difTest_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.difTest_runs (s : IState NS) (L : ℤ) (LL : List Item) (m : ℕ)
    (hL : s.G DL = .int L :: LL) (hm : zsz L ≤ m) :
    IRuns I iprog difTest s (s.upd CC (.int (L + -1) :: s.G CC)) (2000 * (m + 2))

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.