructions exhibit a clean coincidence: the same weighted minimum cut determined the shortest nontrivial intrinsic loop and the exact spacing of a finite phase code. The important general question is not whether more terminology can be attached to this coincidence. It is whether there is an exact boundary between integer systems where the coincidence survives and systems where it fails. A second question is whether the failure disappears in a controlled way as phase resolution increases.

The foundational results answer both questions in the finite-character setting; later sections add sharp inverse and extremal classifications. The first answer is arithmetic and uniform in all positive weights. The second is geometric and asymptotic for fixed weights, but includes a finite, explicit threshold. The distinction between these quantifiers is essential. A modulus can permit a specially weighted counterexample even though a particular fixed weighting behaves regularly at that same, sufficiently large modulus.

\subsection{Integer characters, weights, and faithful phase coordinates}
Throughout, $d\ge1$, $m\ge d$, and $A\in\Z^{m\times d}$ has full column rank. Its $i$th row is denoted $a_i^\top$; thus $a_i\in\Z^d$ is a column vector when used in an outer product. The weights $w_1,\ldots,w_m$ are strictly positive real numbers. Zero rows are allowed and produce constant output coordinates; they do not contribute to the metric. We put
\begin{equation}\label{eq:embedding}
 F_{A,w}(\theta)=\bigl(\sqrt{w_i}\,e^{i a_i\cdot\theta}\bigr)_{i=1}^m,
 \qquad \theta\in\T^d\eqdef\R^d/(2\pi\Z^d).
\end{equation}
Complex Euclidean space is regarded as a real inner-product space, with inner product $\Re\sum_i\overline{u_i}v_i$.

\begin{definition}[Primitive frequency representation]
The matrix $A$ is \emph{primitive}, or \emph{faithful}, if its rows generate the entire lattice $\Z^d$. Equivalently, there exists $C\in\Z^{d\times m}$ with $CA=I_d$. Unless explicitly discussing failures, all results below assume this condition.
\end{definition}
Full real rank alone is insufficient. For example, the single row $(2)$ has real rank one but $\theta$ and $\theta+\pi$ have the same image. Primitivity eliminates this finite kernel, rather than merely eliminating infinitesimal degeneracy.

\begin{lemma}[Equivalent faithfulness tests]\label{lem:faithful}
For a full-column-rank integer $A$, the following are equivalent: its rows generate $\Z^d$; the greatest common divisor of its nonzero maximal minors is one; an integer left inverse exists; and \eqref{eq:embedding} is injective on $\T^d$.
\end{lemma}
\begin{proof}
Smith reduction by integer row and column operations of determinant $\pm1$ transforms $A$ to a rectangular diagonal form with positive nonzero entries $s_1\mid\cdots\mid s_d$. These transformations preserve the row-lattice index. That index is $\prod_j s_j$, which is also the gcd of the maximal minors. Thus the first two conditions are equivalent, and both hold precisely when all $s_j=1$. The same normal form gives an integer left inverse in that case.

For an explicit alternative construction, let $E_I$ select the rows of a nonsingular maximal submatrix $A_I$. Then $\operatorname{adj}(A_I)E_I A=\det(A_I)I_d$. A Bezout combination of the maximal determinants equal to one therefore gives an integer left inverse. This is the construction implemented in the release.

If $CA=I_d$ and $F(\theta)=F(\eta)$, then $A(\theta-\eta)\in2\pi\Z^m$ and multiplication by $C$ gives $\theta-\eta\in2\pi\Z^d$. Conversely, a Smith invariant $s_j>1$ gives a nonzero torus element of order $s_j$ in the kernel. Positivity of weights ensures that no character is lost by multiplication by its amplitude.
\end{proof}

