One invariant null exception for all physical sections and the constraint volume

Proved

BianchiIX.exists_common_borel_success_exception

by trppppp · Nov 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

For real variables $p,s,a,b,c$ , set

K=a^2+b^2+c^2-2(ab+bc+ca),\qquad S=p^2+s^2,\qquad C=S+K.

The shear components and wall rates are

P=Kp-2a^2+b^2+c^2+ab-2bc+ac,

Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac),

r_1=-S+2p,\qquad r_2=-S-p+\sqrt{3}s,\qquad r_3=-S-p-\sqrt{3}s.

These functions define the normalized vacuum Bianchi IX vector field $(P,Q,r_1a,r_2b,r_3c)$ .

Let $X=\mathbb R^5$ with coordinates $x=(p,s,a,b,c)$ and the sup norm. Write

F(x)=(P,Q,r_1a,r_2b,r_3c),\qquad \mathcal C(x)=C(p,s,a,b,c),

and let $W=\{(p,s,a,b,c)\in X:a>0,\ b>0,\ c>0\}$ .

Write $N_1(x)=a$ , $N_2(x)=b$ , and $N_3(x)=c$ . For initial data $(x_0,t_0)$ , let $I_{\max}(x_0,t_0)$ be the union of all open convex domains of solutions with these initial data, and let $\alpha_{x_0,t_0}$ be their compatible curve. A solution $S$ has such a domain $I_S$ , a curve $\alpha_S$ , and $\alpha_S(t_0)=x_0$ .

Let $\mathcal L(X,X)$ be the continuous linear operators on $X$ and set

A(x)=dF_x,\qquad \widetilde F(x,J)=(F(x),A(x)\circ J).

For a fixed initial time $t_0$ , put

\mathcal D_{t_0}=\{(x,t):t\in I_{\max}(x,t_0)\},\qquad \Phi_{t_0}(x,t)=\alpha_{x,t_0}(t),\qquad J_{x,t_0}(t)=d_x\Phi_{t_0}(x,t).

Write $\lambda_x=d\mathcal C_x:X\to\mathbb R$ for the constraint differential.

For $x=(p,s,a,b,c)$ , define

H(x)=(abc)^{2/3},\quad w(x)=abc(1+4H(x)),\quad \delta(x)=2K-3S,\quad \bar\delta(x)=\frac{2(K+4H(x))}{1+4H(x)},

V(x,t_0,t)=\frac{w(x)\det J_{x,t_0}(t)}{w(\alpha_{x,t_0}(t))}.

Write $[L]$ for the matrix of a continuous linear operator $L:X\to X$ in the standard basis. Set $\Omega(v_0,\ldots,v_4)=\det(v_0,\ldots,v_4)$ ,

\omega_x(v_0,\ldots,v_3)=\frac{\Omega(x/2,v_0,\ldots,v_3)}{w(x)},\qquad \varphi_x(v_0,v_1,v_2)=\omega_x(F(x),v_0,v_1,v_2).

A normalized symmetry is a continuous linear equivalence $L:X\to X$ satisfying

\mathcal C(Lx)=\mathcal C(x),\quad Lx\in W\ \Longleftrightarrow\ x\in W, \quad F(Lx)=LF(x),\quad w(Lx)=w(x),\quad\det L=1.

These symmetries include the identity, composition, inverse, and the two maps

R(p,s,a,b,c)=((-p+\sqrt3s)/2,(-\sqrt3p-s)/2,b,c,a),\qquad S_0(p,s,a,b,c)=(p,-s,a,c,b).

Transforming a solution replaces its curve $\alpha$ by $L\alpha$ and keeps its time domain.

Write $\Phi^s_t(x)=\alpha_{x,s}(t)$ , with domain $I(x,s)$ , and $\mathcal X=\{x:\mathcal C(x)=1,\ N_i(x)>0\ (i=1,2,3)\}$ . For $y=(v,l,r)\in\mathbb R^3$ and a wall height $h$ , put

b=e^{-l},\quad c=e^{-r},\quad d=1-v^2-K(h,b,c),\quad U_h=\{y:d>0\},\quad c_h(y)=(-\sqrt d,v,h,b,c).

For a normalized axis symmetry $A$ , the chart is $c_{A,h}=A^{-1}c_h$ , with coordinates $\pi_A(x)=((Ax)_1,-\log (Ax)_3,-\log (Ax)_4)$ . At $p=-\sqrt d$ its flux density and measure are

\rho_h(y)=\frac{p^2+v^2-2p}{2|p|(1+4(hbc)^{2/3})}, \qquad d\mu_h={\bf1}_{U_h}\rho_h^+\,dy.

Write $\varphi$ for the signed section flux and $\bar\delta$ for the corrected divergence. The three primary frames interchange axis zero with the chosen dominant axis.

Let $\mathcal T$ be the union, over primary frames, of the subspaces where the transverse shear and the difference of the other two walls both vanish. Set $\Delta_{ij}(x)=2\sqrt{|N_i(x)N_j(x)|}$ . The Brehm attractor input used here is the following proposition: for each $x\in\mathcal X\setminus\mathcal T$ and $i\ne j$ ,

\Delta_{ij}(\Phi^0_t(x))\longrightarrow0\qquad(t\to+\infty).

For a measurable space $X$ , let $(\mathrm B)_X$ be the following proposition: for every probability measure $\mu$ on $X$ , every ergodic map $F:X\to X$ , and every integrable $f:X\to\mathbb R$ ,

\frac1n\sum_{j=0}^{n-1}f(F^jx)\longrightarrow\int_X f\,d\mu \quad\text{for }\mu\text{-almost every }x.

