Prove2Me

Isotropic Schwarzschild (Cartesian form): vacuum certificate for every M ≥ 0 on the natural domain ρ > 0, ρ ≠ M/2

Proved
ExactVacuumAtlas.Release02.isotropic_cartesian

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real M ≥ 0: VacuumCertificate (gIsoCart M) (giIsoCart M) (domIsoCart M), domIsoCart M = {y : 0 < ρ(y) ∧ ρ(y) ≠ M/2}. gIsoCart M in (t, x, y, z): g_tt = −A(ρ)², g_xx = g_yy = g_zz = B(ρ)⁴, rest 0, ρ = √(x² + y² + z²); giIsoCart the reciprocal diagonal. No polar-axis or azimuthal-cut exclusion. VacuumCertificate g ĝ U (bundle Atlas_Core_Certificate, release I) is the conjunction of eight obligations: U open; U nonempty; g symmetric on U; ĝ(x)·g(x) = I on U; every component of g C^∞ on U; every component of ĝ C^∞ on U; at every x ∈ U an invertible E with Eᵀ g(x) E = diag(−1,1,1,1) (Lorentzian); and all 16 components Ric(g,ĝ)_{bd}(x) = 0 for x ∈ U.

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_Isotropic_CartChart
import Definitions.Def_Atlas_Isotropic_CartCover
import Definitions.Def_Atlas_Isotropic_CartMaps
import Definitions.Def_Atlas_Isotropic_Defs
import Definitions.Def_Atlas_Minkowski_Spherical
import Definitions.Def_Atlas_Schwarzschild_Defs
import Definitions.Def_Atlas_Schwarzschild_Domain
import Definitions.Def_Atlas_Schwarzschild_PGCertParts
import Definitions.Def_Atlas_Schwarzschild_PGDefs
import Definitions.Def_Atlas_Schwarzschild_PGRelation
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_Curvature
import Definitions.Def_Atlas_Transport_MatrixCalc
open Real Matrix
open ExactVacuumAtlas ExactVacuumAtlas.Schwarzschild ExactVacuumAtlas.Controls
Formal statement
theorem ExactVacuumAtlas.Release02.isotropic_cartesian : ∀ M : ℝ, 0 ≤ M →
    VacuumCertificate (Iso.gIsoCart M) (Iso.giIsoCart M) (Iso.domIsoCart M) := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Release02/TransportMilestone.lean#L96-L99; SHA-256 e066de778216dc552a43e41e7967661796a78d9fd40c374cb701b56fdf9c42ae. 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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