Prove2Me

Pull-back identity Jᵀ h(Φ x) J = gCtl λ (x)

Proved
ExactVacuumAtlas.Controls.Shear.pullback

by He Wang · Oct 15, 2026 · Mathlib 0df444a (Lean v4.33.1)

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real λ and x ∈ domEF: (jacShear λ x)ᵀ · gCyl 0 (shear λ x) · jacShear λ x = gCtl λ (x).

Preamble
import Mathlib
import Definitions.Def_Atlas_Controls_CylinderTables
import Definitions.Def_Atlas_Controls_Shear
import Definitions.Def_Atlas_Controls_ShearTables
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Certificate
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_Core_Relation
import Definitions.Def_Atlas_Schwarzschild_Defs
import Definitions.Def_Atlas_Schwarzschild_Domain
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_Curvature
import Definitions.Def_Atlas_Transport_MatrixCalc
open Real
open scoped ContDiff
open Matrix
open ExactVacuumAtlas ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas
open ExactVacuumAtlas.Controls
open ExactVacuumAtlas.Controls.Shear
set_option maxHeartbeats 4000000
variable (lam : ℝ)
Formal statement
theorem ExactVacuumAtlas.Controls.Shear.pullback {x : Pt 4} (hx : x ∈ domEF) :
    (jacShear lam x)ᵀ * metricMatrix (Cylinder.gCyl 0) (shear lam x) * jacShear lam x
      = metricMatrix (gCtl lam) x := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Controls/Shear.lean#L103-L109; SHA-256 219b8cea95eaf75ce7002541a89f6bcd6bdaeacb3ba69ac402eaacd84c242ddc. Mathematical background: See mission/PHYSICS_BOUNDARY.md §A-F for registered definitions and literature references (Wald, Müller-Grave arXiv:0904.4184, Visser arXiv:0706.0622, Martel-Poisson gr-qc/0001069).
Human review

  • Endorsed by Shuze Chen · Oct 18, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 18, 2026

    Confirmed by the mission captain (proposal self-audit).

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