Closed per-round bound for one Proposition 14 round of prop14Ph_kTable

Closed per-round bound for one Proposition 14 round of prop14Ph_kTable.

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

theorem NLogN.KFM.round_cost_closed (A : ℕ) (Kc : ℝ) (hKc : 0 ≤ Kc) : ∃ C₀ : ℝ, 0 < C₀ ∧
    ∀ {K r Z P D S : ℕ} (E dC w kCw : ℕ), 1 ≤ D →
      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)) →
      (kCw : 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) * (Kc * (K + 2) + 1)))) *
            (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 (PsiR r Z) (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) →
      (K : ℝ) ≤ (D : ℝ) ^ ((2 : ℝ)⁻¹ ^ 56) → EwB A D ≤ w + 1 → w + EwB A D + 2 < 2 ^ Z →
      2 * (2 * Z + 4) ^ 3 + 2 ≤ r →
      (K * D + P + 10) * (Z + 2) ≤ vol (Butterfly.layerShape P K D S) * r →
      zsz ((16 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ) ≤ Z →
      ((kCw + cWrapPh P K D S r E (zsz ((16 * (vol (Butterfly.layerShape P K D S) * r) : ℕ) : ℤ)) w dC : ℕ)
        : ℝ) ≤ C₀ * ((vol (Butterfly.layerShape P K D S) * r : ℕ) : ℝ) * (w + 1) *
          (D : ℝ) ^ (1 - (2 : ℝ)⁻¹ ^ 54)

Context. The lemma belongs to the part KFM 4c: the closed per-round bound (MACHINE-PLAN item 4, §5 (C)): The cost of one Proposition 14 round in the merged table (prop14Ph_kTable) is kC w + cWrapPh …, with the row constant F1 = 2((k+1)m(10^5 + k a + 2 k b CR) D^{τ50} + 1) + 10^6(71k+2), k = ⌊log_m D⌋, m = 100^3, CR = Kc(K+2)+1, and the width w' = w + Ew(D) inside Xf r Ψ.

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.