Prove2Me

Weighted transverse matrix and its decreasing budget

Definition
BianchiIX_Analysis_ConnectionTransverseCoefficients

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

bianchi-ix-era-statisticsdynamical-systems

For the positive reflected phase variable mm, put

n(a,b)=a2+b2,j(a,b)=1n(a,b)(n(a,b)+a),n(a,b)=\sqrt{a^2+b^2},\quad j(a,b)=\frac1{n(a,b)(n(a,b)+a)}, ϕ=n(m+1,v)+n(m−1,w)−m,h=m+1n(m+1,v)+m−1n(m−1,w)−1,\phi=n(m+1,v)+n(m-1,w)-m,\quad h=\frac{m+1}{n(m+1,v)}+\frac{m-1}{n(m-1,w)}-1, Q=mh+v2n(m+1,v)+w2n(m−1,w),k=j(m+1,v)−j(m−1,w).Q=mh+\frac{v^2}{n(m+1,v)}+\frac{w^2}{n(m-1,w)},\quad k=j(m+1,v)-j(m-1,w).

Write κ=u(v+w−u)\kappa=u(v+w-u), s=m−1−κs=m-1-\kappa, b=(1−m2−(v−w)2+u2)/2b=(1-m^2-(v-w)^2+u^2)/2, and

Mv=−v2n(m+1,v)h,Mw=−w2n(m−1,w)h.M_v=-\frac{v^2}{n(m+1,v)h},\qquad M_w=-\frac{w^2}{n(m-1,w)h}.

The four coordinate differential operators, in the order (ϕ,log⁡u,log⁡v,log⁡w)(\phi,\log u,\log v,\log w), are

L0=h−1∂m,L1=u∂u,L2=Mv∂m+v∂v,L3=Mw∂m+w∂w.L_0=h^{-1}\partial_m,\quad L_1=u\partial_u,\quad L_2=M_v\partial_m+v\partial_v,\quad L_3=M_w\partial_m+w\partial_w.

Subscripts ii below denote the rational expressions obtained by these rules, with total real-field division. On the regular positive-wall region they are the corresponding coordinate partial derivatives. Set

N=h(v−w)+(v+w−u)Q,R=h(v−w)+(v+w−u)(Q−s),N=h(v-w)+(v+w-u)Q,\quad R=h(v-w)+(v+w-u)(Q-s), A=−N/s,B=k/s,G=uA+(vw)2B,α=b/s,β=−1−2(1+κ)/s.A=-N/s,\quad B=k/s,\quad G=uA+(vw)^2B,\quad \alpha=b/s,\quad \beta=-1-2(1+\kappa)/s.

The coefficient matrix CC has rows (Gi)i(G_i)_i, (αi)i(\alpha_i)_i and (βi)i(\beta_i)_i.

Define

Mμ=(α1α2/μ2μ2β1β2),rμ=64(2μu+v2/μ3+vw/μ3),M_\mu=\begin{pmatrix}\alpha_1&\alpha_2/\mu^2\\ \mu^2\beta_1&\beta_2\end{pmatrix},\quad r_\mu=64(2\mu u+v^2/\mu^3+vw/\mu^3), Bμ(u,v,w)=64(4u+v2/(2μ3)+vw/(2μ2)).B_\mu(u,v,w)=64(4u+v^2/(2\mu^3)+vw/(2\mu^2)).
Definition code
import Definitions.Def_BianchiIX_Analysis_ConnectionAmplitudeDifferential
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.Calculus.ImplicitContDiff
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Tactic.FunProp

noncomputable section

namespace BianchiIX

/-- The homogeneous equation for (X, mu^2 Y). -/
def connectionTransverseMatrix (μ m u v w : ℝ) : Matrix (Fin 2) (Fin 2) ℝ :=
  !![connectionStrongClockPartials m u v w 1, connectionStrongClockPartials m u v w 2 / μ ^ 2;
     μ ^ 2 * connectionWeakClockPartials m u v w 1, connectionWeakClockPartials m u v w 2]

def connectionTransverseRate (μ u v w : ℝ) : ℝ :=
  64 * (2 * μ * u + v ^ 2 / μ ^ 3 + v * w / μ ^ 3)

/-- A decreasing state function pays for the entire transverse growth. -/
def connectionTransverseBudget (μ u v w : ℝ) : ℝ :=
  64 * (4 * u + v ^ 2 / (2 * μ ^ 3) + v * w / (2 * μ ^ 2))











end BianchiIX
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Analysis/ConnectionTransverseCoefficients.lean

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