Round-six lemma `poisson_shift`

Proved

IntMul.HvdH.Res.poisson_shift

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

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.KFMBaseP47):

theorem IntMul.HvdH.Res.poisson_shift (α : ℝ) (hα : 0 < α) (s : ℝ) (hs : 0 < s) (x ρ : ℝ) :
    ∑' j : ℤ, ((Real.exp (-π * α⁻¹ ^ 2 * s ^ 2 * (x - j / s) ^ 2) : ℝ) : ℂ) *
        cexp (2 * π * I * ρ * j / s)
      = α * ∑' n : ℤ, ((Real.exp (-π * α ^ 2 * (n - ρ / s) ^ 2) : ℝ) : ℂ) *
        cexp (-2 * π * I * s * x * (n - ρ / s))

Context. Definitions (HvdH §2.4, §4.1)

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 Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_SApprox
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Mathlib
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.HvdH IntMul.HvdH.Res
open Complex Real

Formal statement

theorem IntMul.HvdH.Res.poisson_shift (α : ℝ) (hα : 0 < α) (s : ℝ) (hs : 0 < s) (x ρ : ℝ) :
    ∑' j : ℤ, ((Real.exp (-π * α⁻¹ ^ 2 * s ^ 2 * (x - j / s) ^ 2) : ℝ) : ℂ) *
        cexp (2 * π * I * ρ * j / s)
      = α * ∑' n : ℤ, ((Real.exp (-π * α ^ 2 * (n - ρ / s) ^ 2) : ℝ) : ℂ) *
        cexp (-2 * π * I * s * x * (n - ρ / s)) := 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.HvdH.Res.poisson_shift`, Solutions/KFMBaseP47.lean (lines 67–111) of the round-six development.

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