Finite-factor subsystem certification

The correlations may equivalently be measured using a Bell input on $QC$ and an independent maximally mixed input $I_S/s$; tracing $R$ before or after the experiment gives the same laws. This does not restrict the conclusion to maximally mixed spectators: the Choi conversion proves a channel bound on all inputs, with the explicit dimension cost $4s$.

For equal counts, best-case generic subsystem acceptance with zero allowances requires

This is another certificate-specific consequence, not a universal subsystem sample lower bound.

Applied to the two-pentagon patch, take $S=C_R$, $s=2$, and

The ideal decoder in Theorem thm:patchrecovery has target ${\mathrm{Tr}}_{C_R}$ exactly. Its implemented version must satisfy the same complete-access and validation obligations as the single block.