The closed energy band controls the core radius

Proved

BianchiIX.coreBootstrap_radius_closed

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

bianchi-ix-era-statisticsdynamical-systems

If $k\geq6$ , $r\geq0$ , and $|r^2-k^2|\leq1/8$ , then

99k/100\leq r\leq101k/100.

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.coreBootstrap_radius_closed {k r : ℝ} (hk : 6 ≤ k) (hr : 0 ≤ r)
    (he : |r ^ 2 - k ^ 2| ≤ 1 / 8) : 99 / 100 * k ≤ r ∧ r ≤ 101 / 100 * k := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Algebra/CoreRadialBounds.lean#L7-L16

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