Proposition 14 in IL for the interface P8Lift.complexIface (W_c roles, m = 100^3, s_c children), with τ =…

Proposition 14 in IL for the interface P8Lift.complexIface (W_c roles, m = 100^3, s_c children), with τ = 1 - 2^{-50}: conditional only on the row additions RowHyp.

The following holds (Lean statement, in the scope of module Solutions.KFNP14):

theorem NLogN.KFN.prop14_il_complex : ∃ A : ℕ, ∀ {K r Z : ℕ} {Ψ : ℕ → ℕ} {CR : ℝ},
    RowHyp Isym iprog K (100 ^ 3) r Z Ψ CR τ50 → 0 ≤ CR → 1 ≤ r → Monotone Ψ →
    (∀ v, Z + 2 ≤ Ψ v) → ∀ {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 Pp) (E dC : ℕ) (kC : ℕ → ℕ),
      E ≤ D + (Nat.log (100 ^ 3) D + 1) * 100 ^ 3 *
        Eb P8Lift.complexIface.net.length P8Lift.sc (Nat.log (100 ^ 3) D) ∧
      dC ≤ 2 * D + (Nat.log (100 ^ 3) D + 1) * 100 ^ 3 *
        DWb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A (Nat.log (100 ^ 3) D) ∧
      (∀ w, (kC w : ℝ) ≤ (2 * ((((Nat.log (100 ^ 3) D + 1) * 100 ^ 3 : ℕ) : ℝ) *
          cb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A (100 ^ 3) CR τ50
            (Nat.log (100 ^ 3) D) + 1) + 1000000 * ((D : 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 *
          DWb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A (Nat.log (100 ^ 3) D) +
          Whb P8Lift.complexIface.net.length P8Lift.sc P8Lift.Wc A (Nat.log (100 ^ 3) D) + 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 iprog (p14WithI P K D S ρ r E Cc w) (encK st f)
            (encK st (Function.update f 0 (gflat zs' ++ l)))
            (kC (w + 2 * D) + cWrapE D E (vol (Butterfly.layerShape P K D S) * r)
              (zsz ((12 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ)) w dC)

Context. The lemma belongs to the part KFN: Proposition 14 in IL around the recursive C-layer.

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_Defs20.