Prove2Me

A quartic stable-phase remainder through second order

Proved
BianchiIX.exists_stableQuadraticRemainder_bounds

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.

The projective vector field in normalized time is

G(y)=6D(y)F(y).G(y)=\frac6{D(y)}F(y).

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.

On the invariant plane with zero active wall, use y=(m,u,w)∈R3y=(m,u,w)\in\mathbb R^3, the inclusion ι(m,u,w)=(m,u,0,w)\iota(m,u,w)=(m,u,0,w) and the restriction πs(m,u,v,w)=(m,u,w)\pi_s(m,u,v,w)=(m,u,w). Define

f(y)=(mu2−(m+1)uw+w2,1−m2−w2+u22u,−(m+1+uw−u2)w),f(y)=\left(mu^2-(m+1)uw+w^2, \frac{1-m^2-w^2+u^2}{2}u,-(m+1+uw-u^2)w\right), Δ(y)=3+m2+(u−w)2,fo(y)=6f(y)/Δ(y).\Delta(y)=3+m^2+(u-w)^2,\qquad f_o(y)=6f(y)/\Delta(y).

Write J=dfJ=df, H=d2fH=d^2f, Jo=dfoJ_o=df_o, Ho=d2foH_o=d^2f_o for their explicit Jacobian and Hessian expressions. The denominator derivatives are dΔ=(2m,2(u−w),2(w−u))d\Delta=(2m,2(u-w),2(w-u)) and d2Δ(v,z)=2v0z0+2(v1−v2)(z1−z2)d^2\Delta(v,z)=2v_0z_0+2(v_1-v_2)(z_1-z_2). The exact original-time boundary flow and its projected variation are

Φ(y,t)=πsΨ0(ιy,t),L(y,t)=πsJιy,0(t)ι.\Phi(y,t)=\pi_s\Psi_0(\iota y,t),\qquad L(y,t)=\pi_s\mathcal J_{\iota y,0}(t)\iota.

For a fixed scale KK, set ΦK(y,s)=Φ(y,Ks)\Phi_K(y,s)=\Phi(y,Ks), LK(y,s)=L(y,Ks)L_K(y,s)=L(y,Ks) and QK(y,s)=dyLK(y,s)Q_K(y,s)=d_yL_K(y,s). For directions v,zv,z, define

SK(y,s;v,z)=K[Ho(ΦK)(LKv,LKz)+Jo(ΦK)QK(v,z)].S_K(y,s;v,z)=K\left[H_o(\Phi_K)(L_Kv,L_Kz)+J_o(\Phi_K)Q_K(v,z)\right].

All vector norms are sup norms, and operator norms are induced by them. Write b(y)=∣u∣+∣w∣b(y)=|u|+|w|, E(y)=e512b(y)E(y)=e^{512b(y)}, vs=(v1,v2)v_s=(v_1,v_2), and

H(y;v,z)=b(y)∥v∥∥z∥+∥vs∥∥z∥+∥v∥∥zs∥,\mathcal H(y;v,z)=b(y)\|v\|\|z\|+\|v_s\|\|z\|+\|v\|\|z_s\|, B2(y)=192b(y)(40000E(y)3)+10000E(y)2.B_2(y)=192b(y)(40000E(y)^3)+10000E(y)^2.

The closed boundary region is Sc={(m,u,0,w):∣m−c∣≤1/10, ∣u∣,∣w∣≤1/100}\mathcal S_c=\{(m,u,0,w):|m-c|\leq1/10,\ |u|,|w|\leq1/100\}. The boundary projection P(m,u,w)P(m,u,w) is the limit of Φ(m,u,w;t)0\Phi(m,u,w;t)_0 as t→∞t\to\infty, using the total limit convention of the formal definitions.

Set r=(m+1)2+w2r=\sqrt{(m+1)^2+w^2} and

A(m)=mm2−1,C(m)=−2m+1,Θ(m,u,w)=r−1+A(m)u2+C(m)uw.A(m)=\frac{m}{m^2-1},\quad C(m)=-\frac2{m+1},\quad \Theta(m,u,w)=r-1+A(m)u^2+C(m)uw.

Its explicit gradient is used together with the residual R=r4u4+r3u3w+r2u2w2+r1uw3R=r_4u^4+r_3u^3w+r_2u^2w^2+r_1uw^3, where

r4=−2m(m2−1)2,r3=2(3m−1)(m2−1)2,r_4=-\frac{2m}{(m^2-1)^2},\quad r_3=\frac{2(3m-1)}{(m^2-1)^2}, r2=−m2+1(m2−1)2−mm2−1+r+1r(r+m+1),r1=2(m+1)2+1m+1−1r+m+1.r_2=-\frac{m^2+1}{(m^2-1)^2}-\frac m{m^2-1} +\frac{r+1}{r(r+m+1)},\quad r_1=\frac2{(m+1)^2}+\frac1{m+1}-\frac1{r+m+1}.

Write L=m+1L=m+1, b=∣u∣+∣w∣b=|u|+|w|, vs=(v1,v2)v_s=(v_1,v_2), and set

Wb(v)=b∥v∥+∥vs∥,Hs(y;v,z)=∣u∣∥v∥∥z∥+∣v0∣∣z1∣+∣v1∣∣z0∣+b(∣v1∣∥z∥+∥v∥∣z1∣).W_b(v)=b\|v\|+\|v_s\|,\quad H_s(y;v,z)=|u|\|v\|\|z\|+|v_0||z_1|+|v_1||z_0| +b(|v_1|\|z\|+\|v\||z_1|).

