The uniform prog_il for a rational circuit

The uniform prog_il for a rational circuit. The single procedure progU (Qc.toCircuit 1) (edgeTplQ Qc) and the constants depend only on Qc; for every prime q, every exponent b with m ^ j ≤ b for the levels j < k, it computes the procedure prog of the circuit Qc.toCircuit (q ^ b) (Lemma 10 factorizations facD (q ^ b)), provided the frame matrices are good at q (GoodAtQ) and the bit size Bb fits the template controls (TOK, SubBnd).

The following holds (Lean statement, in the scope of module Solutions.KIL5Main):

theorem NLogN.KIL.prog_il_uniform_rat (Qc : QCircuit (ZMod 2) m W) (hm : 1 ≤ m) :
    ∃ (iprog : Fin 1 → IStmt HS 1) (K Kd S : ℕ),
      ∀ (q : ℕ), q.Prime → ∀ (b Bb k : ℕ), (∀ j < k, m ^ j ≤ b) →
      (∀ j < k, FramesOK (fun X => GoodAtQ q ((Qc.den : ℚ)⁻¹ • ratM X) ∧
        ∀ T, TStep.sh T ∈ edgeTplQ Qc X → TOK T (q ^ m ^ j) Bb)
        (Qc.toCircuit (q ^ b)).gates (Qc.toCircuit (q ^ b)).Min (Qc.toCircuit (q ^ b)).Mout) →
      (∀ j < k, SubBnd (q ^ m ^ j) Bb) → zsz (k : ℤ) ≤ Bb →
      ∀ (P L : ℕ) (A B Cs : Shape), 1 ≤ P → W ^ k ∣ L → 1 ≤ L →
      (∀ a ∈ A, 1 ≤ a) → (∀ a ∈ B, 1 ≤ a) → (∀ a ∈ Cs, 1 ≤ a) →
      zsz ((vol (P :: L :: swShape A B Cs (q ^ m ^ k)) : ℕ) : ℤ) ≤ Bb →
      ∀ (I : List Sym) (t : IState HS) (R : List Item) (xs : List (ZMod 2)),
        DescIn t q m k P L A B Cs →
        xs.length = vol (P :: L :: swShape A B Cs (q ^ m ^ k)) →
        IRuns I iprog (iprog 0) (t.upd 0 (ints (enc2 xs) ++ R))
          (t.upd 0 (ints (enc2 (AModel.prog (Qc.toCircuit (q ^ b)) q k P L A B Cs xs)) ++ R))
          (ilCostU5 W K Kd S q Bb k (vol (P :: L :: swShape A B Cs (q ^ m ^ k))))

Context. The lemma belongs to the part KIL5: the uniform prog_il: progU_run: the single fixed procedure progU C tpl (KIL5Prog; it depends only on the circuit C and the template function tpl) computes prog C q k P L A B Cs at every level k, for every q, P, L, A, B, Cs and input, from any state carrying the descriptor (DescIn), with cost ilCostU5 (constants Klev C tpl + 10⁹, Kd m, SWn C tpl, independent of q, k and the sizes).

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.