The edge steps of the rational circuit are template-denoted, uniformly in the modulus N ∣ q ^ b (1 < N),…

The edge steps of the rational circuit are template-denoted, uniformly in the modulus N ∣ q ^ b (1 < N), with templates independent of q, b, N.

The following holds (Lean statement, in the scope of module Solutions.KIL5Tpl):

theorem NLogN.KIL.edgeSteps_toCircuit {R : Type*} [CommRing R] {W : ℕ} (C : QCircuit R m W) {q b N : ℕ}
    (hq : q.Prime) (hN : N ∣ q ^ b) (hN1 : 1 < N) (X : IMat m)
    (hg : GoodAtQ q ((C.den : ℚ)⁻¹ • ratM X)) :
    AModel.edgeSteps (C.toCircuit (q ^ b)) N X = (edgeTplQ C X).map (TStep.eval N)

Context. The lemma belongs to the part KIL5: the edge steps of the rational circuits are template-denoted: TStep m is a step template: a shift template (TSh) or a field transposition. The rational circuit QCircuit.toCircuit Q (AModelRat) takes as Lemma 10 data of a frame change X the integer lifts mod Q of the rational Bruhat factors ratFacQ A of A = D⁻¹ • X (facD Q D). Reduced mod N ∣ Q, these lifts are the reductions redQ N of the fixed rationals, so every step of edgeSteps is the evaluation at N of a template whose unit is ±|num|/den and whose controls are num/den of fixed rationals.

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_Defs18.