Prove2Me

A small wall budget controls terminal-tail feedback

Proved
BianchiIX.exists_exitTail_feedback_budget

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

bianchi-ix-era-statisticsdynamical-systems

For y=(m,u,v,w)∈R4y=(m,u,v,w)\in\mathbb R^4, define

K(u,v,w)=u2+v2+w2−2(uv+vw+wu),κ=u(v+w−u),K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u), Fm=−v2+w2+u(v−w)−mκ,Fu=12(1−m2−(v−w)2+u2)u,F_m=-v^2+w^2+u(v-w)-m\kappa,\qquad F_u=\tfrac12(1-m^2-(v-w)^2+u^2)u, Fv=(m−1−κ)v,Fw=(−m−1−κ)w,F=(Fm,Fu,Fv,Fw),F_v=(m-1-\kappa)v,\qquad F_w=(-m-1-\kappa)w,\qquad F=(F_m,F_u,F_v,F_w),

and E(m,v)=(m−1)2+v2E(m,v)=(m-1)^2+v^2.

Write r(m,v)=(m−1)2+v2r(m,v)=\sqrt{(m-1)^2+v^2} and define

g(R,v)=R2−v2,d(R,u,v,w)=g(R,v)+u(v+w−u),g(R,v)=\sqrt{R^2-v^2},\qquad d(R,u,v,w)=g(R,v)+u(v+w-u), fR=−u(v+w−u)R−g(R,v)(u−w)2/Rd,f_R=\frac{-u(v+w-u)R-g(R,v)(u-w)^2/R}{d}, as=2g(R,v)−R2+2vw−w2+u22d,aw=g(R,v)−2−u(v+w−u)d.a_s=\frac{2g(R,v)-R^2+2vw-w^2+u^2}{2d},\qquad a_w=\frac{g(R,v)-2-u(v+w-u)}d.

The exit clock rate is ω(y)=1−m+u(v+w−u)\omega(y)=1-m+u(v+w-u), and its field is Flog⁡(y)=F(y)/ω(y)F_{\log}(y)=F(y)/\omega(y). Division and square roots have their total real-valued Lean meanings.

For Y=(R,u,w)Y=(R,u,w) define Ev(Y)=(fR,asu,aww)E_v(Y)=(f_R,a_su,a_ww), evaluated at (R,u,v,w)(R,u,v,w). The coordinate maps are Q(m,u,v,w)=(r(m,v),u,w)Q(m,u,v,w)=(r(m,v),u,w) and Qv−1(R,u,w)=(1−g(R,v),u,v,w)Q_v^{-1}(R,u,w)=(1-g(R,v),u,v,w). For a curve Y(s)Y(s) set

Mi(Y,t)=exp⁡∫0tai(Y0(s),Y1(s),e−s,Y2(s)) ds(i=s,w).M_i(Y,t)=\exp\int_0^t a_i(Y_0(s),Y_1(s),e^{-s},Y_2(s))\,ds \quad(i=s,w).

Let Rα,b,A,D\mathcal R_{\alpha,b,A,D} consist of the states satisfying

7/4≤R≤17/4,∣u∣,∣w∣≤1/100,∣u∣+∣w∣≤Ab,∣R−(2+2α)∣≤Db.7/4\leq R\leq17/4,\quad |u|,|w|\leq1/100,\quad |u|+|w|\leq Ab,\quad |R-(2+2\alpha)|\leq Db.

Put

B(α,b,A,D,T)=[10(α+Db)+100(Ab)2]T+100Ab+5,B(\alpha,b,A,D,T)=[10(\alpha+Db)+100(Ab)^2]T+100Ab+5, L(η,α,J)=log⁡(1/η)+Jlog⁡(1/α).L(\eta,\alpha,J)=\log(1/\eta)+J\log(1/\alpha).

For A≥1A\geq1 and D≥0D\geq0 there exists 0<ε≤10<\varepsilon\leq1 such that, whenever T,b≥0T,b\geq0, b(1+T)≤εb(1+T)\leq\varepsilon and αT≤K\alpha T\leq K,

B(α,b,A,D,T)≤10K+6,20[Ab+(Ab)2T]≤40Ab.B(\alpha,b,A,D,T)\leq10K+6,\qquad 20[Ab+(Ab)^2T]\leq40Ab.
Preamble
import Definitions.Def_BianchiIX_TerminalConnectionData
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
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arcosh
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

noncomputable section
open BianchiIX
open Set
Formal statement
theorem BianchiIX.exists_exitTail_feedback_budget {A D : ℝ} (hA : 1 ≤ A) (hD : 0 ≤ D) :
    ∃ ε > (0 : ℝ), ε ≤ 1 ∧ ∀ T b α K : ℝ,
      0 ≤ T → 0 ≤ b → b * (1 + T) ≤ ε → α * T ≤ K →
      exitTailRateBudget α b A D T ≤ 10 * K + 6 ∧
      20 * (A * b + (A * b) ^ 2 * T) ≤ 40 * A * b := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/ExitTailBootstrap.lean#L9-L46

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