Manuscript Psy03-A@1 · Psy03-A.tex · theorem · b30eb6e9

Theorem: Exact calibrated score and robust gap

Robust complete-record contrasts and identification › A calibrated forward detector · lines 1897–1940 · proof included or linked

Psy03-A@1#thm:gap, Psy03-A@1#eq:interventiongap

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For successful coherent phase preparation and dephasing,

E f = \frac{1}{2} + A e^{- γ T} .

For its population-matched dephased preparation, $E f = 1/2$ .

For a rectangular coherent family

A \in [A_{-}, A_{+}], γ \in [γ_{-}, γ_{+}], A_{-} > 0,

the exact ideal oriented score gap from the dephased family is

g_{ideal} = A_{-} e^{- γ_{+} T} .

If the two actual complete attempted-run laws have uniform total-variation errors $e_{0}, e_{1}$ , then

g_{robust} = A_{-} e^{- γ_{+} T} - e_{0} - e_{1}

is a lower bound on their oriented expectation gap.

For two rectangular coherent intervention families, a sufficient post-minus-pre gap is

A_{post, -} e^{- γ_{post, +} T} - A_{pre, +} e^{- γ_{pre, -} T} - e_{post} - e_{pre} .

Proof. The ideal signed phase outcome has mean $λ$ . Assignment multiplies it by $1 - 2 ε$ ; preparation and observation multiply it by $p η$ . Neutral scores have zero centered contribution. Take the worst-case rectangular endpoints. A $[0, 1]$ -score expectation changes by at most total variation.

Displayed formulas in this passage

unlabeled · line 1900
$E f = \frac{1}{2} + A e^{- γ T} .$
unlabeled · line 1906
$A \in [A_{-}, A_{+}], γ \in [γ_{-}, γ_{+}], A_{-} > 0,$
unlabeled · line 1911
$g_{ideal} = A_{-} e^{- γ_{+} T} .$
unlabeled · line 1916
$g_{robust} = A_{-} e^{- γ_{+} T} - e_{0} - e_{1}$
Psy03-A@1#eq:interventiongap · line 1924
$A_{post, -} e^{- γ_{post, +} T} - A_{pre, +} e^{- γ_{pre, -} T} - e_{post} - e_{pre} .$

Theorem: Exact calibrated score and robust gap

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