The templates of the network are admissible at `q ^ f` (round six, `tok_edges`)

Proved

IntMul.Kappa.R6.IL.tok_edges

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

The templates of the network are admissible at q ^ f.

The following holds (Lean statement, in the scope of module Solutions.R6ILTok):

theorem IntMul.Kappa.R6.IL.tok_edges (C : Bit.QHNet (ZMod 2) W T m) {q f Bb : ℕ} (hq : q.Prime)
    (hqB : qBound C < q) (hf : 1 ≤ f) (hOK : C.EdgesOK (q ^ f))
    (hBb : zsz ((((q ^ f + 1) ^ 3 : ℕ)) : ℤ) ≤ Bb) :
    (∀ e ∈ C.edges, ∀ T' ∈ tplE e.E₁ ++ tplE e.E₂, TOK T' (q ^ f) Bb) ∧
    (∀ e ∈ C.edges, ∀ i j, Nat.Coprime (e.E₁ i j).den (q ^ f) ∧
      Nat.Coprime (e.E₂ i j).den (q ^ f))

Context. Round six, item language: the shift templates are admissible for large primes

tok_edges: for a prime q above the template constant qBound C and f ≥ 1, if the edges are EdgeOK at q ^ f, then every shift template of every edge satisfies TOK at q ^ f, and all matrix denominators are coprime to q ^ 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_R6_Defs13

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.IL
open IntMul NLogN NLogN.RS NLogN.KIL NLogN.KFD AModel AModelL12 AModelFac Finset
variable {W T m : ℕ}

Formal statement

theorem IntMul.Kappa.R6.IL.tok_edges (C : Bit.QHNet (ZMod 2) W T m) {q f Bb : ℕ} (hq : q.Prime)
    (hqB : qBound C < q) (hf : 1 ≤ f) (hOK : C.EdgesOK (q ^ f))
    (hBb : zsz ((((q ^ f + 1) ^ 3 : ℕ)) : ℤ) ≤ Bb) :
    (∀ e ∈ C.edges, ∀ T' ∈ tplE e.E₁ ++ tplE e.E₂, TOK T' (q ^ f) Bb) ∧
    (∀ e ∈ C.edges, ∀ i j, Nat.Coprime (e.E₁ i j).den (q ^ f) ∧
      Nat.Coprime (e.E₂ i j).den (q ^ f)) := 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.IL.tok_edges`, Solutions/R6ILTok.lean (lines 61–102) of the round-six development.

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