One exchange on the stored records

Proved

NLogN.KFV.xchC_run

by xuanji · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

One exchange on the stored records.

The following holds (Lean statement, in the scope of module Solutions.KFV30LayX):

theorem NLogN.KFV.xchC_run {eSw : Fin Pp} {dW : Fin HS} {Kc : ℝ} (HSw : SwOK I iprog eSw dW Kc)
    (hd0 : dW ≠ 0) (hd56 : dW.val < 56) (hKc : 0 ≤ Kc)
    (S : IState HS) (zs : List GI) (l : List ℤ) {i j K n ℓ W B : ℕ}
    (hij : i + K ≤ j) (hj : j + K ≤ n) (hK : 1 ≤ K)
    (hzs : zs.length = 2 ^ n * 2 ^ ℓ) (hgb : NLogN.KFB.GB W zs)
    {eE eK eM eI eW : Ex HS} (sE : DSafe dW eE) (sK : DSafe dW eK) (sM : DSafe dW eM)
    (sI : DSafe dW eI) (sW : DSafe dW eW)
    (vE : EV (encK S (o0 (gflat zs ++ l))) B eE ((n - j - K + ℓ : ℕ) : ℤ))
    (vK : EV (encK S (o0 (gflat zs ++ l))) B eK (K : ℤ))
    (vM : EV (encK S (o0 (gflat zs ++ l))) B eM ((j - i - K : ℕ) : ℤ))
    (vI : EV (encK S (o0 (gflat zs ++ l))) B eI (i : ℤ))
    (vW : EV (encK S (o0 (gflat zs ++ l))) B eW (W : ℤ))
    (hB : n + ℓ + 5 ≤ B) (hWB : W ≤ B) :
    IRuns I iprog (xchC eSw dW eE eK eM eI eW) (encK S (o0 (gflat zs ++ l)))
      (encK S (o0 (gflat (zMv n (2 ^ ℓ) (bsIdx i j K) zs) ++ l)))
      (cXch Kc eE eK eM eI eW n ℓ K W B)

Context. The lemma belongs to the part KFV §33: the chunk exchange as fixed IL code.

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.unusedSectionVars false
set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
open NLogN
open NLogN.KFV
open IntMul RS KIL KFD KFB KFH KFC KFT Synth AModel AModelL12
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
set_option maxHeartbeats 1000000

Formal statement

theorem NLogN.KFV.xchC_run {eSw : Fin Pp} {dW : Fin HS} {Kc : ℝ} (HSw : SwOK I iprog eSw dW Kc)
    (hd0 : dW ≠ 0) (hd56 : dW.val < 56) (hKc : 0 ≤ Kc)
    (S : IState HS) (zs : List GI) (l : List ℤ) {i j K n ℓ W B : ℕ}
    (hij : i + K ≤ j) (hj : j + K ≤ n) (hK : 1 ≤ K)
    (hzs : zs.length = 2 ^ n * 2 ^ ℓ) (hgb : NLogN.KFB.GB W zs)
    {eE eK eM eI eW : Ex HS} (sE : DSafe dW eE) (sK : DSafe dW eK) (sM : DSafe dW eM)
    (sI : DSafe dW eI) (sW : DSafe dW eW)
    (vE : EV (encK S (o0 (gflat zs ++ l))) B eE ((n - j - K + ℓ : ℕ) : ℤ))
    (vK : EV (encK S (o0 (gflat zs ++ l))) B eK (K : ℤ))
    (vM : EV (encK S (o0 (gflat zs ++ l))) B eM ((j - i - K : ℕ) : ℤ))
    (vI : EV (encK S (o0 (gflat zs ++ l))) B eI (i : ℤ))
    (vW : EV (encK S (o0 (gflat zs ++ l))) B eW (W : ℤ))
    (hB : n + ℓ + 5 ≤ B) (hWB : W ≤ B) :
    IRuns I iprog (xchC eSw dW eE eK eM eI eW) (encK S (o0 (gflat zs ++ l)))
      (encK S (o0 (gflat (zMv n (2 ^ ℓ) (bsIdx i j K) zs) ++ l)))
      (cXch Kc eE eK eM eI eW n ℓ K W B) := 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.KFV.xchC_run` of the Lean development Solutions/KFV30LayX.lean (lines 133–228), integer-multiplication-kappa campaign.

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