Manuscript X-J-ST@1 · X-J-ST.tex · theorem · 8f0d1a36

Theorem: Robust selective certificate

Errors, finite composition, locality, and transport › Input precision and robust positive margins · lines 1713–1748 · proof included or linked

X-J-ST@1#thm:robust, X-J-ST@1#eq:robustTV, X-J-ST@1#eq:enclosuregap

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Suppose the two complete implemented comparisons have trace-distance errors at most $e_{d}, e_{0}$ relative to the complete ideal outputs of Theorem thm:TV. Their actual record gap satisfies

d_{TV} (P_{d}, P_{0}) \geq max {0, Δ (d) - e_{d} - e_{0}} .

If a positive task value is certified by $d \in [L, U]$ , $L > 0$ , and the declared parameter bounds give

r \leq r_{m a x} < 1, ∣ ν ∣ \geq ν_{m i n} > 0, σ \leq σ_{m a x},

then

Δ (d) \geq \frac{1 - r _{m a x}}{2} tanh (\frac{ν _{m i n} L}{σ _{m a x}}) > 0.

Proof. Taking a record marginal is trace-distance contractive. Apply the reverse triangle inequality through the two ideal laws. The second result follows from monotonicity of $tanh$ on nonnegative arguments.

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X-J-ST@1#eq:robustTV · line 1719
$d_{TV} (P_{d}, P_{0}) \geq max {0, Δ (d) - e_{d} - e_{0}} .$
X-J-ST@1#eq:enclosuregap · line 1734
$Δ (d) \geq \frac{1 - r _{m a x}}{2} tanh (\frac{ν _{m i n} L}{σ _{m a x}}) > 0.$
unlabeled · line 1728
$r \leq r_{m a x} < 1, ∣ ν ∣ \geq ν_{m i n} > 0, σ \leq σ_{m a x},$

Theorem: Robust selective certificate

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