Main statement

Main statement. The restricted interface exists for W_c roles, label dimension m = 100³ and s_c child calls, with s_c < W_c m; through the forgetful map the recursive program computes C^{⊗} at every depth.

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

theorem P8Lift.complexIface_main :
    Nonempty (Prop8IfaceR Wc (100 ^ 3) sc) ∧ sc < Wc * 100 ^ 3 ∧
      ∀ u k, recOp complexIface.toIface u k = Butterfly.tens Cm

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.