The closed cost bound of the Lemma 12 program: ≤ K (V + 1) (Bb + 2)² u^τ, V the volume of the bit array,…

The closed cost bound of the Lemma 12 program: ≤ K (V + 1) (Bb + 2)² u^τ, V the volume of the bit array, when the oracle's child count satisfies S / W < m^τ.

The following holds (Lean statement, in the scope of module Solutions.KIL6Cost):

theorem NLogN.KIL.cL12_le {m W S : ℕ} (Kc Kd q : ℕ) (hm : 2 ≤ m) (hW : 1 ≤ W) (hq : 2 ≤ q) {τ : ℝ}
    (hτ : 0 < τ ∧ τ < 1) (hs : (S : ℝ) / W < (m : ℝ) ^ τ) :
    ∃ K : ℝ, 0 ≤ K ∧ ∀ (u : ℕ) (Pre G Suf : Shape) (Bb e Kp : ℕ), 1 ≤ u →
      (∀ a ∈ Pre, 1 ≤ a) → (∀ a ∈ G, 1 ≤ a) → (∀ a ∈ Suf, 1 ≤ a) →
      e = Nat.clog q (2 ^ u) → Kp = Nat.clog m e →
      (cL12 u e (min (W * Kp) e) Kp (e - min (W * Kp) e) (vol (qShape Pre G Suf q e))
        (vol (rowShape (vol Pre) (padRows W Kp (q ^ min (W * Kp) e))
          (G ++ List.replicate (min (W * Kp) e) q) Suf q (e - min (W * Kp) e))) m q
        (fun j => ilCostU5 W Kc Kd S q Bb j
          (vol (rowShape (vol Pre) (padRows W Kp (q ^ min (W * Kp) e))
            (G ++ List.replicate (min (W * Kp) e) q) Suf q (e - min (W * Kp) e)))) Bb : ℝ) ≤
        K * ((vol (binShape Pre G Suf (2 ^ u)) : ℝ) + 1) * ((Bb : ℝ) + 2) ^ 2 * (u : ℝ) ^ τ

Context. The lemma belongs to the part KIL6: closed cost bounds.

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.