The phase step in O(V) as a fine IL run

The phase step in O(V) as a fine IL run.

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

theorem NLogN.KFG.phaseT_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))
    (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 : ClT f) (hb : NLogN.KFB.GB wb zs) (Z : ℕ)
    (hZ : zsz ((4 * (w.val * 2 ^ (m * n) + 2 ^ (m * n)) : ℕ) : ℤ) ≤ Z) :
    ∃ 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 (phaseT m n w.val k₀ k₁ a)) (encK s f)
        (encK s (Function.update f 0 (gflat zs' ++ l)))
        (cPhT wb (m * n) (w.val * 2 ^ (m * n)) Z)

Context. The lemma belongs to the part KFG: the phase step in O(V) (task B): phaseT m n w k₀ k₁ a := phT (mn) (w 2^{mn}) (k₀ + k₁ n) (phE a) builds the exponent table of the 2^{mn} addresses by doubling (O(2^{mn}) total) and rotates every entry of role w by its table entry (O(1) selections per entry). Compared with KFB.phaseProg (one pass per address bit, O(mn · 2^{mn})), the per-element mn = log V factor is gone.

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.