Cost of the recursion through the complex network

Cost of the recursion through the complex network. With σ_c = log_m (s_c / W_c) < 1 (m = 100³), for every λ > σ_c (λ ≥ 0) and leaf width u ≥ 1, the recursive program whose local work is the restricted steps' actual work (nloc · W_c · V) costs ≤ V (K₁ (u m^k)^λ + C₀ (u m^k)^{σ_c} u^{1-σ_c}).

The following holds (Lean statement, in the scope of module Solutions.P8LiftFinal):

theorem P8Lift.complexIface_cost :
    ∃ σc : ℝ, σc < 1 ∧ (sc : ℝ) / Wc = ((100 ^ 3 : ℕ) : ℝ) ^ σc ∧
      ∀ (u : ℕ), 1 ≤ u → ∀ (lam C₀ : ℝ), 0 ≤ lam → σc < lam → 0 ≤ C₀ →
        ∃ K₁ : ℝ, 0 ≤ K₁ ∧ ∀ (Cleaf : ℝ → ℝ), (∀ V, 0 ≤ V → Cleaf V ≤ C₀ * V * u) →
          ∀ k V, 0 ≤ V → recCost complexIface.toIface u
            (fun _ V => (complexIface.nloc * Wc : ℝ) * V) Cleaf k V ≤
            V * (K₁ * ((u : ℝ) * ((100 ^ 3 : ℕ) : ℝ) ^ k) ^ lam +
              C₀ * (((u : ℝ) * ((100 ^ 3 : ℕ) : ℝ) ^ k) ^ σc * (u : ℝ) ^ (1 - σc)))

Context. The lemma belongs to the part The restricted Proposition 8 interface for the complex network (h = 100).

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.