fmulPU_spec (KIL4: the uniform `c` scatter (`c·`) with run-time `Q, P, B`)

The following holds (Lean statement, in the scope of module Solutions.KIL4Mul):

theorem NLogN.KIL.fmulPU_spec (t : IState HS) (xs : List ℤ) (P Q k B Bb : ℕ) (R : List Item)
    (hlen : xs.length = P * Q * B) (hb : Bnd Bb xs) (hk : 1 ≤ k)
    (hQk : Nat.Coprime Q k) (zB : zsz (B : ℤ) ≤ Bb) (zk : zsz ((2 * k : ℕ) : ℤ) ≤ Bb)
    (zQ : zsz (Q : ℤ) ≤ Bb) (zQB : zsz ((Q * B : ℕ) : ℤ) ≤ Bb) (zP : zsz (P : ℤ) ≤ Bb)
    (zV : zsz ((P * (Q * B) : ℕ) : ℤ) ≤ Bb) (h0 : t.G 0 = ints xs ++ R) (hp : StP t P Q B) :
    Sp (fmulPU k) t (t.upd 0 (ints (AModel.pull [P, Q, B] [P, Q, B] (mulAddr Q k) 0 xs) ++ R))
      (cMulPgU P Q k B Bb)

Context. The lemma belongs to the part KIL4: the uniform c scatter (c·) with run-time Q, P, B: mulPgU c depends only on the fixed factor c (as divPgU c, KIL4Div): parameters on stacks 22, 24, 25, 26, 27 (Q, P, B, P·(Q·B), Q·B), residue Q mod c on 28, pass counts on 29. mulPgU_spec: it computes the array of KIL3's mulPg P Q c B. Also the uniform stages fdivPU d, fmulPU k and their specifications.

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.