oprodP_runs (KFM 9d: the `d`-axis chirp array — outer products of the axis tables)

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

theorem NLogN.KFM.oprodP_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) (A B : List GI) (R1 R2 LR : List Item)
    (h1 : s.G pQ1 = ints (gflat A) ++ R1) (h2 : s.G pQ2 = ints (gflat B) ++ R2)
    (hR : s.G pR = .int (2 ^ Δ) :: LR) (hA : NLogN.KFB.GB w A) (hB : NLogN.KFB.GB w B) (hC : NLogN.KFB.GB w (oprod Δ A B)) :
    IRuns I iprog (oprodP pM A.length B.length) s
      ((s.upd pQ1 R1).upd pQ3 (ints (gflat (oprod Δ A B)).reverse ++ s.G pQ3))
      (A.length * (cOb B.length w Δ φ + 64) + 60 + cRpt A.length)

Context. The lemma belongs to the part KFM 9d: the d-axis chirp array — outer products of the axis tables.

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.