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Eves-Impossible-Complex-Frequencies-v5.0.0

\section{Exact arithmetic obstruction at a fixed alphabet}\label{sec:arithmetic}
Write $\mathcal B(A)$ for the sets of $d$ rows forming nonsingular submatrices. These are the real row bases. A basis may be nonsingular over $\R$ yet singular modulo a prime; that loss of information is precisely the obstruction considered here.

\begin{definition}[Safe modulus and exceptional primes]
A modulus $Q\ge2$ is \emph{basis-safe} for $A$ if
\begin{equation}\label{eq:safe}
 \gcd(Q,\det A_I)=1\quad\text{for every }I\in\mathcal B(A).
\end{equation}
The exceptional-prime set is
\begin{equation}\label{eq:badprimes}
 \mathcal P(A)=\bigcup_{I\in\mathcal B(A)}\{p\text{ prime}:p\mid\det A_I\}.
\end{equation}
It is finite. A modulus is safe precisely when none of its prime factors belongs to this set.
\end{definition}

\begin{theorem}[Fixed-alphabet arithmetic classification]\label{thm:fixed}
Let $A$ be primitive and full-column-rank, and fix $Q\ge2$. The following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item For every $w\in(0,\infty)^m$,
\begin{equation}\label{eq:universalbound}
 \DQ^2\ge4\sin^2(\pi/Q)\lambda_A(w).
\end{equation}
\item Every maximal nonsingular row submatrix is invertible over $\Z/Q\Z$.
\item $Q$ is basis-safe in the sense of \eqref{eq:safe}.
\end{enumerate}
If these conditions fail, an explicit positive integer weighting and an explicit nonzero $Q$-phase difference violate \eqref{eq:universalbound} strictly.
\end{theorem}
\begin{proof}
For a square integer matrix, invertibility modulo $Q$ is equivalent to its determinant being a unit modulo $Q$. This follows in one direction from determinants, and in the other from the adjugate formula. It proves the equivalence of (ii) and (iii), for composite as well as prime $Q$.

Suppose (ii) holds. For $0\ne x\in(\Z/Q\Z)^d$, let
\[
 R(x)=\{i:(Ax)_i=0\pmod Q\},\qquad S(x)=R(x)^c.
\]
If the real rank of the rows in $R(x)$ were $d$, these rows would contain $I\in\mathcal B(A)$ with $A_Ix=0\pmod Q$. Invertibility would imply $x=0$, a contradiction. Lemma~\ref{lem:hyperplane} therefore gives a cocircuit $D\subseteq S(x)$. Every nonzero residue $r\pmod Q$ satisfies
\[
 \sin^2(\pi r/Q)\ge\sin^2(\pi/Q).
\]
Together with positive weights this implies
\[
 E_Q(x)\ge4\sin^2(\pi/Q)\sum_{i\in S(x)}w_i
 \ge4\sin^2(\pi/Q)\lambda_A(w).
\]
Taking the minimum over $x$ proves (i).

For the converse, choose a basis $I$ and a prime $p$ dividing both $Q$ and $\det A_I$. There exists a nonzero $y\in(\Z/p\Z)^d$ with $A_Iy=0$. Set
\begin{equation}\label{eq:badwitness}
 x=\frac Qp y\pmod Q,\qquad W=(m+1)Q^2,\qquad
 w_i=\begin{cases}W&i\in I,\\1&i\notin I.\end{cases}
\end{equation}
The residue vector $x$ is nonzero, and $A_Ix=0\pmod Q$. Faithfulness ensures that $Ax$ is not zero on all rows. Only the $m-d$ outside rows can contribute to $E_Q(x)$, so
\[
 E_Q(x)\le4(m-d).
\]
Every real cocircuit intersects $I$: a nonzero real vector annihilating all basis rows would be zero. Thus $\lambda_A(w)\ge W$. Concavity of sine on $[0,\pi/2]$ gives $\sin(\pi/Q)\ge2/Q$, including $Q=2$. Consequently,
\[
 4\sin^2(\pi/Q)\lambda_A(w)
 \ge \frac{16W}{Q^2}=16(m+1)>4(m-d).
\]
The particular codeword difference $x$ is already shorter than the asserted lower bound, proving failure of (i). All steps of the witness construction use integer arithmetic except the sine inequality, whose rational lower bound suffices to certify the strict violation.
\end{proof}

