Every role permutation as fine IL passes: a list of adjacent swaps depending only on σ; the program permP…

Every role permutation as fine IL passes: a list of adjacent swaps depending only on σ; the program permP maps an encoding of x to an encoding of (den (perm σ)) *ᵥ x.

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

theorem NLogN.KFB.perm_iruns {k m n : ℕ} (σ : Equiv.Perm (Fin (k + 1))) :
    ∃ L : List ℕ, (∀ r ∈ L, r + 1 < k + 1) ∧
    ∀ {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp} (s : IState HS) (wb : ℕ)
      (T : Butterfly.Op (Fin n)) (f : Fin HS → List ℤ) (zs : List GI) (l : List ℤ)
      (x : Fin (k + 1) × (Fin m × Fin n → Fin 2) → ℂ), Rep zs x → f 0 = gflat zs ++ l →
      Cl f → NLogN.KFB.GB wb zs →
      ∃ zs', Rep zs' ((P8Lift.RStep.den T
          (P8Lift.RStep.perm σ : P8Lift.RStep (Fin (k + 1)) m)).mulVec x) ∧ NLogN.KFB.GB wb zs' ∧
        IRuns I iprog (fcode (NLogN.KFB.permP (2 ^ (m * n)) L)) (encK s f)
          (encK s (Function.update f 0 (gflat zs' ++ l)))
          (L.length * NLogN.KFB.cPass wb (2 * 2 ^ (m * n)) ((k + 1) * 2 ^ (m * n))
            (zsz (((4 * (6 * 2 ^ (m * n) + (k + 1) * 2 ^ (m * n))) : ℕ) : ℤ)) + 1)

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.