setupC_cost (KFM S4: the cost of the scalar setup program)

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

theorem NLogN.KFM.setupC_cost (n : ℕ) (hn : n0 ≤ n) (s : IState HS) (LD LR : List Item)
    (hD : s.G NLogN.KFM.zD = .int (dP n : ℤ) :: LD) (hR : s.G zR = .int (IntMul.OpenAI.Glue.rP n : ℤ) :: LR) :
    setupC.cost s ≤ 4 * 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.