total_le (KFM Final: `PhiNeed ∧ LineNeed ∧ ChirpNeedR → KappaBound (1/2^182)`)

Proved

NLogN.KFM.total_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.KFMFinal):

theorem NLogN.KFM.total_le {n D : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (hD : IntMul.OpenAI.Glue.dP n = D + 1) {CΦ CL CC : ℝ} (h1 : 0 ≤ CΦ)
    (h2 : 0 ≤ CL) (h3 : 0 ≤ CC) (AL AC : ℕ) (x y : List Bool) (hx : x.length = n) :
    (NLogN.KFM.setC x y : ℝ) + kBigF n D (dzB (descPhiN n)) (dzB (descLineN n)) (dzB (descChirpN n)) AL AC
      CΦ CL CC + 2 ≤ Ctot CΦ CL CC * cLineU n (AL + AC + 2)

Context. The lemma belongs to the part KFM Final: PhiNeed ∧ LineNeed ∧ ChirpNeedR → KappaBound (1/2^182).

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 maxRecDepth 100000
set_option maxHeartbeats 8000000

Formal statement

theorem NLogN.KFM.total_le {n D : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (hD : IntMul.OpenAI.Glue.dP n = D + 1) {CΦ CL CC : ℝ} (h1 : 0 ≤ CΦ)
    (h2 : 0 ≤ CL) (h3 : 0 ≤ CC) (AL AC : ℕ) (x y : List Bool) (hx : x.length = n) :
    (NLogN.KFM.setC x y : ℝ) + kBigF n D (dzB (descPhiN n)) (dzB (descLineN n)) (dzB (descChirpN n)) AL AC
      CΦ CL CC + 2 ≤ Ctot CΦ CL CC * cLineU n (AL + AC + 2) := 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.total_le` of the Lean development Solutions/KFMFinal.lean (lines 104–248), integer-multiplication-kappa campaign.

View graph