The gate on two adjacent roles as one fine IL pass

The gate on two adjacent roles as one fine IL pass.

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

theorem NLogN.KFB.gate2_iruns {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    (s : IState HS) (wb : ℕ) {W m n : ℕ} (T : Butterfly.Op (Fin n)) (r : ℕ) (hr : r + 1 < W)
    (M : M2G) (hM : M.Ok) (f : Fin HS → List ℤ) (zs : List GI) (l : List ℤ)
    (x : Fin W × (Fin m × Fin n → Fin 2) → ℂ) (hx : Rep zs x) (h0 : f 0 = gflat zs ++ l)
    (hC : Cl f) (hb : NLogN.KFB.GB wb zs) :
    ∃ zs', Rep zs' ((P8Lift.RStep.den T
        (P8Lift.RStep.gate (g2Mat W r M) : P8Lift.RStep (Fin W) m)).mulVec x) ∧
      NLogN.KFB.GB (wb + 2) zs' ∧ zs' = g2L (2 ^ (m * n)) r M zs ∧
      IRuns I iprog (fcode (qopW [M] (2 ^ (m * n)) 1 (r * 2 ^ (m * n)))) (encK s f)
        (encK s (Function.update f 0 (gflat zs' ++ l)))
        (NLogN.KFB.cPass wb (2 * 2 ^ (m * n)) (r * 2 ^ (m * n))
          (zsz (((4 * (3 * (2 * 2 ^ (m * n)) + r * 2 ^ (m * n))) : ℕ) : ℤ)))

Context. The lemma belongs to the part KFB: role gates on adjacent roles and role permutations.

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.