The correction step (KFL_WIP.corrI_spec with the linear term V: every record is moved once, M · B = V item…

Proved

NLogN.KFL.corrI_spec

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

complexity-theoryinteger-multiplicationkappa-improvement

The correction step (KFL_WIP.corrI_spec with the linear term V: every record is moved once, M · B = V item moves).

The following holds (Lean statement, in the scope of module Solutions.KFLCorr4):

theorem NLogN.KFL.corrI_spec :
    ∃ Kc : ℝ, 0 < Kc ∧
      ∀ (K ρ f : ℕ), 6 ≤ K → ρ < K → 1 ≤ f →
      ∀ (g : Geo), g.Pos → g.Q = 2 ^ (f * K) →
      ∀ (I : List Sym) (iprog : Fin 1 → IStmt HS 1) (t : IState HS) (zs : List ℤ) (R : List Item),
      Bits zs → t.G 0 = ints zs ++ R → zs.length = g.V →
      ∃ c : ℕ, IRuns I iprog (corrI K ρ f g) t
          (t.upd 0 (ints (rixT g (corrc K ρ f) zs) ++ R)) c ∧
        (c : ℝ) ≤ Kc * (((g.V / g.B : ℕ) : ℝ) * ((zsz (g.V : ℤ) : ℝ) + 1) ^ 3 + (g.V : ℝ) +
          min 1 (80 * ((f : ℝ) - 1) * 2⁻¹ ^ K) * ((g.V / g.B : ℕ) : ℝ) *
            ((zsz (g.V : ℤ) : ℝ) + 1) * ((g.B : ℝ) + (zsz (g.V : ℤ) : ℝ) + 1))

Context. The lemma belongs to the part KFL §15: the number of moved records and the IL correction step: card_moved: the records moved by the correction (keyR m ≠ m) are bad, and the bad records number at most 80 (f-1) 2^{-K} of all (two guards per index, twenty bad values of 2^{K-1} each, by count_digit / count_bad_one).

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_Defs18.

Preamble

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

import Definitions.Def_IntMul_Kappa_K_Defs18

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
open NLogN
open NLogN.KFL
open IntMul RS KIL KFD Butterfly Finset

Formal statement

theorem NLogN.KFL.corrI_spec :
    ∃ Kc : ℝ, 0 < Kc ∧
      ∀ (K ρ f : ℕ), 6 ≤ K → ρ < K → 1 ≤ f →
      ∀ (g : Geo), g.Pos → g.Q = 2 ^ (f * K) →
      ∀ (I : List Sym) (iprog : Fin 1 → IStmt HS 1) (t : IState HS) (zs : List ℤ) (R : List Item),
      Bits zs → t.G 0 = ints zs ++ R → zs.length = g.V →
      ∃ c : ℕ, IRuns I iprog (corrI K ρ f g) t
          (t.upd 0 (ints (rixT g (corrc K ρ f) zs) ++ R)) c ∧
        (c : ℝ) ≤ Kc * (((g.V / g.B : ℕ) : ℝ) * ((zsz (g.V : ℤ) : ℝ) + 1) ^ 3 + (g.V : ℝ) +
          min 1 (80 * ((f : ℝ) - 1) * 2⁻¹ ^ K) * ((g.V / g.B : ℕ) : ℝ) *
            ((zsz (g.V : ℤ) : ℝ) + 1) * ((g.B : ℝ) + (zsz (g.V : ℤ) : ℝ) + 1)) := 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.KFL.corrI_spec` of the Lean development Solutions/KFLCorr4.lean (lines 251–332), integer-multiplication-kappa campaign.

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