Lemma 19, FPApprox form

Lemma 19, FPApprox form. Sapprox approximates 2^p A and the permutation-free B0P approximates 2^p B₀ (B₀ = P_s B P_t⁻¹) in the sense of FPApprox (outputs stay in the disc), and both lifted maps are sup-norm 1-Lipschitz.

The following holds (Lean statement, in the scope of module Solutions.KFMR19):

theorem NLogN.KFM.lemma19_fp (p s t α : ℕ) [NeZero s] [NeZero t] (hst : s < t) (hcop : Nat.Coprime s t)
    (hα : 2 ≤ α) (hαp : α ^ 2 ≤ p) (hx : s ≤ α ^ 2 * (t - s))
    (hp : 100 * (IntMul.HvdH.jIter p s t α * (10 * IntMul.HvdH.eWidth p α + 12) + 1) ≤ 2 ^ p)
    (hpA : 4 * (6 * IntMul.HvdH.sWidth p α + 4) ≤ 2 ^ p) :
    ∃ (A : (ZMod s → ℂ) →L[ℂ] (ZMod t → ℂ)) (B₀ : (ZMod t → ℂ) →L[ℂ] (ZMod s → ℂ)),
      ‖A‖ ≤ 1 ∧ ‖B₀‖ ≤ 1 ∧
      (∀ u, IntMul.OpenAI.RH.permS s t ((s : ℂ)⁻¹ • ZMod.dft u) =
        (2 : ℂ) ^ (2 * α ^ 2) • B₀ (IntMul.OpenAI.RH.permT s t ((t : ℂ)⁻¹ • ZMod.dft (A u)))) ∧
      IntMul.HvdH.FPApprox p (IntMul.HvdH.Sapprox p s t α) (NLogN.KFM.liftL A)
        (6 * IntMul.HvdH.sWidth p α + 4) ∧
      IntMul.HvdH.FPApprox p (B0P p s t α) (NLogN.KFM.liftL B₀)
        (IntMul.HvdH.jIter p s t α * (10 * IntMul.HvdH.eWidth p α + 12) + 4) ∧
      IntMul.HvdH.SupLip (NLogN.KFM.liftL A) ∧ IntMul.HvdH.SupLip (NLogN.KFM.liftL B₀)

Context. The lemma belongs to the part Item 12 (math side): Lemma 19/20 with the base witnesses and a permutation-free B̃₀: The resampling identities of Resample19/ResampleTensor/ResampleFinal are existential in the maps A, B₀. The machine computes the base thread's fixed-point maps Sapprox and Bapprox. This file pins the witnesses to those maps.

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.