setupB_run (KFM S3: the scalar setup parameters as counts (math side))

The following holds (Lean statement, in the scope of module Solutions.KFMS3):

theorem NLogN.KFM.setupB_run (n : ℕ) (hn : n0 ≤ n) (s : IState HS) (LN LB LD : List Item)
    (hN : s.G zN = .int (n : ℤ) :: LN) (hB : s.G zB = .int (bP n : ℤ) :: LB)
    (hD : s.G NLogN.KFM.zD = .int (dP n : ℤ) :: LD) :
    setupB.ok s ∧ setupB.run s = (((s.upd zL (.int (LP n : ℤ) :: s.G zL)).upd zEl
      (.int (ℓP n : ℤ) :: s.G zEl)).upd zR (.int (IntMul.OpenAI.Glue.rP n : ℤ) :: s.G zR)).upd zDp
      (.int (mcnt (LP n) (dP n) : ℤ) :: s.G zDp)

Context. The lemma belongs to the part KFM S3: the scalar setup parameters as counts (math side).

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_Defs22.