Closed cost bound O(n log n log log n) for one multiplication of total bit length n (from SS mainCost_le)

Closed cost bound O(n log n log log n) for one multiplication of total bit length n (from SS mainCost_le).

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

theorem NLogN.KFM.mulCostH_le (n : ℕ) :
    mulCostH n ≤ (8 * fmA + 10 ^ 7 + 10 ^ 5) * (n + 1) * (zsz ((n + 1 : ℕ) : ℤ) + 2) *
      (zsz (((zsz ((n + 1 : ℕ) : ℤ) + 2) ^ 4 : ℕ) : ℤ) + 1)

Context. The lemma belongs to the part KFM 6: the Schönhage–Strassen FM as a MulOK multiplication procedure.

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.