fmUnpar_runs (`FM`: parameter computation)

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

theorem NLogN.fmUnpar_runs (s : IState NS) (n k Q : ℕ) (LN LF LP L2 LH LE LQ : List Item)
    (hkn : k ≤ n) (hQ : Q ≤ n)
    (hN : s.G RN = .int (2 ^ n) :: LN) (hF : s.G RF = .int (2 ^ n + 1) :: LF)
    (hP : s.G PRM = .int n :: LP) (h2 : s.G W2 = .int (2 ^ (n - k)) :: L2)
    (hH : s.G HB = .int (2 ^ (n - 1)) :: LH) (hE : s.G NLogN.DE = .int (2 ^ (2 * Q)) :: LE)
    (hRQ : s.G RQ = .int (2 ^ Q) :: LQ) :
    IRuns I iprog fmUnpar s
      ((((((((s.upd RN LN).upd RF LF).upd PRM LP).upd W2 L2).upd HB LH).upd NLogN.DE LE).upd RQ LQ))
      (100 * (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.