inv_sum (KFY 13: one axis of the `B₀` pass)

Proved

NLogN.KFY.inv_sum

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.KFY13AxB):

theorem NLogN.KFY.inv_sum (s t : ℕ) (ht : 2 ≤ t) (hc : Nat.Coprime s t) :
    (∑ y ∈ Finset.Icc 1 t, (y : ℤ) * (if ((y : ℤ) * (s : ℤ)) % (t : ℤ) = 1 then 1 else 0)) =
      (((s : ZMod t)⁻¹).val : ℤ)

Context. The lemma belongs to the part KFY 13: one axis of the B₀ pass.

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
open NLogN
open NLogN.KFY
open IntMul NLogN IntMul.HvdH.Ref NLogN.GL NLogN.AX NLogN.Rsp
open NLogN.KFM (shPad bsN)

Formal statement

theorem NLogN.KFY.inv_sum (s t : ℕ) (ht : 2 ≤ t) (hc : Nat.Coprime s t) :
    (∑ y ∈ Finset.Icc 1 t, (y : ℤ) * (if ((y : ℤ) * (s : ℤ)) % (t : ℤ) = 1 then 1 else 0)) =
      (((s : ZMod t)⁻¹).val : ℤ) := 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.KFY.inv_sum` of the Lean development Solutions/KFY13AxB.lean (lines 30–70), integer-multiplication-kappa campaign.

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