\subsection{Identifiability and stability of inferred constraints}

SFT treats target inference as a certificate-producing step. To prevent over-reading, the inference tool reports two quantitative objects:
\begin{itemize}[leftmargin=*]
\item an \emph{excitation summary} computed from the centered increment matrix $\Delta y_i := y_{i+1}-y_i$ after lift-unwrapping, including a numerical rank and a smallest nonzero singular value (a conservative persistent-excitation proxy),
\item a \emph{stability margin} $m$ that measures how far the inferred constraint residuals sit below the declared tolerance.
\end{itemize}

The identifiability claim below is deliberately narrow: it is about \emph{stability within the declared finite search space} (bounded coefficient size, bounded number of constraints), and it is conditional on a declared observation-noise budget.

\begin{theorem}[Conservative identifiability in the declared small-integer regime (tool-matched)]
Fix a period $P>0$ and a dimension $k\ge 1$.
Fix declared inference parameters
\begin{itemize}[leftmargin=*]
\item coefficient bound $M\in\mathbb{N}$ (the tool enumerates $a\in[-M,M]^k\cap\mathbb{Z}^k\setminus\{0\}$),
\item relation cap $L\in\mathbb{N}$ (the tool selects up to $L$ non-multiple relation rows),
\item tolerance fraction $\alpha\in(0,1)$ and absolute tolerance $\tau:=\alpha P$.
\end{itemize}
Assume the tool's basic regime checks are satisfied: the sample size satisfies $n\ge \max(25,8k)$ and not all coordinates are weak at threshold $4\tau$ (otherwise the tool returns WOB-14 or WOB-13).

Let $y_0,\dots,y_{n-1}\in\mathbb{T}_P^k$ be sampled phase vectors and suppose a lift choice produces unwrapped representatives $\tilde y_i\in\mathbb{R}^k$.
Assume a bounded observation model
\[
\|\tilde y_i-\tilde y_i^{\mathrm{true}}\|_{\infty}\le \varepsilon
\qquad\text{for all }i,
\]
for some declared noise budget $\varepsilon\ge 0$.

Suppose the inference tool returns a CST candidate $(A^\star,\Psi^\star)$ with:
\begin{itemize}[leftmargin=*]
\item constraint rank $r=\mathrm{rank}(A^\star)$ and induced target rank $d:=k-r$,
\item joint residual bound $\rho := \max_i \|A^\star \tilde y_i-\Psi^\star\|_{\infty}$,
\item stability margin $m := \tau-\rho>0$,
\item excitation flag \texttt{excitation\_ok=true}, meaning the reported increment-rank proxy is at least $d$.
\end{itemize}
Let the tool's best single-relation residual be $\rho_{\mathrm{best}}$ with first non-multiple runner-up residual $\rho_{\mathrm{ru}}$ (both reported), and define the ranking gap
\[
\Delta := \rho_{\mathrm{ru}}-\rho_{\mathrm{best}}\ \ge 0.
\]

If \(\varepsilon \le \min(m/2,\ \Delta/2)\), then the inference outcome is stable within the declared finite search:
\begin{itemize}[leftmargin=*]
\item the CST feasibility decision cannot flip under perturbations of size $\varepsilon$ (constraints remain feasible at tolerance $\tau$), and
\item the tool's best-relation ranking used in row selection is stable, hence the returned $(A^\star,\Psi^\star)$ does not change under such perturbations.
\end{itemize}
Equivalently, the tool's reported envelopes \(\varepsilon \le m/2\) and \(\varepsilon \le \Delta/2\) provide an auditable safety region for trusting the inferred constraint in the declared small-integer regime.
\end{theorem}

\noindent\textbf{Interpretation.}
The theorem does not claim that all couplings are identifiable from all data.
It says: within the declared finite coefficient regime, the tool reports two concrete safety envelopes: $\varepsilon\le m/2$ for feasibility stability, and $\varepsilon\le \Delta/2$ for stability of the best-relation ranking used in row selection.
When excitation is weak or the margin is small, SFT treats the correct output as an inference obstruction (typically WOB-14, WOB-13, or WOB-15).
\begin{theorem}[Null-model upper bound for spurious small-integer constraints (tool gate)]
Fix a period $P>0$, a tolerance $\tau\in(0,P/2]$, and a dimension $k\ge 1$.
Let $y_1,\dots,y_n$ be independent Haar samples on $\mathbb{T}_P^k$.
Fix a nonzero integer vector $a\in\mathbb{Z}^k\setminus\{0\}$ and set
\[
z_i := a\cdot y_i \ \ (\mathrm{mod}\ P)\in\mathbb{T}_P.
\]
Then $(z_i)_{i=1}^n$ are independent Haar samples on $\mathbb{T}_P$.
Moreover, the event
\[
\exists \Psi\in\mathbb{T}_P\ \text{such that}\ \max_i \mathrm{dist}(z_i,\Psi)\le \tau
\]
is equivalent to all $z_i$ lying in some arc of length $2\tau$, and satisfies the bound
\[
\mathbb{P}\Big(\exists \Psi:\max_i \mathrm{dist}(z_i,\Psi)\le \tau\Big)\ \le\ n\Big(\frac{2\tau}{P}\Big)^{n-1}.
\]
If the inference search enumerates at most $N$ distinct relation directions $a$ (after collapsing integer multiples), then by a union bound the probability of finding \emph{any} such relation within tolerance obeys
\[
p_{\mathrm{upper}} \ \le\ N\,n\Big(\frac{2\tau}{P}\Big)^{n-1}.
\]
If, in addition, the selected relation is required to satisfy the same tolerance on an independent holdout set of size $n_{\mathrm{ho}}$, then the combined null-model bound multiplies:
\[
p_{\mathrm{upper,combined}}\ \le\ \Big(N\,n\Big(\frac{2\tau}{P}\Big)^{n-1}\Big)\cdot \Big(N\,n_{\mathrm{ho}}\Big(\frac{2\tau}{P}\Big)^{n_{\mathrm{ho}}-1}\Big).
\]
\end{theorem}

\noindent\textbf{Interpretation.}
The bound is conservative and uses only Haar geometry and a union bound on the declared finite search.
It supplies an auditable gate: a candidate constraint is accepted only when it is holdout-stable at tolerance $\tau$ and the reported $p_{\mathrm{upper,combined}}$ is below a declared threshold.

