Remark: What is and is not completed

The shift in (eq:arithrepair) is essential unless positivity of the numerical $A_k$ has been independently certified. Scaling an arbitrary rounded matrix alone cannot repair a negative eigenvalue.

The displayed effects are exact mathematical matrices defined by the stored centers and certified bounds. Exact rational centers and rational outward bounds permit an exact rational completion. If the completed effects or reported probabilities are subsequently rounded, those errors require additional enclosures.

One possible finite instrument uses update operators $\sqrt{F_k}$. The effect-law estimate above does not, by itself, certify an approximation of a particular post-measurement instrument. Because the exact outside effect is included, its probability is retained rather than postselected away. The domain-tail bound is needed when outside events are omitted or when a truncated raw quadrature is compared as in (eq:bosonbudget).