A uniform register program from the spec of its literal instance

A uniform register program from the spec of its literal instance.

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

theorem NLogN.KFW.FSp.uni {σ : Fin HS → Option ℤ} (hσ : SClean σ) {U O : RS} (hw : NoW σ U) {X : ℕ}
    (hO : HRef O (RS.sub σ U) X) {s t : IState HS} (hs : Agree σ s) {c : ℕ} (H : FSp O s t c) :
    FSp U s t (400 * (c + X))

Context. The lemma belongs to the part KFW: header refinement, uniform leaves, parameters in the background.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs20.