padAll_spec (KFU §2: padding / deletion of every field)

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

theorem NLogN.KFU.padAll_spec (S T : Shape) (hF : List.Forall₂ (· ≤ ·) S T) (hS : ∀ a ∈ S, 1 ≤ a)
    (w : ℕ) (xs : List ℤ) (hxs : xs.length = AModel.vol S) (hb : Bnd w xs) (s : IState HS)
    (R : List Item) (h0 : s.G 0 = ints xs ++ R) :
    ∀ k ≤ S.length, IRuns I iprog (padAllC S T k) s
      (s.upd 0 (ints (AModelL12.embed S (T.take k ++ S.drop k) 0 xs) ++ R)) (k * cScan (AModel.vol T) w + 1)

Context. The lemma belongs to the part KFU §2: padding / deletion of every field: padAllC S T n runs one padding scan per field k < n (S ≤ T pointwise); after all |S| scans stack 0 holds embed S T 0 xs (padAll_spec). unpadAllC S T n runs the deletion scans and computes extract S T 0 ys (unpadAll_spec). Cost ≤ |S| · 10^4 (w+1)(2 vol T + 1).

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_Defs18.