src_facts (KFM S8: the source and inverse lists for the parameters of `n`)

Proved

NLogN.KFM.src_facts

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.KFMS8):

theorem NLogN.KFM.src_facts {n : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (j : ℕ) (hj : j < IntMul.OpenAI.Glue.dP n) :
    (j < dpN n → j ≤ srcM (IntMul.OpenAI.Glue.dP n) (IntMul.OpenAI.Glue.rP n) (IntMul.OpenAI.Glue.rP n / 2)) ∧
    (dpN n ≤ j → j - dpN n ≤ srcM (IntMul.OpenAI.Glue.dP n) (IntMul.OpenAI.Glue.rP n) (IntMul.OpenAI.Glue.rP n)) ∧
    2 ≤ srcN n j ∧ srcN n j ≤ IntMul.OpenAI.Glue.rP n ∧ Nat.Coprime (prodN n j) (srcN n j)

Context. The lemma belongs to the part KFM S8: the source and inverse lists for the parameters of n.

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
open NLogN
open NLogN.KFM
open IntMul IntMul.OpenAI.Glue IntMul.OpenAI.RP RS

Formal statement

theorem NLogN.KFM.src_facts {n : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (j : ℕ) (hj : j < IntMul.OpenAI.Glue.dP n) :
    (j < dpN n → j ≤ srcM (IntMul.OpenAI.Glue.dP n) (IntMul.OpenAI.Glue.rP n) (IntMul.OpenAI.Glue.rP n / 2)) ∧
    (dpN n ≤ j → j - dpN n ≤ srcM (IntMul.OpenAI.Glue.dP n) (IntMul.OpenAI.Glue.rP n) (IntMul.OpenAI.Glue.rP n)) ∧
    2 ≤ srcN n j ∧ srcN n j ≤ IntMul.OpenAI.Glue.rP n ∧ Nat.Coprime (prodN n j) (srcN n j) := 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.src_facts` of the Lean development Solutions/KFMS8.lean (lines 21–63), integer-multiplication-kappa campaign.

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