General boundary-bank error

For a finite set of boundary tests, suppose $p(f)\le 1-\kappa$, $0<\kappa <1$. Project empirical probabilities into $[0,1-\kappa /2]$. The inverse response $F(p)=p/(1-p)$ has derivative at most $4/{\kappa }^2$ on that interval. Therefore

If every energy used by polarization has error at most ${\epsilon }_Q$, the resulting complex pairing has error at most ${\epsilon }_Q$.

For $K$ predeclared probability tests, independent $M$-shot blocks and a union bound give simultaneous probability error

with coverage at least $1-\alpha$. A calibration certificate valid only with probability $1-\beta$ reduces unconditional coverage to at least $1-\alpha -\beta$.

These bounds concern finitely many form values. They do not give metric stability, exact recovery from finitely many noisy records, or a resource bound for a general inverse algorithm.