Manuscript Hawk01-A@1 · Hawk01-A.tex · proposition · 11979653

Proposition: Finite-cover null certificate

Observer access, collection, and robust certification › Certified astrophysical nulls · lines 2007–2039 · proof included or linked

Hawk01-A@1#prop:nullcover, Hawk01-A@1#eq:nullcover

Suppose cells $Θ_{k}$ cover $Θ_{sc}$ . For each cell there is a regulated representative $N_{k}$ and a bound

θ \in Θ_{k} sup D_{⋄} (N^{sc}_{u, θ}, N_{k}) \leq r_{k} .

If $Q_{k}$ is a computed score with certified absolute numerical error at most $e_{k}$ , then

b_{sc} (u) \leq min {q_{Q}, k max (Q_{k} + e_{k} + r_{k})} .

Proof. For $θ \in Θ_{k}$ , Proposition prop:scorecontinuity gives

Q_{opt} (N^{sc}_{u, θ}) \leq Q_{k} + e_{k} + r_{k} .

Take the supremum over the covering cells and impose the trusted-input ceiling.

Displayed formulas in this passage

Hawk01-A@1#eq:nullcover · line 2019
$b_{sc} (u) \leq min {q_{Q}, k max (Q_{k} + e_{k} + r_{k})} .$
unlabeled · line 2012
$θ \in Θ_{k} sup D_{⋄} (N^{sc}_{u, θ}, N_{k}) \leq r_{k} .$
unlabeled · line 2033
$Q_{opt} (N^{sc}_{u, θ}) \leq Q_{k} + e_{k} + r_{k} .$