\begin{corollary}[Arbitrarily severe weighted failure]\label{cor:severe}
For a fixed unsafe $A,Q$, the ratio
\[
 \frac{D_Q(A,w)^2}{4\sin^2(\pi/Q)\lambda_A(w)}
\]
can be made arbitrarily close to zero by positive weights.
\end{corollary}
\begin{proof}
Keep weight $W$ on the bad basis and weight one outside. The numerator is bounded by $4(m-d)$, while the denominator is at least $4W\sin^2(\pi/Q)$. Let $W$ tend to infinity. Alternatively, normalize all weights by their sum; the ratio is unchanged because numerator and denominator are homogeneous of degree one in weights.
\end{proof}

\subsection{The Smith kernel formula and what it measures}
The preceding proof only needs a bad basis. A fuller certificate records how many grid configurations a subsystem fails to distinguish.
\begin{proposition}[Subsystem torsion count]\label{prop:smith}
Let $A_R$ have real rank $r$, with nonzero Smith invariants $s_1\mid\cdots\mid s_r$. Then
\begin{equation}\label{eq:smithkernel}
 \#\ker(A_R:(\Z/Q\Z)^d\to(\Z/Q\Z)^{|R|})
 =Q^{d-r}\prod_{j=1}^r\gcd(Q,s_j).
\end{equation}
\end{proposition}
\begin{proof}
Integer unimodular row and column operations remain bijections modulo $Q$, so they do not change the kernel cardinality. In Smith form, $r$ independent equations are $s_jz_j=0\pmod Q$, each with exactly $\gcd(Q,s_j)$ solutions. The remaining $d-r$ coordinates are arbitrary. Multiplication gives \eqref{eq:smithkernel}.
\end{proof}
The $Q^{d-r}$ factor reflects an ordinary real-rank deficit. The additional gcd factors reflect arithmetic aliasing beyond that deficit. Arithmetic matroids and toric arrangements were introduced to track closely related lattice-index and torsion information \cite{dadderio,mociquasi}. The new assertion proposed here is the exact equivalence between a weighted \emph{Euclidean quantization lower bound} and the maximal-basis unit condition, in the specified spectral realization; the use of Smith invariants is classical.

\subsection{Safe does not mean exact, unsafe does not mean always failing}
These two qualifications avoid opposite errors. A safe alphabet guarantees the lower bound for every weighting, but the shortest nonzero character values may not all be $\pm1$ modulo that alphabet. Equality can fail upward. Conversely, an unsafe alphabet admits a carefully chosen weighting that fails, but it need not fail at unit weights or at every fixed weighting.

For example, $A=(2,3)^\top$ is primitive, with exceptional primes $2,3$. Its only cocircuit is both rows, so $\lambda_A(w)=w_1+w_2$. At $Q=2$ and $w=(1,1)$, the squared minimum distance is $4$, while the putative universal formula would give $8$. At $Q=7$, which is safe, equality in the lower bound would require both $2x$ and $3x$ to be $\pm1$ modulo $7$. No such nonzero residue exists: multiplying the first equation by $3$ and the second by $2$ would force $3\equiv\pm2\pmod7$. The lower bound is therefore strict for all positive weights in this example.

The set of safe moduli is periodic with respect to divisibility by the squarefree product of exceptional primes. That statement concerns the validity of a uniform-in-weights theorem, not periodicity of the actual numerical minimum distance.

