K = KP n is the integer 2^56-th root of d

K = KP n is the integer 2^56-th root of d.

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

theorem NLogN.KFM.KP_spec (n : ℕ) : KP n ^ (2 ^ 56) ≤ IntMul.OpenAI.Glue.dP n ∧ IntMul.OpenAI.Glue.dP n < (KP n + 1) ^ (2 ^ 56)

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.