\section{Worked examples and exact consequences}\label{sec:examples}
\subsection{Graph incidence and the preceding cut theory}
Let $\mathcal G=(V,E)$ be a connected simple graph and fix a root vertex. Orient its edges arbitrarily and delete the root column of the signed incidence matrix to obtain $A\in\Z^{|E|\times(|V|-1)}$. A row basis is the edge set of a spanning tree; its determinant is $\pm1$. A non-tree row subset of the same size is singular. These facts follow by repeatedly removing leaves from a tree incidence matrix and by observing that cycles give dependent rows and disconnection lowers rank. Thus $A$ is unimodular.

For an integer or real potential $y$, the coordinates of $Ay$ are edge differences. A minimal nonempty support of these differences is a bond: the two constant-value regions give a minimal cut, and any non-minimal cut contains a smaller such support. Equivalently, the complement rank-$|V|-2$ condition in Lemma~\ref{lem:hyperplane}, interpreted through connected components, says the complement has exactly two connected components with all internal edges retained. Hence the weighted cogirth is the graph's minimum weighted cut $\kappa_w(\mathcal G)$. Cauchy--Binet reduces to the weighted spanning-tree sum
\[
 \det G=\tau_w(\mathcal G)=\sum_{T\text{ spanning tree}}\prod_{e\in T}w_e.
\]
Therefore
\begin{align*}
 D_Q&=2\sqrt{\kappa_w(\mathcal G)}\sin(\pi/Q),\\
 \Vol&=(2\pi)^{|V|-1}\sqrt{\tau_w(\mathcal G)},\\
 \sys&=2\pi\sqrt{\kappa_w(\mathcal G)},\qquad
 \inj=\pi\sqrt{\kappa_w(\mathcal G)}.
\end{align*}
These recover the earlier graph results as an exact special case rather than as an unproved premise. The shortest-vector/min-cut identity has prior literature \cite{mckilliam}; the present advance is the arithmetic classification of how far this law extends beyond graph incidence systems.

\subsection{A cographic example: cycles rather than cuts}
Take the complete graph on vertices $0,1,2,3,4$, orient edges increasingly, and use the star $(0,j)$ as a spanning tree. The six remaining edges determine the fundamental cycles $0\to u\to v\to0$, $1\le u<v\le4$. Listing their edge-flow vectors as columns gives
\begin{equation}\label{eq:K5matrix}
 A=\begin{pmatrix}
1&1&1&0&0&0\\-1&0&0&1&1&0\\0&-1&0&-1&0&1\\0&0&-1&0&-1&-1\\
1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1
 \end{pmatrix}.
\end{equation}
The final identity block proves primitivity. Of its $\binom{10}{6}=210$ maximal minors, $64$ are $+1$, $61$ are $-1$, and $85$ are zero; the release verifies every determinant exactly. This proves unimodularity for this explicit example without relying on a software matroid label.

The real image of $A$ is the circulation space of $K_5$. A support-minimal nonzero circulation is an oriented simple cycle, so the cocircuit supports of this row representation are simple cycles of the underlying graph. At unit weights the minimum supports are its ten triangles. Hence
\begin{equation}\label{eq:K5numbers}
 d=6,\quad m=10,\quad\lambda_A(1)=3,\quad N_*=10,\quad\det(A^\top A)=125.
\end{equation}
The determinant also follows by summing the $125$ unit squared maximal minors. Thus the code has $Q^6$ words and
\[
 D_Q=2\sqrt3\sin(\pi/Q),\quad
 \Vol=(2\pi)^6\sqrt{125},\quad\sys=2\pi\sqrt3.
\]
It has ten nearest neighbors per word for $Q=2$ and twenty for every $Q\ge3$. Its first normalized intrinsic heat correction is $20e^{-3\pi^2/t}(1+o(1))$. The spectral adapter needs $21$ real moments of matrices of order $55$ when $M=5$.

