twC_runs (KFM 9c: the per-axis root-table program (`chirpL` recurrence in IL))

Proved

NLogN.KFM.twC_runs

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

complexity-theoryinteger-multiplicationkappa-improvement

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

theorem NLogN.KFM.twC_runs (n k j : ℕ) (hk : 0 < k) (X : Fin HS) (h51 : X ≠ 51) (h52 : X ≠ 52) (s : IState HS) :
    IRuns I iprog (twC n k j X) s
      (s.upd X (.int (HvdH.twiddle n k j).1 :: .int (HvdH.twiddle n k j).2 :: s.G X)) (cTw n k j)

Context. The lemma belongs to the part KFM 9c: the per-axis root-table program (chirpL recurrence in IL).

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_Defs22.

Preamble

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

import Definitions.Def_IntMul_Kappa_K_Defs22

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.KFM
open IntMul KFC KFB KFT KIL
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem NLogN.KFM.twC_runs (n k j : ℕ) (hk : 0 < k) (X : Fin HS) (h51 : X ≠ 51) (h52 : X ≠ 52) (s : IState HS) :
    IRuns I iprog (twC n k j X) s
      (s.upd X (.int (HvdH.twiddle n k j).1 :: .int (HvdH.twiddle n k j).2 :: s.G X)) (cTw n k j) := 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.twC_runs` of the Lean development Solutions/KFMTabP.lean (lines 525–546), integer-multiplication-kappa campaign.

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