o0_ne (KFV §23: uniform layers in a context)

The following holds (Lean statement, in the scope of module Solutions.KFV19Layer):

theorem NLogN.KFV.o0_ne (v : List ℤ) {j : Fin HS} (h : j ≠ 0) : o0 v j = []

Context. The lemma belongs to the part KFV §23: uniform layers in a context: LayerU Ctx code T N p W cost: on any state encK s (o0 (gflat zs ++ l)) (only stack 0 managed) whose base s satisfies Ctx, code replaces the array by one representing T xs. The round's context (layout registers, flags) lives in the base, so a layer may call procedures that need every other managed stack empty (Proposition 14).

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.