Dyadic Power Comparison

Proved

LevyStochCalc.Brownian.Continuity.rpow_half_pow_le_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

This is a deterministic comparison between dyadic scales and Hölder powers. Let $\alpha,d\in\mathbb R$ and $m\in\mathbb N$ . If $\alpha>0$ and $(1/2)^{m+1}<d$ , then

\bigl((1/2)^\alpha\bigr)^m\le 2^\alpha d^\alpha.

It converts a geometric level factor into a power of the underlying distance, with no probabilistic assumptions.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Continuity. Shown in its original module context. Proof omitted.

lemma rpow_half_pow_le {α : ℝ} (hα : 0 < α) {d : ℝ} (m : ℕ)
    (hd : (1 / 2 : ℝ) ^ (m + 1) < d) :
    ((1 / 2 : ℝ) ^ α) ^ m ≤ (2 : ℝ) ^ α * d ^ α

Module navigation

Module: LevyStochCalc/Brownian/ContinuityDyadicChaining.

Previous: exists_dyadic_scale · Next: kc_exponent_identity.

Source excerpt, lines 359–361 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ContinuityDyadicChaining

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/ContinuityDyadicChaining.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)

Formal statement


lemma LevyStochCalc.Brownian.Continuity.rpow_half_pow_le_readable {α : ℝ} (hα : 0 < α) {d : ℝ} (m : ℕ)
    (hd : (1 / 2 : ℝ) ^ (m + 1) < d) :
    ((1 / 2 : ℝ) ^ α) ^ m ≤ (2 : ℝ) ^ α * d ^ α := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ContinuityDyadicChaining.lean#L357-L375 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ContinuityDyadicChaining`, declaration `LevyStochCalc.Brownian.Continuity.rpow_half_pow_le`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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