mvH_runs (The forward FFT (decimation in frequency) over `ℤ/(2^n+1)`, as a recursive proce)

The following holds (Lean statement, in the scope of module NLogN.SSDif):

theorem NLogN.mvH_runs {X Y : Fin NS} (hXY : X ≠ Y) (hXC : X ≠ CC) (hYC : Y ≠ CC) (hX3 : X ≠ NLogN.T3)
    (hY3 : Y ≠ NLogN.T3) (hXH : X ≠ HH) (hYH : Y ≠ HH) (s : IState NS) (A : List ℤ) (R LH : List Item)
    (m : ℕ) (hX : s.G X = ints A ++ R) (hH : s.G HH = .int A.length :: LH) (hA : NLogN.Bnd m A) :
    IRuns I iprog (mvH X Y) s ((s.upd X R).upd Y ((ints A).reverse ++ s.G Y))
      (A.length * (1000 * (m + zsz (A.length : ℤ) + 3)) + 300 * (zsz (A.length : ℤ) + 1) + 40)

Context. The lemma belongs to the part The forward FFT (decimation in frequency) over ℤ/(2^n+1), as a recursive procedure.

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.