cPhiU_le (KFM Cost: the cost of `bigE` in the unit `ZU`)

Proved

NLogN.KFM.cPhiU_le

by xuanji · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

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

theorem NLogN.KFM.cPhiU_le {n D : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (hD : IntMul.OpenAI.Glue.dP n = D + 1) :
    cPhiU (pP n + 2) (ℓP n - 1) (IntMul.OpenAI.Glue.dP n) (bLn n) (srcN n) ≤
      10 ^ 9 * ((IntMul.OpenAI.Glue.TP n : ℝ) * pP n * NcN n) + 10 ^ 4 * ((bP n + 1) ^ 6 : ℕ)

Context. The lemma belongs to the part KFM Cost: the cost of bigE in the unit ZU.

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$ : the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.

Preamble

-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs33

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
open NLogN
open NLogN.KFM
open Finset IntMul KIL KFB KFT KFU RS IntMul.OpenAI.Glue IntMul.OpenAI.RCost
set_option maxHeartbeats 2000000

Formal statement

theorem NLogN.KFM.cPhiU_le {n D : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (hD : IntMul.OpenAI.Glue.dP n = D + 1) :
    cPhiU (pP n + 2) (ℓP n - 1) (IntMul.OpenAI.Glue.dP n) (bLn n) (srcN n) ≤
      10 ^ 9 * ((IntMul.OpenAI.Glue.TP n : ℝ) * pP n * NcN n) + 10 ^ 4 * ((bP n + 1) ^ 6 : ℕ) := by sorry

Source

OpenAI, Integer multiplication below n log n (preprint, 23 Sep 2026), https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026. Lemma `NLogN.KFM.cPhiU_le` of the Lean development Solutions/KFMCost.lean (lines 466–617), integer-multiplication-kappa campaign.

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