The phase step as a fine IL run

The phase step as a fine IL run. On a clean managed state whose data stack holds an encoding zs of x (entries of ≤ wb bits per component), fcode (phaseProg …) leaves an encoding of (RStep.den T (phase w k₀ k₁ a)) *ᵥ x (same bit bound), at cost (mn + 1) · cPass wb 2^{mn} (w 2^{mn}) Z.

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

theorem NLogN.KFB.phase_iruns {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    (s : IState HS) (wb : ℕ) {W m n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : Butterfly.Op (Fin n))
    (w : Fin W) (k₀ k₁ : ℕ) (a : Fin m → ℕ) (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.phase w k₀ k₁ a)).mulVec x) ∧ NLogN.KFB.GB wb zs' ∧
      IRuns I iprog (fcode (phaseProg m n w.val k₀ k₁ a)) (encK s f)
        (encK s (Function.update f 0 (gflat zs' ++ l)))
        ((m * n + 1) * NLogN.KFB.cPass wb (2 ^ (m * n)) (w.val * 2 ^ (m * n))
          (zsz (((4 * (3 * 2 ^ (m * n) + w.val * 2 ^ (m * n))) : ℕ) : ℤ)))

Context. The lemma belongs to the part KFB: the phase step as a fine-cost item-language program: The state x : Fin W × Addr m (Fin n) → ℂ of the restricted interface is encoded role-major as a Gaussian array: role w, address b at index w · 2^{mn} + aidx b, where aidx b = Σ_{(r,c)} b(r,c) · 2^{c + n r} (finProdFinEquiv).

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.