gmulP_runs (KFM 9a: the pointwise Gaussian product pass `Q_Δ(θ · f)` with `MulOK` products)

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

theorem NLogN.KFM.gmulP_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 : IState HS) (zs ts : List GI) (R0 RW LR : List Item) (hlen : ts.length = zs.length)
    (h0 : s.G 0 = ints (gflat zs) ++ R0) (hW : s.G pW = ints (gflat ts) ++ RW)
    (hR : s.G pR = .int (2 ^ Δ) :: LR) (hz : NLogN.KFB.GB w zs) (ht : NLogN.KFB.GB w ts) :
    IRuns I iprog (gmulP pM zs.length) s
      ((s.upd 0 (ints (gflat (List.zipWith (gmulT Δ) zs ts)) ++ R0)).upd pW RW)
      (cGmul zs.length w Δ φ)

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.