The corrected and quadratic remainders are

Rc=P−Θ,Rq=P−(m+A(m)u2+C(m)uw+w22L).R_c=P-\Theta,\quad R_q=P-\left(m+A(m)u^2+C(m)uw+\frac{w^2}{2L}\right).

For K∈RK\in\mathbb R, let ai(K,y)=6Kr4−i(m,w)/Δ(y)a_i(K,y)=6K r_{4-i}(m,w)/\Delta(y), where r4,r3r_4,r_3 are independent of ww, and put

M(y)=(u3,u2w,uw2,w3),CK(y)=∑i=03ai(K,y)Mi(y),M(y)=(u^3,u^2w,uw^2,w^3),\quad C_K(y)=\sum_{i=0}^3a_i(K,y)M_i(y), gK,s(y)=6KΔ(ΦK(y,s))R(ΦK(y,s)).g_{K,s}(y)=\frac{6K}{\Delta(\Phi_K(y,s))}R(\Phi_K(y,s)).

The cubic derivative budgets are

B1(y;v)=b3∥v∥+3b2∥vs∥,B_1(y;v)=b^3\|v\|+3b^2\|v_s\|, B2c(y;v,z)=b3∥v∥∥z∥+3b2(∥v∥∥zs∥+∥vs∥∥z∥)+6b∥vs∥∥zs∥.B_2^c(y;v,z)=b^3\|v\|\|z\| +3b^2(\|v\|\|z_s\|+\|v_s\|\|z\|)+6b\|v_s\|\|z_s\|.

A direction is radial if its last two coordinates vanish, and transverse if its zeroth coordinate vanishes. There is C>0C>0 such that, for m≥2m\geq2 and ∣u∣,∣w∣<1/200|u|,|w|<1/200, RqR_q is C2C^2 and ∣Rq∣≤Cb4/L3|R_q|\leq Cb^4/L^3. Its radial and transverse first derivatives are bounded by Cb4/L3Cb^4/L^3 and Cb3/L3Cb^3/L^3 times the direction norm. Its radial-radial, radial-transverse and transverse-transverse second derivatives are bounded by Cb4/L3Cb^4/L^3, Cb3/L3Cb^3/L^3, and Cb2/L3Cb^2/L^3, respectively, times the product of direction norms.

Preamble
import Definitions.Def_BianchiIX_Algebra_Flow
import Definitions.Def_BianchiIX_Dynamics_Normalized
import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_StableProjectionRefinementData
import Definitions.Def_BianchiIX_Algebra_Coordinates
import Definitions.Def_BianchiIX_Dynamics_WallEvolution
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_CompactPhaseExpansionData
import Definitions.Def_BianchiIX_Algebra_MatchingPhase
import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_ProjectiveClock
import Definitions.Def_BianchiIX_Dynamics_ProjectiveOriginalTime
import Definitions.Def_BianchiIX_Dynamics_ProjectiveFlowVariation
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.Deriv.Inv
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.SpecialFunctions.Arsinh
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.Arcosh
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Tactic.GCongr
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Tactic.NormNum
import Mathlib.Topology.Order.Compact
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.Calculus.ParametricIntervalIntegral
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

noncomputable section
open BianchiIX
Formal statement
theorem BianchiIX.exists_stableQuadraticRemainder_bounds :
    ∃ C > (0 : ℝ), ∀ y : StableState, 2 ≤ y 0 → |y 1| < 1 / 200 → |y 2| < 1 / 200 →
      let b := |y 1| + |y 2|
      let L := y 0 + 1
      ContDiffAt ℝ 2 stableQuadraticRemainder y ∧
      |stableQuadraticRemainder y| ≤ C / L ^ 3 * b ^ 4 ∧
      (∀ v, v 1 = 0 → v 2 = 0 → |fderiv ℝ stableQuadraticRemainder y v| ≤ C / L ^ 3 * b ^ 4 * ‖v‖) ∧
      (∀ v, v 0 = 0 → |fderiv ℝ stableQuadraticRemainder y v| ≤ C / L ^ 3 * b ^ 3 * ‖v‖) ∧
      (∀ v r, v 1 = 0 → v 2 = 0 → r 1 = 0 → r 2 = 0 →
        |fderiv ℝ (fderiv ℝ stableQuadraticRemainder) y v r| ≤ C / L ^ 3 * b ^ 4 * ‖v‖ * ‖r‖) ∧
      (∀ v r, v 1 = 0 → v 2 = 0 → r 0 = 0 →
        |fderiv ℝ (fderiv ℝ stableQuadraticRemainder) y v r| ≤ C / L ^ 3 * b ^ 3 * ‖v‖ * ‖r‖) ∧
      (∀ v r, v 0 = 0 → r 0 = 0 →
        |fderiv ℝ (fderiv ℝ stableQuadraticRemainder) y v r| ≤ C / L ^ 3 * b ^ 2 * ‖v‖ * ‖r‖) := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Dynamics/StableQuadraticRemainderBounds.lean#L11-L115

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