setup_St (KFZ Full: the whole chirp procedure body)

Proved

NLogN.KFZ.setup_St

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.KFZFull):

theorem NLogN.KFZ.setup_St (s : IState HS) (f : Fin HS → List ℤ) (hf : EmptyEx f) (K p α h d : ℕ)
    (bs ss : List ℕ) (hbl : bs.length = d) (hsl : ss.length = d) (hb1 : ∀ b ∈ bs, b ≤ 1)
    (hst : ∀ (i : ℕ) (hi : i < d), ss[i]'(by omega) ≤ 2 ^ (h + bs[i]'(by omega)))
    (hc : f NLogN.KFZ.cS = (K : ℤ) :: (p : ℤ) :: (α : ℤ) :: (h : ℤ) :: (d : ℤ) :: (NLogN.KFZ.zl bs ++ NLogN.KFZ.zl ss))
    (B : ℕ) (hB : NLogN.KFZ.eS h bs d + h + 6 ≤ B) (hdB : d + 2 ≤ B) (hK : zsz (K : ℤ) ≤ B)
    (hp : zsz (p : ℤ) ≤ B) (hα : zsz (α : ℤ) ≤ B) (td : List ℤ)
    (hdesc : descA 1 (axD h bs ss) ++ [(2 * 2 ^ NLogN.KFZ.eS h bs d : ℤ)] = td) :
    Sp setupP (NLogN.KIL.encK s f) (NLogN.KIL.encK s (St f p h d bs.sum K td (f 0) [] [] [] [] []))
      (10 ^ 10 * (d + 1) * (B + 2) ^ 2)

Context. The lemma belongs to the part KFZ Full: the whole chirp procedure body: body j d18 = setupP; T1; …; T7; wP setC (PP j d18); popsL regL.

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 autoImplicit false
set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
set_option linter.unusedSectionVars false
set_option linter.unnecessarySeqFocus false
open NLogN
open NLogN.KFZ
open IntMul NLogN.KIL KFB NLogN.KFM IntMul.OpenAI.Glue

Formal statement

theorem NLogN.KFZ.setup_St (s : IState HS) (f : Fin HS → List ℤ) (hf : EmptyEx f) (K p α h d : ℕ)
    (bs ss : List ℕ) (hbl : bs.length = d) (hsl : ss.length = d) (hb1 : ∀ b ∈ bs, b ≤ 1)
    (hst : ∀ (i : ℕ) (hi : i < d), ss[i]'(by omega) ≤ 2 ^ (h + bs[i]'(by omega)))
    (hc : f NLogN.KFZ.cS = (K : ℤ) :: (p : ℤ) :: (α : ℤ) :: (h : ℤ) :: (d : ℤ) :: (NLogN.KFZ.zl bs ++ NLogN.KFZ.zl ss))
    (B : ℕ) (hB : NLogN.KFZ.eS h bs d + h + 6 ≤ B) (hdB : d + 2 ≤ B) (hK : zsz (K : ℤ) ≤ B)
    (hp : zsz (p : ℤ) ≤ B) (hα : zsz (α : ℤ) ≤ B) (td : List ℤ)
    (hdesc : descA 1 (axD h bs ss) ++ [(2 * 2 ^ NLogN.KFZ.eS h bs d : ℤ)] = td) :
    Sp setupP (NLogN.KIL.encK s f) (NLogN.KIL.encK s (St f p h d bs.sum K td (f 0) [] [] [] [] []))
      (10 ^ 10 * (d + 1) * (B + 2) ^ 2) := 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.KFZ.setup_St` of the Lean development Solutions/KFZFull.lean (lines 132–163), integer-multiplication-kappa campaign.

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