Q_Δ(S ·) pass (one MulOK product per item) on N items of ≤ w bits

Q_Δ(S ·) pass (one MulOK product per item) on N items of ≤ w bits.

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

theorem NLogN.KFM.smulP_runs {M : MulSpec} {pM : Fin P} (hM : MulOK I iprog pM M) (hdom : ∀ x y, M.dom x y)
    {w φ : ℕ} (hΦ : ∀ x y : ℤ, zsz x ≤ w → zsz y ≤ w → M.Φ x y ≤ φ) (Δ : ℕ) (S : ℤ)
    (s : IState HS) (L : List ℤ) (R0 LR LS : List Item)
    (h0 : s.G 0 = ints L ++ R0) (hR : s.G pR = .int (2 ^ Δ) :: LR) (hS : s.G pX3 = .int S :: LS)
    (hw : ∀ x ∈ L, zsz x ≤ w) (hSw : zsz S ≤ w) :
    IRuns I iprog (itemP (smulB pM) L.length) s
      (s.upd 0 (ints (L.map fun x => (S * x).tdiv (2 ^ Δ)) ++ R0))
      (cItem L.length (φ + 5000 * (w + Δ + 3)) (2 * w))

Context. The lemma belongs to the part KFM 9a: the pointwise Gaussian product pass Q_Δ(θ · f) with MulOK products.

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_Defs22.