α = αP n iff (α-1)^4 < 12 d² b ≤ α^4

α = αP n iff (α-1)^4 < 12 d² b ≤ α^4.

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

theorem NLogN.KFM.alphaP_eq (n a : ℕ) (ha : 1 ≤ a) (h1 : (a - 1) ^ 4 < 12 * IntMul.OpenAI.Glue.dP n ^ 2 * bP n)
    (h2 : 12 * IntMul.OpenAI.Glue.dP n ^ 2 * bP n ≤ a ^ 4) : αP n = a

Context. The lemma belongs to the part KFM 14: the setup arithmetic (§9.2), in the integer form the machine computes.

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_Defs20.