Theorem: Full-model near-obstruction certificate

Choose $b_0=b_1+\lambda A_{r_{\mathrm{c}}}c$ in the calibration interval. Replace the gravitational quadratic term by $\lambda g_s(t){\mathcal{G}}_{r_{\mathrm{c}}}$. Suppose bounded-aperture comparison models have $\Vert \hat{x}\Vert \le L_s(t)$, and the actual complete laws are within total variation $e_h$ of those comparison laws.

With uniform history-wise bounds when feedback is used,

where

The sums denote the stages of an attempted run, not an unweighted sum over alternative experimental designs. A bound exceeding one is replaced by the trivial bound one.

For $N$ independent attempted runs,

Thus uniformly small errors ${\alpha }_N,{\beta }_N\le \alpha <1/2$ require $N\zeta \ge 1-2\alpha$.

Proof. After matching the scalar shift, the generator difference is

The first norm is bounded by Proposition prop:aperture. The second is at most $4\lambda A_{r_{\mathrm{c}}}{L_s}^2|g_s-c{f}_s|$. Variation of constants and CPTP contractivity bound the state trace distance by half the integrated generator norms. Complete instruments and terminal processing contract the same norm. Add the two law-approximation errors.

For product laws, $\mathrm{TV}(p^{\otimes N},q^{\otimes N})\le N\mathrm{TV}(p,q)$ by telescoping. Every test has error sum at least one minus product-law total variation.