A finite verification programme

For finite input dimension $d$, a normalized Choi comparison gives

A separate already-certified diamond allowance is added without another factor $d$. This is the certificate accounting emphasized in Cheung Hawk02-E and Hawk04-A CheungHawk02E,CheungHawk04A; see also Watrous.

Flagged truncations must remain deterministic. For a projection $P$, the map

is CPTP. For states commuting with $P$, the trace-distance error in a common enlarged output space is exactly the discarded probability. That equality is not asserted for arbitrary states with low–high coherences. Cheung En02-E gives this accounting for its specified Gaussian packet states CheungEn02E (thm:cutoff).

A finite-graph certificate is neither a continuum limit nor evidence that the operational, algebraic, or covariance construction gaps have been discharged.