Round-six lemma `poisson_shift`
ProvedIntMul.OpenAI.RH.poisson_shiftAn 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.ResampleHvdH):
theorem IntMul.OpenAI.RH.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. Copy (renamed namespace IntMul.OpenAI.RH) of the base thread's HvdH Prop 4.7(i)
(intmul/base/Sol_prop_4_7_i.lean): Gaussian resampling identity, maps S, T, C, D, P_s, P_t.
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*).
-- 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 Mathlib set_option autoImplicit false set_option relaxedAutoImplicit false set_option maxHeartbeats 1000000 open IntMul IntMul.OpenAI IntMul.OpenAI.RH open Complex Real
theorem IntMul.OpenAI.RH.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