itemP_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.itemP_runs (body : IStmt HS P) (f : ℤ → ℤ) (c B : ℕ) (s : IState HS) (L : List ℤ)
    (R0 : List Item) (h0 : s.G 0 = ints L ++ R0) (hB : ∀ x ∈ L, zsz (f x) ≤ B)
    (hb : ∀ (st : IState HS) (x : ℤ) (L0 : List Item), Fixd s st → st.G 0 = .int x :: L0 →
      x ∈ L → IRuns I iprog body st ((st.upd 0 L0).upd pO (.int (f x) :: st.G pO)) c) :
    IRuns I iprog (itemP body L.length) s (s.upd 0 (ints (L.map f) ++ R0))
      (L.length * (c + 64) + 60 + cRpt L.length +
        (L.length * (110 * (B + 1) + 64) + 60 + cRpt L.length))

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.