A small radial drift strictly improves the energy band

Proved

BianchiIX.core_radius_energy_improvement

by trppppp · Nov 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

Assume $k\geq6$ , $0\leq r\leq101k/100$ , $0\leq u,w\leq1/(10000k)$ , and $|r-k|\leq16(u+w^2)/k$ . Then

|r^2-k^2|<1/16.

Preamble

import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.FieldSimp

namespace BianchiIX
end BianchiIX

noncomputable section
open BianchiIX

Formal statement

theorem BianchiIX.core_radius_energy_improvement {k r u w : ℝ} (hk : 6 ≤ k)
    (hr : 0 ≤ r) (hrhi : r ≤ 101 / 100 * k)
    (hu : 0 ≤ u ∧ u ≤ (1 / 10000) / k) (hw : 0 ≤ w ∧ w ≤ (1 / 10000) / k)
    (hR : |r - k| ≤ 16 / k * (u + w ^ 2)) : |r ^ 2 - k ^ 2| < 1 / 16 := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Algebra/CoreRadialBounds.lean#L68-L84

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