parE_runs (`FM`: parameter computation)

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

theorem NLogN.parE_runs (pP : Fin P) (hP : iprog pP = pow2Body pP) (s : IState NS) (N n k Q ℓ : ℕ)
    (L1 L2 L3 LZ : List Item) (hn : 1 ≤ n) (hkn : k ≤ n) (hnN : n ≤ N) (hQ : Q ≤ N)
    (hℓ : ℓ ≤ N)
    (hT1 : s.G T1 = .int k :: L1) (hT2 : s.G T2 = .int Q :: L2) (hT3 : s.G NLogN.T3 = .int n :: L3)
    (hZ : s.G ZZ = .int ℓ :: LZ) :
    IRuns I iprog (parE pP) s
      ((((((((s.upd W2 (.int (2 ^ (n - k)) :: s.G W2)).upd HB (.int (2 ^ (n - 1)) :: s.G HB)).upd
        PRM (.int n :: s.G PRM)).upd NLogN.T3 L3).upd T2 L2).upd T1 L1).upd ZZ LZ))
      (60000 * (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.