The uniform Proposition 18, from the uniform Proposition 14 at A and the width condition at A

The uniform Proposition 18, from the uniform Proposition 14 at A and the width condition at A.

The following holds (Lean statement, in the scope of module Solutions.KFV62Top):

theorem NLogN.KFV.p18IL_at (A : ℕ) (HU : P14Uat A) (HE : EwB18 A) : KFM.P18IL

Context. The lemma belongs to the part KFV §62: the uniform Proposition 18 at a gate-class constant A.

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.