powL_runs (KFUU §2h: the set-up of phase 2 (volume, powers))

Proved

NLogN.KFUU.powL_runs

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.KFUURev8):

theorem NLogN.KFUU.powL_runs (u : ℕ) (t : IState HS) (n : ℕ) {L2 LN : List Item}
    (h2 : t.G rU2 = .int ((2 ^ u : ℕ) : ℤ) :: L2) (hN : t.G nR = .int (n : ℤ) :: LN) :
    IRuns I iprog (powL (P := P)) t (t.upd NLogN.KFUU.rM (.int (((2 ^ u) ^ n : ℕ) : ℤ) :: t.G NLogN.KFUU.rM))
      ((n + 1) * (200 * (u + 1) + 3000) + 3)

Context. The lemma belongs to the part KFUU §2h: the set-up of phase 2 (volume, powers).

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 linter.unnecessarySeqFocus false
set_option maxRecDepth 20000
open NLogN
open NLogN.KFUU
open IntMul KIL KFB KFC KFT KFU AModel AModelL12
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem NLogN.KFUU.powL_runs (u : ℕ) (t : IState HS) (n : ℕ) {L2 LN : List Item}
    (h2 : t.G rU2 = .int ((2 ^ u : ℕ) : ℤ) :: L2) (hN : t.G nR = .int (n : ℤ) :: LN) :
    IRuns I iprog (powL (P := P)) t (t.upd NLogN.KFUU.rM (.int (((2 ^ u) ^ n : ℕ) : ℤ) :: t.G NLogN.KFUU.rM))
      ((n + 1) * (200 * (u + 1) + 3000) + 3) := 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.KFUU.powL_runs` of the Lean development Solutions/KFUURev8.lean (lines 153–173), integer-multiplication-kappa campaign.

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