Prove2Me

Entrance directions control the projected variation throughout the first core return

Proved
BianchiIX.exists_large_core_entry_variation

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

bianchi-ix-era-statisticsdynamical-systems

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

K=a2+b2+c2−2(ab+bc+ca),S=p2+s2,C=S+K.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−2a2+b2+c2+ab−2bc+ac,P=Kp-2a^2+b^2+c^2+ab-2bc+ac, Q=Ks+3(−b2+c2+ab−ac),Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac), r1=−S+2p,r2=−S−p+3s,r3=−S−p−3s.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,r1a,r2b,r3c)(P,Q,r_1a,r_2b,r_3c). 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.

For x=(p,s,a,b,c)x=(p,s,a,b,c), put

ψ(p,z)=3 z1+p,Π(x)=(ψ(p,s),ψ(p,a),ψ(p,b),ψ(p,c)).\psi(p,z)=\frac{\sqrt3\,z}{1+p},\qquad \Pi(x)=(\psi(p,s),\psi(p,a),\psi(p,b),\psi(p,c)).

For y=(m,u,v,w)y=(m,u,v,w), set D(y)=3+m2+K(u,v,w)D(y)=3+m^2+K(u,v,w) and

ΞA(y)=(A−1,Am3,Au3,Av3,Aw3),Ξ(y)=Ξ6/D(y)(y).\Xi_A(y)=\left(A-1,\frac{Am}{\sqrt3},\frac{Au}{\sqrt3}, \frac{Av}{\sqrt3},\frac{Aw}{\sqrt3}\right),\qquad \Xi(y)=\Xi_{6/D(y)}(y).

Write N(x)=(P,Q,r1a,r2b,r3c)N(x)=(P,Q,r_1a,r_2b,r_3c) for the normalized vector field and C(x)=C(p,s,a,b,c)\mathcal C(x)=C(p,s,a,b,c) for its constraint.

The normalized and projective state spaces are R5\mathbb R^5 and R4\mathbb R^4 with their finite-product norms. The maps Π,Ξ,D,F\Pi,\Xi,D,F act on these state spaces.

Define the core radius, angle, and clock factor by

r(m,v)=(m−1)2+v2,θ(m,v)=arsinh⁡1−mv,d(r,u,w)=1−(u−w)2r2.r(m,v)=\sqrt{(m-1)^2+v^2},\qquad \theta(m,v)=\operatorname{arsinh}\frac{1-m}{v},\qquad d(r,u,w)=1-\frac{(u-w)^2}{r^2}.

For y=(m,u,v,w)y=(m,u,v,w), write ry=r(m,v)r_y=r(m,v), θy=θ(m,v)\theta_y=\theta(m,v), and dy=d(ry,u,w)d_y=d(r_y,u,w). All quotients use the total convention x/0=0x/0=0.

The core state space is R3\mathbb R^3 with coordinates Y=(r,u,p)Y=(r,u,p). Define

η(m,u,v,w)=(r(m,v),u,vw),ιz(r,u,p)=(1−rtanh⁡z,u,rcosh⁡z,pr/cosh⁡z).\eta(m,u,v,w)=(r(m,v),u,vw),\qquad \iota_z(r,u,p)=\left(1-r\tanh z,u,\frac r{\cosh z}, \frac p{r/\cosh z}\right).

For fixed (z,Y)(z,Y) put v=r/cosh⁡zv=r/\cosh z, w=p/vw=p/v, d=d(r,u,w)d=d(r,u,w), and define

V(z,Y)=(−κ(u,v,w)−tanh⁡z (u−w)2/rd,u[−r/2+tanh⁡z+w/cosh⁡z+(u2−w2)/(2r)]d,−2p(1+κ(u,v,w))rd).\mathcal V(z,Y)=\left( \frac{-\kappa(u,v,w)-\tanh z\,(u-w)^2/r}{d},\quad \frac{u[-r/2+\tanh z+w/\cosh z+(u^2-w^2)/(2r)]}{d},\quad \frac{-2p(1+\kappa(u,v,w))}{rd}\right).

For real kk, put εk=1/(10000k)\varepsilon_k=1/(10000k) and define

Kk={(m,u,v,w):∣E(m,v)−k2∣≤1/8,v≥1,0≤u≤εk,0≤vw≤εk}.\mathcal K_k=\{(m,u,v,w): |E(m,v)-k^2|\leq1/8,\quad v\geq1, \quad0\leq u\leq\varepsilon_k,\quad0\leq vw\leq\varepsilon_k\}.

Write X=R5X=\mathbb R^5, Y=R4Y=\mathbb R^4, and let αx,t0\alpha_{x,t_0} be the maximal solution of α′=N(α)\alpha'=N(\alpha) with initial value xx at t0t_0 and domain Imax⁡(x,t0)I_{\max}(x,t_0). Put

Jx,t0(t)=dxαx,t0(t),Ψt0(y,t)=Π(αΞ(y),t0(t)),J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad \Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)), Jy,t0(t)=dΠαΞ(y),t0(t)∘JΞ(y),t0(t)∘dΞy.\mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ J_{\Xi(y),t_0}(t)\circ d\Xi_y.

