Proposition 14 in IL, unconditional: for the Lemma 12 table KFL.l12prog, with λ = τ50, CR = Kc (K+2) + 1…

Proposition 14 in IL, unconditional: for the Lemma 12 table KFL.l12prog, with λ = τ50, CR = Kc (K+2) + 1 and Ψ = PsiR r Z.

The following holds (Lean statement, in the scope of module Solutions.KFMProp14U):

theorem NLogN.KFM.prop14Ph_l12 : ∃ (A : ℕ) (Kc : ℝ), 0 < Kc ∧
    ∀ (Isym : List Sym) {K r Z : ℕ}, 6 ≤ K → 1 ≤ r → 80 * Z * (Z + 2) ≤ 2 ^ K →
    let CR : ℝ := Kc * (K + 2) + 1
    let Ψ : ℕ → ℕ := KFR.PsiR r Z
    ∀ {P D S ρ : ℕ} (hρ : ρ < K), 1 ≤ S → 1 ≤ P → 1 ≤ D →
    zsz ((32 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ) ≤ Z →
    ∃ (Cc : ℕ → IStmt HS 1) (E dC : ℕ) (kC : ℕ → ℕ),
      E ≤ D + (Nat.log (100 ^ 3) D + 1) * 100 ^ 3 * ((Nat.log (100 ^ 3) D + 1) *
        NLogN.KFN.Gb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A ^ Nat.log (100 ^ 3) D) ∧
      dC ≤ 2 * D + (Nat.log (100 ^ 3) D + 1) * 100 ^ 3 * (4 * (Nat.log (100 ^ 3) D + 1) ^ 2 *
        NLogN.KFN.Gb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A ^ (Nat.log (100 ^ 3) D + 2)) ∧
      (∀ w, (kC w : ℝ) ≤ (2 * ((((Nat.log (100 ^ 3) D + 1) * 100 ^ 3 : ℕ) : ℝ) *
          ((100000 + (Nat.log (100 ^ 3) D : ℝ) *
            (((P8Lift.sc + P8Lift.complexIface.net.length) * alN P8Lift.Wc A + 1 : ℕ) : ℝ) +
            (Nat.log (100 ^ 3) D : ℝ) *
              (2 * (((P8Lift.sc + P8Lift.complexIface.net.length : ℕ) : ℝ) * CR))) *
            (D : ℝ) ^ τ50) + 1) + 1000000 * (((Nat.log (100 ^ 3) D * 71 : ℕ) : ℝ) + 2)) *
        ((vol (Butterfly.layerShape P K D S) : ℕ) : ℝ) *
        (Xf r Ψ (w + 2 * D + (Nat.log (100 ^ 3) D + 1) * 100 ^ 3 *
          (4 * (Nat.log (100 ^ 3) D + 1) ^ 2 *
            NLogN.KFN.Gb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A ^ (Nat.log (100 ^ 3) D + 2)) +
          20 * (Nat.log (100 ^ 3) D + 1) ^ 3 *
            NLogN.KFN.Gb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A ^ (Nat.log (100 ^ 3) D + 2) +
          2) : ℝ)) ∧
      ∀ (st : IState HS) (f : Fin HS → List ℤ) (zs : List GI) (l : List ℤ)
        (xs : List (Fin r → ℂ)) (p w : ℕ), RecRep r p zs xs →
        xs.length = vol (Butterfly.layerShape P K D S) → f 0 = gflat zs ++ l → ClN f →
        NLogN.KFB.GB w zs → (∀ j (h : j < xs.length), ‖xs[j]‖ ≤ 1) →
        ∃ zs', RecRep r p zs' (Butterfly.prop14Proc (P := P) (S := S) hρ p
            (Butterfly.tens Butterfly.Cm : Butterfly.Op (Fin D)) xs) ∧
          NLogN.KFB.GB (p + 1) zs' ∧
          (∀ j (h : j < (Butterfly.prop14Proc (P := P) (S := S) hρ p
              (Butterfly.tens Butterfly.Cm : Butterfly.Op (Fin D)) xs).length)
            (h' : j < (Butterfly.layerH P K D S ρ hρ xs).length), ∀ c,
            ‖(Butterfly.prop14Proc (P := P) (S := S) hρ p
              (Butterfly.tens Butterfly.Cm : Butterfly.Op (Fin D)) xs)[j] c -
              (Butterfly.layerH P K D S ρ hρ xs)[j] c‖ < Real.sqrt 2 * 2⁻¹ ^ p) ∧
          IRuns Isym KFL.l12prog (p14Ph P K D S ρ r E dC Cc w) (encK st f)
            (encK st (Function.update f 0 (gflat zs' ++ l)))
            (kC w + cWrapPh P K D S r E
              (zsz ((16 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ)) w dC)

Context. The lemma belongs to the part KFM 4b: Proposition 14 (one-pass phases) for the Lemma 12 table, unconditional: KFMProp14.prop14Ph_complex_closed with its RowHyp hypothesis discharged by KFR.closedHyps_uncond (KFL's proof of Lemma 13 in IL, for the table KFL.l12prog), with CR = Kc (K+2) + 1, Ψ = KFR.PsiR r Z, and the exponent λ = τ50 (KFMLam.lam_complex_eq). The gate-class constant A is chosen once, before Isym, K, r, Z. The only side conditions left are 6 ≤ K, 1 ≤ r and 80 Z (Z+2) ≤ 2^K.

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.