Manuscript Psy02-B@2 · Psy02-B-v2.tex · theorem · cfd06e2e

Theorem: Complete forward expectation and robust gap

Track A6–A7: forward instrument and robust viability · lines 1104–1141 · proof included or linked

Psy02-B@2#thm:gap

Let $v = 1 - 2 ε$ , $A = p η v /2$ , and $r = κ + γ$ . Then

E_{R} f = \frac{1}{2} - A e^{- (r + R) t} .

For rectangular calibrated families

A \in [A_{-}, A_{+}], r \in [r_{-}, r_{+}], R \in [R_{-}, R_{+}], A_{-} > 0, R_{-} > 0,

the null $R = 0$ and alternative have exact ideal oriented gap

g_{ideal} (t) = A_{-} e^{- r_{+} t} - A_{+} e^{- (r_{-} + R_{-}) t} .

For nonrectangular families enclosed by these intervals, the same expression is a lower bound, not necessarily the exact gap. Uniform complete-law errors $e_{0}, e_{1}$ give

g_{robust} = g_{ideal} - e_{0} - e_{1} .

For $t > 0$ , the displayed ideal lower bound is positive exactly when

R_{-} > r_{+} - r_{-} + t^{- 1} lo g (A_{+} / A_{-}) .

Proof. The ideal signed outcome mean is $q c$ . Assignment, observation, and preparation multiply it by $v, η, p$ . Neutral outcomes have zero centered score. For a rectangle, the null supremum is $1/2 - A_{-} e^{- r_{+} t}$ , and the alternative infimum is $1/2 - A_{+} e^{- (r_{-} + R_{-}) t}$ . Subtract and apply total-variation control for a bounded score.

Displayed formulas in this passage

unlabeled · line 1108
$E_{R} f = \frac{1}{2} - A e^{- (r + R) t} .$
unlabeled · line 1112
$A \in [A_{-}, A_{+}], r \in [r_{-}, r_{+}], R \in [R_{-}, R_{+}], A_{-} > 0, R_{-} > 0,$
unlabeled · line 1117
$g_{ideal} (t) = A_{-} e^{- r_{+} t} - A_{+} e^{- (r_{-} + R_{-}) t} .$
unlabeled · line 1124
$g_{robust} = g_{ideal} - e_{0} - e_{1} .$
unlabeled · line 1128
$R_{-} > r_{+} - r_{-} + t^{- 1} lo g (A_{+} / A_{-}) .$

Theorem: Complete forward expectation and robust gap

Displayed formulas in this passage

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