Manuscript Psy03-C@1 · Psy03-C-v1.tex · lemma · d3380b90

Lemma: Pointwise cell-parameter allowance

Noise, parameter, and control certificates › An explicit cell-parameter residual · lines 879–900 · proof included or linked

Psy03-C@1#lem:paramcell, Psy03-C@1#eq:paramcell

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The cell difference satisfies

\norm F_{θ} (u, h) - F_{θ^{'}} (u, h)_{2} \leq q_{n} + \frac{\norm w _{n}^{'} _{\infty}}{4} q_{r} + \frac{R + 1}{4} q_{ζ} =: ϵ_{F} (u, h) .

Proof. Gate differences are at most $q_{r} /4, q_{ζ} /4$ . The candidate argument difference is bounded by $q_{n} + \norm w_{n}^{'}_{\infty} q_{r} /4$ , using $r \circ (w - w^{'}) + (r - r^{'}) \circ w^{'}$ . Tanh is $1$ -Lipschitz. Apply the output decomposition from Theorem thm:celllip at common $h$ .

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Psy03-C@1#eq:paramcell · line 882
$\norm F_{θ} (u, h) - F_{θ^{'}} (u, h)_{2} \leq q_{n} + \frac{\norm w _{n}^{'} _{\infty}}{4} q_{r} + \frac{R + 1}{4} q_{ζ} =: ϵ_{F} (u, h) .$

Lemma: Pointwise cell-parameter allowance

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