Good primes for finitely many rational matrices (`good_mats`)

Proved

IntMul.Kappa.R6.Bit.NB.good_mats

by xuanji · Oct 19, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

Good primes for finitely many rational matrices: all denominators coprime to q ^ f.

The following holds (Lean statement, in the scope of module Solutions.R6BitNBPrime):

theorem IntMul.Kappa.R6.Bit.NB.good_mats {ι κ : Type*} [Fintype ι] [Fintype κ] (S : Finset (Matrix ι κ ℚ)) :
    ∃ q₀ : ℕ, ∀ q, q.Prime → q₀ < q → ∀ f : ℕ, ∀ A ∈ S, ∀ i j,
      Nat.Coprime (A i j).den (q ^ f) ∧ (A i j ≠ 0 → IsUnit (redQ (q ^ f) (A i j)))

Context. Round six, bit network: good primes

Finitely many rationals have denominators coprime to every power of every prime q > q₀, and the nonzero ones are units mod q ^ f. Consequently a finite set of edges with EdgeQ data is EdgeOK (q ^ f) for all large primes q and all f.

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Mathlib
import Definitions.Def_IntMul_KappaFramework

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open Matrix Finset AModelFac

Formal statement

theorem IntMul.Kappa.R6.Bit.NB.good_mats {ι κ : Type*} [Fintype ι] [Fintype κ] (S : Finset (Matrix ι κ ℚ)) :
    ∃ q₀ : ℕ, ∀ q, q.Prime → q₀ < q → ∀ f : ℕ, ∀ A ∈ S, ∀ i j,
      Nat.Coprime (A i j).den (q ^ f) ∧ (A i j ≠ 0 → IsUnit (redQ (q ^ f) (A i j))) := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `IntMul.Kappa.R6.Bit.NB.good_mats`, Solutions/R6BitNBPrime.lean (lines 40–45) of the round-six development.

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