\subsection{The finite code and its exact energy}
For an integer $Q\ge2$, quantize the phase coordinates to $2\pi(\Z/Q\Z)^d/Q$. Define
\begin{equation}\label{eq:code}
 \mathcal C_{A,w,Q}=\{F_{A,w}(2\pi x/Q):x\in(\Z/Q\Z)^d\}.
\end{equation}
Faithfulness gives exactly $Q^d$ codewords. Since multiplication of output coordinate $i$ by $e^{ia_i\cdot\eta}$ is a Euclidean isometry, all codewords have the same distance spectrum relative to the rest of the code. It is sufficient to measure distances to $F(0)$. The squared minimum distance is
\begin{equation}\label{eq:distance}
 \DQ^2=\min_{0\ne x\in(\Z/Q\Z)^d}
 E_Q(x),\qquad
 E_Q(x)=4\sum_{i=1}^m w_i\sin^2\!\left(\frac{\pi(Ax)_i}{Q}\right).
\end{equation}
The sine-squared expression is invariant under changing the integer representatives of $x$. It is not an approximation. This is the standard chordal distance mechanism underlying commutative group codes and flat-torus codes; those subjects have a substantial prior literature \cite{torezzan,siqueira}.

\subsection{Real supports and weighted cogirth}
\begin{definition}[Cocircuit and weighted cogirth]
A \emph{cocircuit support} of the row configuration $A$ is an inclusion-minimal set among the nonempty supports of vectors $Ay$, $y\in\R^d$. Let $\mathcal D(A)$ be the finite collection of these supports, and define
\begin{equation}\label{eq:cogirth}
 \lambda_A(w)=\min_{D\in\mathcal D(A)}\sum_{i\in D}w_i.
\end{equation}
The number of supports attaining this minimum is denoted $N_*(A,w)$.
\end{definition}
The notation $D$ denotes a support, while $D_Q$ denotes a Euclidean distance. In graph incidence systems these supports are bonds, that is, inclusion-minimal nonempty cuts. In general they need not be graph cuts.

\begin{lemma}[Hyperplane characterization]\label{lem:hyperplane}
Every nonzero support $\supp(Ay)$ contains a cocircuit. If $D$ is a cocircuit, the rows indexed by $D^c$ have rank exactly $d-1$. Conversely, the complement of all rows lying in a row-generated hyperplane of rank $d-1$ is a cocircuit.
\end{lemma}
\begin{proof}
A finite nonempty family of subsets contains an inclusion-minimal member. For the rank statement, put $R=D^c$. All rows in $R$ annihilate $y$, so their rank is at most $d-1$. If it were at most $d-2$, their nullspace would have dimension at least two. Choose $j\in D$. The additional equation $a_j\cdot z=0$ still leaves a nonzero solution $z$ annihilating all rows of $R$. Full column rank implies $Az\ne0$. Its support is a proper subset of $D$, a contradiction.

Conversely, the nullspace of a rank-$d-1$ hyperplane is one-dimensional. A normal $z$ vanishes on exactly the rows in the hyperplane. A strictly smaller nonempty support would require a nonzero normal annihilating that hyperplane and an additional row outside it, which would force full rank and hence a zero normal.
\end{proof}
A useful consequence is that if a set $R$ of zero rows has real rank below $d$, then $R^c$ contains a cocircuit. This elementary support observation is the bridge between modular aliasing and a real combinatorial invariant.

\subsection{The scope of the two main results}
The first main result, Section~\ref{sec:arithmetic}, is an exact equivalence for a fixed modulus and \emph{all} positive weights. Its all-moduli version is the classification in Section~\ref{sec:rigidity}. The second main result, Section~\ref{sec:finitejet}, fixes $A,w$ and describes \emph{every sufficiently large} modulus, including composite moduli and multiples of exceptional primes. It does not require unimodularity.

The word ``perfect'' in the title refers to a named family of integer alphabet sizes and to the search for sharp formulas. It does not identify these codes as perfect sphere-packing codes. ``Impossible complex frequencies'' is the project name; the mathematics uses ordinary integer characters and complex exponentials. The geometry is Riemannian but flat intrinsically. Its extrinsic curvature supplies a nontrivial higher-order invariant.