The scalar setup program: from n on zN and b = lg n on zB, it pushes the §9.2 parameters, in O((lg n)^4) steps

The scalar setup program: from n on zN and b = lg n on zB, it pushes the §9.2 parameters, in O((lg n)^4) steps.

The following holds (Lean statement, in the scope of module Solutions.KFMS4):

theorem NLogN.KFM.setupS_all (n : ℕ) (hn : n0 ≤ n) (s : IState HS) (LN LB : List Item)
    (hN : s.G zN = .int (n : ℤ) :: LN) (hB : s.G zB = .int (bP n : ℤ) :: LB) :
    setupS.ok s ∧ setupS.run s = setupSt n s ∧
      setupS.cost s ≤ 12 * 10 ^ 15 * (bP n + 1) ^ 4

Context. The lemma belongs to the part KFM S4: the cost of the scalar setup program: For n ≥ n₀, with b = lg n, every atom the setup expressions read is at most A = 2^{b+2}. The expressions have degree at most 8, so all values have fewer than B = 8b+17 bits (KFMSm). This gives

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.