Manuscript Psy04-A@1 · Psy04-A-v1.tex · theorem · d5800e2a

Theorem: Robust family separation

Robust model and apparatus allowances › Complete-law stability · lines 985–1025 · proof included or linked

Psy04-A@1#thm:robustgap, Psy04-A@1#eq:robustgap, Psy04-A@1#eq:robustthreshold

Suppose:

an actual quantum complete law is within total variation $e_{Q}$ of the specified law in Theorem thm:detector;
every actual classical null law is within total variation $e_{C}$ of some complete law satisfying Theorem thm:abstention.

Then

E_{Q} S \geq Q_{score} (T) - e_{Q}, null sup E_{C} S \leq \frac{3}{4} + e_{C} .

Therefore

g_{robust} = \frac{1}{2} + \frac{p v ( 1 - 2 ϵ ) r _{T}}{2 2} - e_{Q} - (\frac{3}{4} + e_{C})

is a certified lower separation allowance. A positive value separates the declared families.

Equivalently, positivity requires

p v (1 - 2 ϵ) r_{T} > \frac{1}{2} + 22 (e_{Q} + e_{C}) .

Proof. Apply Lemma lem:tv to the common score. Subtract the null upper bound from the quantum lower bound and rearrange.

Displayed formulas in this passage

Psy04-A@1#eq:robustgap · line 1001
$g_{robust} = \frac{1}{2} + \frac{p v ( 1 - 2 ϵ ) r _{T}}{2 2} - e_{Q} - (\frac{3}{4} + e_{C})$
unlabeled · line 996
$E_{Q} S \geq Q_{score} (T) - e_{Q}, null sup E_{C} S \leq \frac{3}{4} + e_{C} .$
Psy04-A@1#eq:robustthreshold · line 1013
$p v (1 - 2 ϵ) r_{T} > \frac{1}{2} + 22 (e_{Q} + e_{C}) .$