	\section{Exact arithmetic obstruction at a fixed alphabet}\label{sec:arithmetic}
	Write $\mathcal B(A)$ for the sets of $d$ rows forming nonsingular submatrices. These are the real row bases. A basis may be nonsingular over $\R$ yet singular modulo a prime; that loss of information is precisely the obstruction considered here.

	\begin{definition}[Safe modulus and exceptional primes]
	A modulus $Q\ge2$ is \emph{basis-safe} for $A$ if
	\begin{equation}\label{eq:safe}
	\gcd(Q,\det A_I)=1\quad\text{for every }I\in\mathcal B(A).
	\end{equation}
	The exceptional-prime set is
	\begin{equation}\label{eq:badprimes}
	\mathcal P(A)=\bigcup_{I\in\mathcal B(A)}\{p\text{ prime}:p\mid\det A_I\}.
	\end{equation}
	It is finite. A modulus is safe precisely when none of its prime factors belongs to this set.
	\end{definition}

	\begin{theorem}[Fixed-alphabet arithmetic classification]\label{thm:fixed}
	Let $A$ be primitive and full-column-rank, and fix $Q\ge2$. The following are equivalent:
	\begin{enumerate}[label=(\roman*)]
	\item For every $w\in(0,\infty)^m$,
	\begin{equation}\label{eq:universalbound}
	\DQ^2\ge4\sin^2(\pi/Q)\lambda_A(w).
	\end{equation}
	\item Every maximal nonsingular row submatrix is invertible over $\Z/Q\Z$.
	\item $Q$ is basis-safe in the sense of \eqref{eq:safe}.
	\end{enumerate}
	If these conditions fail, an explicit positive integer weighting and an explicit nonzero $Q$-phase difference violate \eqref{eq:universalbound} strictly.
	\end{theorem}
	\begin{proof}
	For a square integer matrix, invertibility modulo $Q$ is equivalent to its determinant being a unit modulo $Q$. This follows in one direction from determinants, and in the other from the adjugate formula. It proves the equivalence of (ii) and (iii), for composite as well as prime $Q$.

	Suppose (ii) holds. For $0\ne x\in(\Z/Q\Z)^d$, let
	\[
	R(x)=\{i:(Ax)_i=0\pmod Q\},\qquad S(x)=R(x)^c.
	\]
	If the real rank of the rows in $R(x)$ were $d$, these rows would contain $I\in\mathcal B(A)$ with $A_Ix=0\pmod Q$. Invertibility would imply $x=0$, a contradiction. Lemma~\ref{lem:hyperplane} therefore gives a cocircuit $D\subseteq S(x)$. Every nonzero residue $r\pmod Q$ satisfies
	\[
	\sin^2(\pi r/Q)\ge\sin^2(\pi/Q).
	\]
	Together with positive weights this implies
	\[
	E_Q(x)\ge4\sin^2(\pi/Q)\sum_{i\in S(x)}w_i
	\ge4\sin^2(\pi/Q)\lambda_A(w).
	\]
	Taking the minimum over $x$ proves (i).

	For the converse, choose a basis $I$ and a prime $p$ dividing both $Q$ and $\det A_I$. There exists a nonzero $y\in(\Z/p\Z)^d$ with $A_Iy=0$. Set
	\begin{equation}\label{eq:badwitness}
	x=\frac Qp y\pmod Q,\qquad W=(m+1)Q^2,\qquad
	w_i=\begin{cases}W&i\in I,\\1&i\notin I.\end{cases}
	\end{equation}
	The residue vector $x$ is nonzero, and $A_Ix=0\pmod Q$. Faithfulness ensures that $Ax$ is not zero on all rows. Only the $m-d$ outside rows can contribute to $E_Q(x)$, so
	\[
	E_Q(x)\le4(m-d).
	\]
	Every real cocircuit intersects $I$: a nonzero real vector annihilating all basis rows would be zero. Thus $\lambda_A(w)\ge W$. Concavity of sine on $[0,\pi/2]$ gives $\sin(\pi/Q)\ge2/Q$, including $Q=2$. Consequently,
	\[
	4\sin^2(\pi/Q)\lambda_A(w)
	\ge \frac{16W}{Q^2}=16(m+1)>4(m-d).
	\]
	The particular codeword difference $x$ is already shorter than the asserted lower bound, proving failure of (i). All steps of the witness construction use integer arithmetic except the sine inequality, whose rational lower bound suffices to certify the strict violation.
	\end{proof}