This proposition is an explicit hypothesis in the results that use it.

Put $\mathcal E=\mathcal X\cap(\mathcal T\cup\{x:\exists\epsilon>0,\ d_T(x_t)\ge\epsilon\text{ eventually}\})$ , where $d_T$ is distance to the nearest Taub point. A transverse chart is a map $C:\mathbb R^3\to\mathbb R^5$ of class $C^1$ on an open $U$ , with $C(U)\subset\mathcal X$ and nonzero section flux there. Four-coordinate nullity means $\lambda_4(Q(S))=0$ for every differentiable $Q:\mathbb R^5\to\mathbb R^4$ .

Take $0<h\le1/16$ and $0<d<h/8$ . On the three primary wall charts let $E_i=U_h\cap\{|v|\le d,\ e^{-l},e^{-r}\le d\}$ . Give their disjoint union the labelled measure $\nu_h=\sum_i(y\mapsto(i,y))_*\mu_h$ . For a Borel set $F$ of labels and coordinates with $\nu_h(F)<\infty$ , the target consists of the chart images of the points $(i,y)\notin F$ with $y\in E_i$ .

Assume $(\mathrm B)_{\mathbb R}$ and the Brehm attractor input. There exists a Borel $N$ with $\mathcal E\subset N\subset\mathcal X$ such that $x_t\in N\iff x\in N$ whenever $t\in I(x,0)$ . Every transverse chart satisfies $\lambda_3(U\cap C^{-1}N)=0$ , and $N$ is four-coordinate null. Every $x\in\mathcal X\setminus N$ reaches the target at an existing nonnegative time.

Preamble

import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Literature
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_AxisAndOrbitData
import Definitions.Def_BianchiIX_Dynamics_AxisMaps
import Definitions.Def_BianchiIX_AxisSectionMeasureData
import Definitions.Def_BianchiIX_Dynamics_WallEvolution
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_FlowComposition
import Definitions.Def_BianchiIX_Dynamics_FlowRegularity
import Definitions.Def_BianchiIX_Dynamics_SectionFlux
import Definitions.Def_BianchiIX_Dynamics_AxisSymmetry
import Definitions.Def_BianchiIX_GeometricExceptionData
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.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.SpecialFunctions.Pow.Deriv
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.LinearAlgebra.Determinant
import Mathlib.Topology.Instances.Matrix
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.LinearAlgebra.Alternating.Curry
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.Topology.Compactness.Lindelof
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Compactness.Compact
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.MeasurableSpace.Prod
import Mathlib.MeasureTheory.Integral.Lebesgue.Markov
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Arsinh
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.Data.Matrix.Mul
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Tactic.GCongr
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Order.Compact
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Tactic.Positivity
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.Normed.Operator.Basic
import Mathlib.LinearAlgebra.Matrix.Notation
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Data.List.Induction
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.Dynamics.BirkhoffSum.Average
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Data.PNat.Basic
import Mathlib.Data.Rat.Floor
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Order.OrderIsoNat
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.JacobianOneDim
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Topology.UniformSpace.HeineCantor
import Mathlib.Dynamics.BirkhoffSum.NormedSpace
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Analysis.Normed.Group.Tannery
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.ContinuousOn
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Order.Lattice
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.Topology.Algebra.InfiniteSum.ENNReal
import Mathlib.Algebra.Order.Archimedean.Basic

noncomputable section
open BianchiIX
open Set Filter MeasureTheory
open scoped Topology ENNReal

Formal statement

theorem BianchiIX.exists_common_borel_success_exception
    (hBirkhoff : Literature.BirkhoffErgodicInput ℝ)
    (hBrehm : Literature.BrehmIXAttractorInput) {h d : ℝ}
    (hh : 0 < h) (hhmax : h ≤ 1 / 16) (hd : 0 < d) (hdh : d < h / 8)
    {F : Set (Fin 3 × SectionCoordinates)} (hF : MeasurableSet F)
    (hfinite : axisFamilyMeasure (fun _ : Fin 3 => h) F ≠ ⊤) :
    let E := fun _ : Fin 3 => mainWallSmallSection d ∩ mainWallChartDomain h
    ∃ N : Set NormalizedState, MeasurableSet N ∧ N ⊆ physicalPhaseSpace ∧
      geometricException ⊆ N ∧
      (∀ x : NormalizedState, ∀ t ∈ normalizedMaxDomain x 0,
        normalizedMaxCurve x 0 t ∈ N ↔ x ∈ N) ∧
      (∀ (C : SectionCoordinates → NormalizedState) (U : Set SectionCoordinates),
        IsOpen U → (∀ y ∈ U, ContDiffAt ℝ 1 C y) →
        (∀ y ∈ U, positiveWalls (C y)) → (∀ y ∈ U, stateConstraint (C y) = 1) →
        (∀ y ∈ U, normalizedSectionFlux (C y) (fun i => fderiv ℝ C y (Pi.single i 1)) ≠ 0) →
        volume (U ∩ C ⁻¹' N) = 0) ∧ NullInFourCoordinates N ∧
      ∀ x ∈ physicalPhaseSpace, x ∉ N →
        x ∈ globalAxisWallHittingDomain primaryAxisFrame (fun _ => h)
          (axisFamilySuccessTargets E F) (Ici 0) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Measure/CommonSuccessException.lean#L48-L101

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