The final reduction with slack

The final reduction with slack. Time C₀ (V · Ncost · F + n) with F ≤ (6 lg n)^κ, beyond any threshold n₁, suffices for KappaBound (2⁻¹⁸²).

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

theorem NLogN.KFM.kappaBound_of_machine_slack (M : IntMul.MultitapeTM)
    (hcorr : ∀ n : ℕ, 1 ≤ n → ∃ T : ℝ, IntMul.MultipliesAt M n T)
    (n₁ : ℕ) (C₀ : ℝ) (hC₀ : 0 ≤ C₀) (F : ℕ → ℝ)
    (hF0 : ∀ n, IntMul.OpenAI.Glue.n0 ≤ n → n₁ ≤ n → 0 ≤ F n)
    (hF : ∀ n, IntMul.OpenAI.Glue.n0 ≤ n → n₁ ≤ n → F n ≤ (6 * (IntMul.lg n : ℝ)) ^ κ)
    (htime : ∀ n, IntMul.OpenAI.Glue.n0 ≤ n → n₁ ≤ n → IntMul.MultipliesAt M n
      (C₀ * (IntMul.OpenAI.Glue.VP n * Ncost (6 * (IntMul.lg n : ℝ)) (IntMul.OpenAI.Glue.dP n) (KP n) (ℓP n) (αP n) * F n + n))) :
    IntMul.KappaBound (1 / 2 ^ 182)

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.