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

Theorem: Global stress-accuracy certificate

Stress, margins, and complete perturbation budgets › A general path lemma and the damped certificate · lines 647–678 · proof included or linked

Psy03-A@1#thm:stressaccuracy

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Let

H_{tot} = 2 + 512 + 512 + 256 = 1282, R_{δ} = 2 lo g (2 H_{tot} / δ), 0 < δ < 1,

and put

C (ξ, σ) = ξ L_{x} 2 + σ r = 1 \sum^{3} a^{4 - r} P_{r} n_{r} .

For the clean margin $m_{0} = m (z_{0, 0} (X), Y)$ ,

Q (ξ, σ) \geq P {m_{0} > 2 R_{δ} C (ξ, σ)} - δ .

The right side may be truncated below at zero.

Proof. The scalar Gaussian tail bound $P (\abs G > u) \leq 2 e^{- u^{2} /2}$ , followed by a union bound, shows that all $1282$ noise coordinates have magnitude at most $R_{δ}$ with probability at least $1 - δ$ . Equation (eq:pathbound) is then at most $R_{δ} C (ξ, σ)$ . The margin condition and (eq:marginlip) preserve correctness.

Displayed formulas in this passage

unlabeled · line 650
$H_{tot} = 2 + 512 + 512 + 256 = 1282, R_{δ} = 2 lo g (2 H_{tot} / δ), 0 < δ < 1,$
unlabeled · line 656
$C (ξ, σ) = ξ L_{x} 2 + σ r = 1 \sum^{3} a^{4 - r} P_{r} n_{r} .$
unlabeled · line 662
$Q (ξ, σ) \geq P {m_{0} > 2 R_{δ} C (ξ, σ)} - δ .$

Theorem: Global stress-accuracy certificate

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