Any fixed power of log p is eventually below p^κ (p = 6 lg n)

Any fixed power of log p is eventually below p^κ (p = 6 lg n).

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

theorem NLogN.KFM.polylog_slack (A : ℕ) (c : ℝ) : ∃ n₁ : ℕ, ∀ n, n₁ ≤ n →
    c * (Real.log (6 * (IntMul.lg n : ℝ)) + 1) ^ A ≤ (6 * (IntMul.lg n : ℝ)) ^ κ

Context. The lemma belongs to the part KFM 1: the final reduction with a raised threshold and a polylog slack factor.

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.