Radius loss through an entrance and its full reflection

Proved

BianchiIX.entrancePartialCore_reflection_values

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

bianchi-ix-era-statisticsdynamical-systems

In projective coordinates $y=(m,u,v,w)$ put $k(y)=\sqrt{(m-1)^2+v^2}$ . Write $Z(y)$ for the state at the first positive unit entrance and $\kappa(y)=k(Z(y))$ . At the partial core's downward unit hit, let $p$ be the three exit parameters, $G$ the reflected core state after its connection, and $t_R$ the absolute arrival time on the original orbit. Write $g=G_0$ and $X=2/g$ . The physical endpoint identity says that the ascending unit entry with core $G$ equals the reflection of the projective state of this orbit at $t_R$ . Let $k(y)\ge8$ , $m\ge1$ , $0<u,w\le10^{-4}$ , $0<v<1$ , $0<\eta\le10^{-4}$ , $u+vw\le\eta/(2k(y))$ . Then $p_0,g\ge\kappa/2$ , $1\le\kappa-g\le3$ , $|g-(\kappa-2)|\le64(Z_1+Z_3^2)/\kappa$ and $0<G_1\le Z_1$ , $0<G_2\le Z_3$ .

Preamble

import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Dynamics_IntegralClock
import Definitions.Def_BianchiIX_Analysis_CoreGrowthBudget
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Algebra_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_WallEvolution
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Algebra_ConnectionBounds
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_ProjectiveRegularity
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_ProjectiveClock
import Definitions.Def_BianchiIX_Dynamics_ProjectiveOriginalTime
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_CoreCoordinateDifferential
import Definitions.Def_BianchiIX_Dynamics_Events
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_CoreFlowVariation
import Definitions.Def_BianchiIX_Dynamics_ConnectionClock
import Definitions.Def_BianchiIX_Dynamics_ConnectionClockDifferential
import Definitions.Def_BianchiIX_Dynamics_ProjectiveReflection
import Definitions.Def_BianchiIX_Dynamics_CoreEntry
import Definitions.Def_BianchiIX_Dynamics_ConnectionRegion
import Definitions.Def_BianchiIX_Dynamics_EntranceRegion
import Definitions.Def_BianchiIX_Dynamics_ConnectionExistence
import Definitions.Def_BianchiIX_Dynamics_CoreExitFactor
import Definitions.Def_BianchiIX_Dynamics_ConnectionEvents
import Definitions.Def_BianchiIX_Dynamics_CoreEntryDifferential
import Definitions.Def_BianchiIX_Dynamics_ConnectionReturn
import Definitions.Def_BianchiIX_Dynamics_EntranceReturn
import Definitions.Def_BianchiIX_Dynamics_ConnectionReturnValues
import Definitions.Def_BianchiIX_EraReturnData
import Definitions.Def_BianchiIX_Dynamics_ConnectionPhysicalReturn
import Definitions.Def_BianchiIX_Dynamics_CoreReturnDifferential
import Definitions.Def_BianchiIX_Dynamics_CoreConnectionChart
import Definitions.Def_BianchiIX_Dynamics_CompactCoreReturn
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.Calculus.ImplicitContDiff
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.Topology.Order.IntermediateValue
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.Module
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Tactic.Positivity
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.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
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

noncomputable section
open BianchiIX
open Set

Formal statement

theorem BianchiIX.entrancePartialCore_reflection_values {y : ProjectiveState} {η : ℝ}
    (hk : 8 ≤ coreRadius (y 0) (y 2)) (hm : 1 ≤ y 0)
    (hu : 0 < y 1 ∧ y 1 ≤ 1 / 10000) (hv : 0 < y 2 ∧ y 2 < 1)
    (hw : 0 < y 3 ∧ y 3 ≤ 1 / 10000) (hη : 0 < η ∧ η ≤ 1 / 10000)
    (hsmall : y 1 + y 2 * y 3 ≤ η / (2 * coreRadius (y 0) (y 2))) :
    let z := entranceReturnState y
    let κ := coreRadius (z 0) (z 2)
    let p := coreExitConnectionParameters (partialCoreDownwardReturnState y)
    let G := connectionReturnPhysicalState p
    κ / 2 ≤ p 0 ∧ (1 ≤ κ - G 0 ∧ κ - G 0 ≤ 3) ∧ κ / 2 ≤ G 0 ∧ G 0 ≤ κ ∧
      |G 0 - (κ - 2)| ≤ 64 * (z 1 + (z 3) ^ 2) / κ ∧
      (0 < G 1 ∧ G 1 ≤ z 1) ∧ (0 < G 2 ∧ G 2 ≤ z 3) := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/EntranceReflectionValues.lean#L10-L66

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