Explicit inverse of the Gaussian Toeplitz matrix via q-LU

Proved

J6QLU.qLU_inv

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

complexity-theoryinteger-multiplicationkappa-improvement

Under the hypotheses of the q-LU factorization ( $K$ a field, $z \ne 0$ , $q = x^2$ , $(q;q)_i \ne 0$ for $i < n$ ), with $a^{+}_h = (-xz)^h/(q;q)_h$ and $a^{-}_h = (-x z^{-1})^h/(q;q)_h$ ,

T^{-1} = \Gamma^{-1}\, U(a^{+})\, \Gamma\, L(a^{-})\, \Gamma^{-1},

where $T$ is the Gaussian Toeplitz matrix, $\Gamma$ the diagonal q-Pochhammer matrix and $L, U$ triangular Toeplitz matrices.

In Lean:

theorem J6QLU.qLU_inv (x z : K) (n : ℕ) (hz : z ≠ 0) (hq : ∀ i < n, qpoch (x ^ 2) i ≠ 0) :
    (gaussToep x z n)⁻¹ =
      GamInv (x ^ 2) n * upT (aP x z) * Gam (x ^ 2) n * lowT (aP x z⁻¹) * GamInv (x ^ 2) n

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_Defs1

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6QLU
open Finset Matrix
variable {K : Type*} [Field K]

Formal statement

theorem J6QLU.qLU_inv (x z : K) (n : ℕ) (hz : z ≠ 0) (hq : ∀ i < n, qpoch (x ^ 2) i ≠ 0) :
    (gaussToep x z n)⁻¹ =
      GamInv (x ^ 2) n * upT (aP x z) * Gam (x ^ 2) n * lowT (aP x z⁻¹) * GamInv (x ^ 2) n := 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 `J6QLU.qLU_inv`, Solutions/J6QLUMain.lean (lines 130–133) of the round-six development.

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