Write G(y)=6F(y)/D(y)G(y)=6F(y)/D(y), λ(y)=6rydy/D(y)\lambda(y)=6r_yd_y/D(y), and Sk(R,U,P)=(R,kU,P)S_k(R,U,P)=(R,kU,P). Define

Pyξ=dηyξ−dθyξ V(θy,η(y)),Wy,t0,w(t)=PΨt0(y,t)(Jy,t0(t)w).\mathcal P_y\xi=d\eta_y\xi-d\theta_y\xi\,\mathcal V(\theta_y,\eta(y)), \qquad \mathcal W_{y,t_0,w}(t)= \mathcal P_{\Psi_{t_0}(y,t)}(\mathcal J_{y,t_0}(t)w).

The entrance parametrization and radial exit factor are

e(k,u,w)=(1+k2−1,u,1,w),e(k,u,w)=(1+\sqrt{k^2-1},u,1,w), fy=V(θy,η(y))0,cy=−tanh⁡θy,hy=cy+fy/ry,ρy=cy/hy.f_y=\mathcal V(\theta_y,\eta(y))_0,\qquad c_y=-\tanh\theta_y, \qquad h_y=c_y+f_y/r_y,\qquad \rho_y=c_y/h_y.

For projective initial data yy, put

Ay,t0(t)=1+(αΞ(y),t0(t))0.A_{y,t_0}(t)=1+(\alpha_{\Xi(y),t_0}(t))_0.

Coordinates of three-vectors are indexed by 0,1,20,1,2; projective coordinates are indexed by m,u,v,wm,u,v,w.

For real k,zk,z, define

Γk(z)=2arctan⁡(sinh⁡z)+2410000kz,C∗=exp⁡(2π+4810000).\Gamma_k(z)=2\arctan(\sinh z)+\frac{24}{10000k}z, \qquad C_* = \exp\left(2\pi+\frac{48}{10000}\right).

Division at zero has its total real-field meaning.

Let Y=(k,u0,w0)Y=(k,u_0,w_0) satisfy k≥6k\geq6 and 0≤u0,w0≤εk0\leq u_0,w_0\leq\varepsilon_k.

There is a least positive return time TT to (Ψ0(e(Y),T))v=1(\Psi_0(e(Y),T))_v=1, with Ψ0(e(Y),t)∈Kk\Psi_0(e(Y),t)\in\mathcal K_k for every t∈[0,T]t\in[0,T]. For every such tt and v∈R3v\in\mathbb R^3,

∥SkWe(Y),0,deYv(t)∥∞≤2C∗(∣v0∣+k∣v1∣+∣v2∣).\|S_k\mathcal W_{e(Y),0,de_Yv}(t)\|_\infty \leq2C_*(|v_0|+k|v_1|+|v_2|).
Preamble
import Definitions.Def_BianchiIX_Analysis_CoreGrowthBudget
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_CoreAngle
import Definitions.Def_BianchiIX_Dynamics_CoreWeightedVariation
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
import Definitions.Def_BianchiIX_Dynamics_CoreFlowVariation
import Definitions.Def_BianchiIX_Dynamics_CoreEntry
import Definitions.Def_BianchiIX_Dynamics_CoreEntryDifferential
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Tactic.Module
import Mathlib.Analysis.Calculus.Deriv.Pow
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.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.Data.Matrix.Mul
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.Analysis.Calculus.ImplicitContDiff
import Mathlib.Data.Real.Basic
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.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.Order.IntermediateValue

noncomputable section
open BianchiIX
open Set
Formal statement
theorem BianchiIX.exists_large_core_entry_variation {Y : CoreAngleState} (hk : 6 ≤ Y 0)
    (hu : 0 ≤ Y 1 ∧ Y 1 ≤ (1 / 10000) / Y 0)
    (hw : 0 ≤ Y 2 ∧ Y 2 ≤ (1 / 10000) / Y 0) :
    ∃ T > 0,
      IsLeast {t | 0 < t ∧ projectiveOriginalFlow (coreRadiusEntry Y) 0 t 2 = 1} T ∧
      (∀ t ∈ Icc 0 T, coreRegion (Y 0) (projectiveOriginalFlow (coreRadiusEntry Y) 0 t)) ∧
      ∀ t ∈ Icc 0 T, ∀ v : CoreAngleState,
        ‖scaleCoreVariation (Y 0) (coreProjectedFlowVariation (coreRadiusEntry Y) 0 t
          (fderiv ℝ coreRadiusEntry Y v))‖ ≤
          (2 * coreVariationBound) * (|v 0| + Y 0 * |v 1| + |v 2|) := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/CoreEntryBounds.lean#L180-L199

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