This cographic instance is an explicit non-incidence extension of the cut-based starting point. It does not constitute a claim that cographic representations or cycle lattices themselves are new.

\begin{figure}[htbp]\centering
\includegraphics[width=.86\textwidth]{04_cographic_quantization.pdf}
\caption{The cographic $K_5$ code's exact distance law. Markers at $Q=2,3,4,6$ come from exhaustive code enumeration using rational trigonometric energy tables; the curve evaluates the theorem.}
\end{figure}

\subsection{Coprime harmonics: a nonunimodular model with a certified endpoint}
For $A=(2,3)^\top$ and $w=(1,1)$, an integer left inverse is $C=(-1,1)$. The metric is $G=13$, its shortest energy is $s=13$, and its shortest vectors are $\pm1$. The shell \eqref{eq:candidates} also contains only these two vectors because $13k^2\le65/2$ implies $|k|=1$. There are no signature comparisons to make. Its quartic moment is $2^4+3^4=97$.

The residue bounds in \eqref{eq:resbounds} are $R_1=R_2=6$, and
\[
 AC-I_2=\begin{pmatrix}-3&2\\-3&2\end{pmatrix}.
\]
Thus $L=30$, and \eqref{eq:Q0} gives $Q_0=31$. We obtain the exact statement
\begin{equation}\label{eq:23exact}
 D_Q^2=4\sin^2(2\pi/Q)+4\sin^2(3\pi/Q),\qquad Q\ge31,
\end{equation}
with exactly the nearest residues $\pm1\pmod Q$. The threshold is conservative. Its proof certifies all integers beyond it, not only the finitely many values evaluated in the experiments.

The asymptotic expansion becomes
\[
 D_Q^2=\frac{52\pi^2}{Q^2}-\frac{388\pi^4}{3Q^4}+O(Q^{-6}).
\]
The real-support prediction uses $\lambda=2$, so its leading ratio approaches $13/2$ rather than one. The graph-style exact formula does not reappear outside the rigid class; instead, the general lattice energy supplies the correct leading term.

\begin{figure}[htbp]\centering
\includegraphics[width=.86\textwidth]{01_arithmetic_recovery.pdf}
\caption{Exhaustive finite-grid minima for the two-character model $(2,3)^\top$ at unit weights, divided by the cocircuit lower-bound expression. Unsafe moduli permit failing weights but need not fail at every weighting. Fine-resolution behavior tends to the intrinsic energy ratio $13/2$.}
\end{figure}
\begin{figure}[htbp]\centering
\includegraphics[width=.86\textwidth]{02_quartic_correction.pdf}
\caption{The quartic correction for the same model. The horizontal target is $-4\pi^4(97)/3$. Numerical finite-grid minima are independent of the asymptotic formula; the limit and the certified $Q\ge31$ identity are proved in the text.}
\end{figure}

\subsection{Perfect-number alphabets: an exact but limited arithmetic role}
The known even perfect numbers $6,28,496,8128$ have factorizations
\[
 6=2\cdot3,\quad28=4\cdot7,\quad496=16\cdot31,\quad8128=64\cdot127.
\]
Their status as perfect numbers can be checked directly by summing proper divisors. More generally, for an alphabet of the form $Q=2^{r-1}(2^r-1)$ with $2^r-1$ prime, the prime factors are $2$ and $2^r-1$. Theorem~\ref{thm:fixed} says that the uniform cocircuit lower bound is valid at this alphabet precisely when neither of those primes belongs to $\mathcal P(A)$.

For a unimodular representation, every alphabet is safe and the exact law holds without using any special divisor-sum property. For a nonunimodular representation, an even perfect alphabet can be unsafe merely because a basis determinant is even. No arithmetic adjective overrules this obstruction. Increasing $Q$ also decreases the exact rigid-class minimum distance as $Q^{-1}$, so increased phase resolution is not increased worst-case noise tolerance.

As a transparent example, the cographic model at $Q=6$ has $D_6=\sqrt3$, while its value at $Q=28$ is $2\sqrt3\sin(\pi/28)$. The theorem applies equally to the nonperfect choices $Q=5,7,9$. There is no claim about odd perfect numbers or about perfect error-correcting codes.