prepStep_runs (`FM`: correctness of the preprocessing segments)

The following holds (Lean statement, in the scope of module NLogN.SSFM1):

theorem NLogN.prepStep_runs (s : IState NS) (n M Q e : ℕ) (y c : ℤ) (LX LW LC LM LQ LN LF : List Item)
    (hM : M ≤ n) (he : e ≤ n) (hQ : Q ≤ n) (hc : zsz c ≤ n)
    (hX : s.G XA = .int y :: LX) (hW : s.G W = .int (2 ^ e) :: LW) (hC : s.G CC = .int c :: LC)
    (hRM : s.G RMs = .int (2 ^ M) :: LM) (hRQ : s.G RQ = .int (2 ^ Q) :: LQ)
    (hN : s.G RN = .int (2 ^ n) :: LN) (hF : s.G RF = .int (2 ^ n + 1) :: LF) :
    IRuns I iprog prepStep s
      ((((s.upd XA (.int (sg y (y.natAbs / 2 ^ M)) :: LX)).upd QS
        (.int ((sg y (y.natAbs % 2 ^ M) * 2 ^ e) % (2 ^ n + 1)) :: s.G QS)).upd W
          (.int (2 ^ (e + Q)) :: LW)).upd CC (.int (c + -1) :: LC))
      (30000 * (n + 3))

Context. The lemma belongs to the part FM: correctness of the preprocessing segments.

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.