	\begin{corollary}[Arbitrarily severe weighted failure]\label{cor:severe}
	For a fixed unsafe $A,Q$, the ratio
	\[
	\frac{D_Q(A,w)^2}{4\sin^2(\pi/Q)\lambda_A(w)}
	\]
	can be made arbitrarily close to zero by positive weights.
	\end{corollary}
	\begin{proof}
	Keep weight $W$ on the bad basis and weight one outside. The numerator is bounded by $4(m-d)$, while the denominator is at least $4W\sin^2(\pi/Q)$. Let $W$ tend to infinity. Alternatively, normalize all weights by their sum; the ratio is unchanged because numerator and denominator are homogeneous of degree one in weights.
	\end{proof}

	\subsection{The Smith kernel formula and what it measures}
	The preceding proof only needs a bad basis. A fuller certificate records how many grid configurations a subsystem fails to distinguish.
	\begin{proposition}[Subsystem torsion count]\label{prop:smith}
	Let $A_R$ have real rank $r$, with nonzero Smith invariants $s_1\mid\cdots\mid s_r$. Then
	\begin{equation}\label{eq:smithkernel}
	\#\ker(A_R:(\Z/Q\Z)^d\to(\Z/Q\Z)^{\|R\|})
	=Q^{d-r}\prod_{j=1}^r\gcd(Q,s_j).
	\end{equation}
	\end{proposition}
	\begin{proof}
	Integer unimodular row and column operations remain bijections modulo $Q$, so they do not change the kernel cardinality. In Smith form, $r$ independent equations are $s_jz_j=0\pmod Q$, each with exactly $\gcd(Q,s_j)$ solutions. The remaining $d-r$ coordinates are arbitrary. Multiplication gives \eqref{eq:smithkernel}.
	\end{proof}
	The $Q^{d-r}$ factor reflects an ordinary real-rank deficit. The additional gcd factors reflect arithmetic aliasing beyond that deficit. Arithmetic matroids and toric arrangements were introduced to track closely related lattice-index and torsion information \cite{dadderio,mociquasi}. The new assertion proposed here is the exact equivalence between a weighted \emph{Euclidean quantization lower bound} and the maximal-basis unit condition, in the specified spectral realization; the use of Smith invariants is classical.

	\subsection{Safe does not mean exact, unsafe does not mean always failing}
	These two qualifications avoid opposite errors. A safe alphabet guarantees the lower bound for every weighting, but the shortest nonzero character values may not all be $\pm1$ modulo that alphabet. Equality can fail upward. Conversely, an unsafe alphabet admits a carefully chosen weighting that fails, but it need not fail at unit weights or at every fixed weighting.

	For example, $A=(2,3)^\top$ is primitive, with exceptional primes $2,3$. Its only cocircuit is both rows, so $\lambda_A(w)=w_1+w_2$. At $Q=2$ and $w=(1,1)$, the squared minimum distance is $4$, while the putative universal formula would give $8$. At $Q=7$, which is safe, equality in the lower bound would require both $2x$ and $3x$ to be $\pm1$ modulo $7$. No such nonzero residue exists: multiplying the first equation by $3$ and the second by $2$ would force $3\equiv\pm2\pmod7$. The lower bound is therefore strict for all positive weights in this example.

	The set of safe moduli is periodic with respect to divisibility by the squarefree product of exceptional primes. That statement concerns the validity of a uniform-in-weights theorem, not periodicity of the actual numerical minimum distance.