The source lengths of §9.2.2 exist for n ≥ n₀ (pairwise coprime odd s_i in the resampling windows of the…

The source lengths of §9.2.2 exist for n ≥ n₀ (pairwise coprime odd s_i in the resampling windows of the axis lengths r/2 or r).

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

theorem IntMul.OpenAI.Glue.sources_ok {n : ℕ} (hn : n0 ≤ n) (d' : ℕ) (hd' : d' ≤ dP n) :
    ∃ s : Fin (dP n) → ℕ, (∀ i, Odd (s i)) ∧ (∀ i j, i ≠ j → Nat.Coprime (s i) (s j)) ∧
      ∀ i : Fin (dP n), (4 * dP n - 2) * (if (i : ℕ) < d' then rP n / 2 else rP n) < 4 * dP n * s i ∧
        4 * dP n * s i ≤ (4 * dP n - 1) * (if (i : ℕ) < d' then rP n / 2 else rP n)

Context. The lemma belongs to the part Glue 3: explicit parameters n ↦ (b, L, T, d, K, ℓ, α, p) of §9.2 and the threshold n₀: For an input length n (§9.2.1):

• b = lg n (bP), p = 6b (pP), d = ⌊b^ε⌋ (dP), K = ⌊d^c⌋ (KP), α = ⌈(12 d² b)^{1/4}⌉ (αP);
• L (LP) = the least L with 4n ≤ 2^L b, T = 2^L (TP), so 4n ≤ T b < 8n;
• ℓ = ⌈L/d⌉ (ℓP, RP.ell), r = 2^ℓ (rP), bit volume V = T p (VP), γ = 2dα² (γP).

Formalization Note. This lemma belongs to the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the array model of tape data (mixed-radix arrays and pull maps), the interchange networks of §3–4 (Propositions 7, 8, 11, Lemmas 9–12), the butterfly layers of §5 (Lemma 13, Proposition 14), the synthesis of §6–7 (Lemma 16, Proposition 18), the resampling and parameter choices of §8–9, and the final glue IntMul.OpenAI.Glue.kappaBound_of_machine, which derives the platform bound KappaBound (1/2^182) (from Def_IntMul_MultitapeModel) for any correct multitape machine meeting the paper's time recurrence. All names refer to the definition bundles IntMul_Kappa_K_Defs1–IntMul_Kappa_K_Defs9.