Complex exchange, signed network form

Complex exchange, signed network form. For any labelling, every gate is OK and the scalar circuit routes w to ρ w with sign ε w.

The following holds (Lean statement, in the scope of module Solutions.NetComplexLemma5):

theorem NetComplex.complex_exchange_net {M : Type*} [AddGroup M] (hcard : ∀ t, #(ind t) = 3)
    (hinj : Function.Injective ind) (lab : GName T → M) :
    (∀ g ∈ cxGatesL R ind lab, g.OK) ∧
      ∀ (v : Wr ind → R) w, scalarRun (cxGatesL R ind lab) v (NetComplex.ρ ind w) = NetComplex.ε R ind w * v w

Context. The lemma belongs to the part Lemma 5 (complex construction) as a scalar gate network over a ring with 1/2: Scalars lie in a commutative ring R in which 2 is invertible (the paper uses ℤ[i, 1/2]). Parameters as in NetScalar: a ground set Fin h, a finite type T of triples with ind : T → Finset (Fin h), #(ind t) = 3, ind injective.

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.