The explicit sources for the parameters of n ≥ n₀

Proved

NLogN.KFM.srcS_ok_n

by xuanji · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

The explicit sources for the parameters of n ≥ n₀.

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

theorem NLogN.KFM.srcS_ok_n {n : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (d' : ℕ) (hd' : d' ≤ IntMul.OpenAI.Glue.dP n) :
    (∀ i : Fin (IntMul.OpenAI.Glue.dP n), Odd (srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i)) ∧
    (∀ i j : Fin (IntMul.OpenAI.Glue.dP n), i ≠ j → Nat.Coprime (srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i) (srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) j)) ∧
    ∀ i : Fin (IntMul.OpenAI.Glue.dP n), (4 * IntMul.OpenAI.Glue.dP n - 2) * (if (i : ℕ) < d' then IntMul.OpenAI.Glue.rP n / 2 else IntMul.OpenAI.Glue.rP n) <
        4 * IntMul.OpenAI.Glue.dP n * srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i ∧
      4 * IntMul.OpenAI.Glue.dP n * srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i ≤ (4 * IntMul.OpenAI.Glue.dP n - 1) * (if (i : ℕ) < d' then IntMul.OpenAI.Glue.rP n / 2 else IntMul.OpenAI.Glue.rP n)

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.

Preamble

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

import Definitions.Def_IntMul_Kappa_K_Defs20

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN
open NLogN.KFM
open IntMul.OpenAI.Glue IntMul.OpenAI.RCost

Formal statement

theorem NLogN.KFM.srcS_ok_n {n : ℕ} (hn : IntMul.OpenAI.Glue.n0 ≤ n) (d' : ℕ) (hd' : d' ≤ IntMul.OpenAI.Glue.dP n) :
    (∀ i : Fin (IntMul.OpenAI.Glue.dP n), Odd (srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i)) ∧
    (∀ i j : Fin (IntMul.OpenAI.Glue.dP n), i ≠ j → Nat.Coprime (srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i) (srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) j)) ∧
    ∀ i : Fin (IntMul.OpenAI.Glue.dP n), (4 * IntMul.OpenAI.Glue.dP n - 2) * (if (i : ℕ) < d' then IntMul.OpenAI.Glue.rP n / 2 else IntMul.OpenAI.Glue.rP n) <
        4 * IntMul.OpenAI.Glue.dP n * srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i ∧
      4 * IntMul.OpenAI.Glue.dP n * srcS (IntMul.OpenAI.Glue.dP n) d' (IntMul.OpenAI.Glue.rP n) i ≤ (4 * IntMul.OpenAI.Glue.dP n - 1) * (if (i : ℕ) < d' then IntMul.OpenAI.Glue.rP n / 2 else IntMul.OpenAI.Glue.rP 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.srcS_ok_n` of the Lean development Solutions/KFMSetup.lean (lines 226–234), integer-multiplication-kappa campaign.

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