%%==================================================================
\section{An exact-arithmetic check}\label{app:check}
%%==================================================================

No statement of the paper depends on a computation. The program described here
was written to confirm, on instances small enough to be summed directly, that
the algebraic identities of Sections \ref{sec:routine} and \ref{sec:design}
hold verbatim over a truncated polynomial ring, and it is supplied with the
archive \cite{Bha26arch}.

The program builds a design with the parameters $r=2$, $Q=3$, $R=2$, $A_0=0$,
far below the asymptotic regime of Lemma~\ref{lem:params}, which is
permissible because the identity $H(0)=Z$ of Proposition~\ref{prop:routine}
needs only the five conditions of Definition~\ref{def:design} and not the size
bounds. The auxiliary set has $D=r^{2Q}=64$ elements; the measurement tuples
are drawn from $\{0,\dots,R-1\}^{\tau n}$ with $\tau n=2$; the labels of
Lemma~\ref{lem:labels} are chosen at random from a fixed seed and the
conditions \eqref{eq:balance} and \eqref{eq:kernel} are asserted, as is the
fixed-point property of $M_0$ and the normalisation $P_u(0)=\e_u$ of
Lemma~\ref{lem:rows}. Every number is a rational in lowest terms, every element
of $\A_N$ a list of $N$ rationals, and every element of $\A_N[z]/(z^{J})$ a list
of $J$ such lists; the Hermite system is solved by exact Gaussian elimination.

Seven sums were computed on $n=6$ variables and compared with the direct sum over
the $64$ assignments: the number of satisfying assignments of a random $2$-CNF
with seven clauses over $\A=\Q$; the Max-$2$-SAT profile of the same formula over
$\Q[W]/(W^{8})$, which came out as
$4W^{3}+8W^{4}+28W^{5}+16W^{6}+8W^{7}$; the independence polynomial
$1+6W+8W^{2}+2W^{3}$ of the hexagon with one chord, over $\Q[W]/(W^{7})$, from
the factor system of Theorem~\ref{thm:indep}; the cut profile
$2+12W^{2}+18W^{3}+18W^{4}+12W^{5}+2W^{7}$ of the same graph, from the factor
system of Corollary~\ref{cor:maxcut} with $U=1$; the weight enumerator
$1+4W^{3}+3W^{4}$ of the binary code of length $6$ with the parity checks
$\{1,2,3\}$, $\{3,4,5\}$, $\{5,6,1\}$, from the factor system of
Theorem~\ref{thm:codes}; the number $2$ of nowhere-zero $\Z_3$-flows of
$K_{2,3}$, from the factor system of Theorem~\ref{thm:edges}\ref{it:flows} over
$\Q$, which agrees with Lemma~\ref{lem:flows} since the flow polynomial of
$K_{2,3}$ is $(q-1)(q-2)$; and the objective profile
$2W+W^{2}+4W^{3}+3W^{4}+2W^{5}+3W^{6}+W^{8}$ of the $0$-$1$ program with the
rows $2x_1-x_2+3x_3\le3$, $x_3+x_4+x_5=1$, $x_1+x_5-x_6\ge0$ and the cost
vector $(1,2,1,3,1,2)$, from the factor system of Theorem~\ref{thm:ip}, over
$\Q[W]/(W^{11})$. All seven agreed with the direct sums. The run took
about twenty minutes in exact arithmetic, almost all of it in the sums
\eqref{eq:Htsum} over the pairs $(E,U)$, whose number is the quantity $C_q$ of
\eqref{eq:Cq} with $q$ between $8$ and $13$.

A second program, also in exact integer and rational arithmetic and supplied
with the archive, checks the finite statements behind Section~\ref{ssec:density}:
the inequality $C_q\,q^q(n-q)^{n-q}\le2^qn^n$ of Lemma~\ref{lem:entropy} for
$1\le n\le200$ and $0\le q\le n/2$; the identity \eqref{eq:complementcut} and the
resulting transfer of cut profiles for every graph on at most six vertices, $33868$
graphs, and for $200$ weighted instances; the inequalities of the proof of
Theorem~\ref{thm:explicit} at $32$ pairs $(\lambda,a)$ with $\lambda$ between $2^{40}$
and $2^{128}$ and $\theta\le2^{-15}$, where the asymptotic estimates are replaced
by exact comparisons of integers; the degree $2m$ and the top coefficient
$(-2)^m(W-1)^m$ of Remark~\ref{rem:densityend} for $m\le5$; the two values of
the footnote to the proof of Theorem~\ref{thm:linear}; and the identity
\eqref{eq:lowrankcut} with the profile read off the generating function of
Theorem~\ref{thm:lowrank}, on $300$ random weighted instances, together with the
representation of Corollary~\ref{cor:blowup} for $200$ perturbed blow-ups. All eight
checks passed in a few seconds.