The pass cost against cLineU n 6

Proved

NLogN.KFY.NG.cost_real

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

complexity-theoryinteger-multiplicationkappa-improvement

The pass cost against cLineU n 6.

The following holds (Lean statement, in the scope of module Solutions.KFY20Real):

theorem NLogN.KFY.NG.cost_real (Kc : ℝ) (hK : 0 ≤ Kc) (Cm Mn : ℕ)
    (hm : Mn ≤ Cm * (SN n * αP n) * 2 ^ IntMul.OpenAI.Glue.LP n * ZN n) (c : ℕ)
    (hc : c ≤ IntMul.OpenAI.Glue.dP n * (⌈trBound Kc (pP n + 2) (ℓP n - 1) (IntMul.OpenAI.Glue.dP n) (2 ^ IntMul.OpenAI.Glue.LP n)⌉₊ +
        trOver (pP n + 2) (ℓP n - 1) (IntMul.OpenAI.Glue.dP n)) + IntMul.OpenAI.Glue.dP n * Mn + (2 * IntMul.OpenAI.Glue.dP n + 1) * overN n + 2) :
    (c : ℝ) ≤ (17 * Kc + 1 + (Cm : ℝ) * NLogN.KFY.cZ * 4 * 58 ^ 6) * NLogN.KFM.cLineU n 6

Context. The lemma belongs to the part KFY 20: the cost of the two passes against cLineU n 6.

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
set_option linter.unusedSectionVars false
set_option maxRecDepth 100000
open NLogN
open NLogN.KFY
open IntMul NLogN IntMul.OpenAI.Glue NLogN.GL NLogN.AX NLogN.Rsp IntMul.OpenAI.RCost
attribute [local irreducible] IntMul.OpenAI.Glue.dP IntMul.OpenAI.Glue.αP IntMul.OpenAI.Glue.bP
  IntMul.OpenAI.Glue.LP IntMul.OpenAI.Glue.ℓP IntMul.OpenAI.Glue.KP cZ ZN overN
variable {n : ℕ} (h : NG n)
include h
set_option maxHeartbeats 4000000

Formal statement

theorem NLogN.KFY.NG.cost_real (Kc : ℝ) (hK : 0 ≤ Kc) (Cm Mn : ℕ)
    (hm : Mn ≤ Cm * (SN n * αP n) * 2 ^ IntMul.OpenAI.Glue.LP n * ZN n) (c : ℕ)
    (hc : c ≤ IntMul.OpenAI.Glue.dP n * (⌈trBound Kc (pP n + 2) (ℓP n - 1) (IntMul.OpenAI.Glue.dP n) (2 ^ IntMul.OpenAI.Glue.LP n)⌉₊ +
        trOver (pP n + 2) (ℓP n - 1) (IntMul.OpenAI.Glue.dP n)) + IntMul.OpenAI.Glue.dP n * Mn + (2 * IntMul.OpenAI.Glue.dP n + 1) * overN n + 2) :
    (c : ℝ) ≤ (17 * Kc + 1 + (Cm : ℝ) * NLogN.KFY.cZ * 4 * 58 ^ 6) * NLogN.KFM.cLineU n 6 := 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.KFY.NG.cost_real` of the Lean development Solutions/KFY20Real.lean (lines 140–274), integer-multiplication-kappa campaign.

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