mulC_runs (KFUU §0: tools for the uniform `Φ` / `Φ⁻¹`)

Proved

NLogN.KFUU.mulC_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.KFUUBase):

theorem NLogN.KFUU.mulC_runs {pM : Fin P} (hM : MulOK I iprog pM NLogN.KFUU.MS) {a b r : Fin HS} (ha : a ≠ ARGM)
    (hb : b ≠ ARGM) (hr : r ≠ ARGM) (t : IState HS) {x y : ℤ} {La Lb : List Item}
    (hx : t.G a = .int x :: La) (hy : t.G b = .int y :: Lb) :
    IRuns I iprog (mulC pM a b r) t (t.upd r (.int (y * x) :: t.G r)) (cMul x y)

Context. The lemma belongs to the part KFUU §0: tools for the uniform Φ / Φ⁻¹.

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 KFC KFT AModel AModelL12
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem NLogN.KFUU.mulC_runs {pM : Fin P} (hM : MulOK I iprog pM NLogN.KFUU.MS) {a b r : Fin HS} (ha : a ≠ ARGM)
    (hb : b ≠ ARGM) (hr : r ≠ ARGM) (t : IState HS) {x y : ℤ} {La Lb : List Item}
    (hx : t.G a = .int x :: La) (hy : t.G b = .int y :: Lb) :
    IRuns I iprog (mulC pM a b r) t (t.upd r (.int (y * x) :: t.G r)) (cMul x y) := 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.mulC_runs` of the Lean development Solutions/KFUUBase.lean (lines 55–70), integer-multiplication-kappa campaign.

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