parB_runs (`FM`: parameter computation)

The following holds (Lean statement, in the scope of module NLogN.SSFM4):

theorem NLogN.parB_runs (pP : Fin P) (hP : iprog pP = pow2Body pP) (s : IState NS) (N k : ℕ)
    (LP LL : List Item) (hk : k ≤ N) (hPRM : s.G PRM = .int N :: LP)
    (hDL : s.G DL = .int (2 ^ k) :: LL) :
    IRuns I iprog (parB pP) s
      ((s.upd T2 (.int (N / 2 ^ k : ℕ) :: s.G T2)).upd RMs (.int (2 ^ (N / 2 ^ k)) :: s.G RMs))
      (25000 * (N + 3))

Context. The lemma belongs to the part FM: parameter computation.

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.