setupA_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.setupA_run (n : ℕ) (hn : n0 ≤ n) (s : IState HS) (LB : List Item)
    (hB : s.G zB = .int (bP n : ℤ) :: LB) :
    setupA.ok s ∧ setupA.run s = ((s.upd NLogN.KFM.zD (.int (dP n : ℤ) :: s.G NLogN.KFM.zD)).upd zK
      (.int (KP n : ℤ) :: s.G zK)).upd zAl (.int (αP n : ℤ) :: s.G zAl)

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.