Decay bound for the coefficients a^- of the Gaussian Toeplitz inverse

Proved

J6QLU.aminus_bound

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

complexity-theoryinteger-multiplicationkappa-improvement

Let $\alpha \ge 2$ , $\sigma \ge 1$ and $\theta \le \sigma - 2|b|$ . Then for every $h \in \mathbb N$ the coefficient $a^{-}_h$ of the explicit inverse of the real Gaussian Toeplitz block satisfies

|a^{-}_h| \le 1.0001\, e^{-\pi \alpha^2 \theta h}.

In Lean:

theorem J6QLU.aminus_bound (α σ b θ : ℝ) (hα : 2 ≤ α) (hσ : 1 ≤ σ) (hθ : θ ≤ σ - 2 * |b|) (h : ℕ) :
    |aminus α σ b h| ≤ 1.0001 * exp (-(π * α ^ 2 * θ * h))

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 Real

Formal statement

theorem J6QLU.aminus_bound (α σ b θ : ℝ) (hα : 2 ≤ α) (hσ : 1 ≤ σ) (hθ : θ ≤ σ - 2 * |b|) (h : ℕ) :
    |aminus α σ b h| ≤ 1.0001 * exp (-(π * α ^ 2 * θ * h)) := 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.aminus_bound`, Solutions/J6QLUBound.lean (lines 83–86) of the round-six development.

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