Dyadic Scale Selection

Proved

LevyStochCalc.Brownian.Continuity.exists_dyadic_scale_readable

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

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

This is a deterministic dyadic-scale selection lemma. For a real number $d$ satisfying $0<d\le 1$ , there exists $m\in\mathbb N$ such that

(1/2)^{m+1}<d\le (1/2)^m.

The result provides the discrete scale used to compare an arbitrary distance with dyadic levels in chaining arguments.

Original Lean declaration header

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

lemma exists_dyadic_scale {d : ℝ} (hd0 : 0 < d) (hd1 : d ≤ 1) :
    ∃ m : ℕ, (1 / 2 : ℝ) ^ (m + 1) < d ∧ d ≤ (1 / 2) ^ m

Module navigation

Module: LevyStochCalc/Brownian/ContinuityDyadicChaining.

Previous: dyadicTrunc_eventually_eq · Next: rpow_half_pow_le.

Source excerpt, lines 341–342 · 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.exists_dyadic_scale_readable {d : ℝ} (hd0 : 0 < d) (hd1 : d ≤ 1) :
    ∃ m : ℕ, (1 / 2 : ℝ) ^ (m + 1) < d ∧ d ≤ (1 / 2) ^ m := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ContinuityDyadicChaining.lean#L339-L355 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ContinuityDyadicChaining`, declaration `LevyStochCalc.Brownian.Continuity.exists_dyadic_scale`; 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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