Explicit inverse of the real Gaussian Toeplitz block

Proved

J6QLU.realToep_inv

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

complexity-theoryinteger-multiplicationkappa-improvement

Let $\alpha, \sigma, b \in \mathbb R$ with $\alpha^2 \sigma > 0$ and $T =$ realToep α σ b n the real Gaussian Toeplitz block. With $q = e^{-2\pi\alpha^2\sigma}$ and the coefficient sequences $a^{\pm}$ ,

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

In Lean:

theorem J6QLU.realToep_inv (h : 0 < α ^ 2 * σ) (n : ℕ) :
    (realToep α σ b n)⁻¹ = GamInv (qR α σ) n * upT (aplus α σ b) * Gam (qR α σ) n *
      lowT (aminus α σ b) * GamInv (qR α σ) 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 Real
variable (α σ b : ℝ)

Formal statement

theorem J6QLU.realToep_inv (h : 0 < α ^ 2 * σ) (n : ℕ) :
    (realToep α σ b n)⁻¹ = GamInv (qR α σ) n * upT (aplus α σ b) * Gam (qR α σ) n *
      lowT (aminus α σ b) * GamInv (qR α σ) 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.realToep_inv`, Solutions/J6QLUReal.lean (lines 89–94) of the round-six development.

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