Prove2Me

Isotropic Schwarzschild (spherical form): vacuum certificate for every M ≥ 0 on ρ > 0, ρ ≠ M/2, 0 < θ < π

Proved
ExactVacuumAtlas.Release02.isotropic_main

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real M ≥ 0: VacuumCertificate (gIso M) (giIso M) (domIso M), domIso M = {ρ > 0, ρ ≠ M/2, 0 < θ < π} (coordinates (t, ρ, θ, φ) = (x 0, x 1, x 2, x 3)). Isotropic data (bundle Atlas_Isotropic_Defs): B(ρ) = 1 + M/(2ρ), A(ρ) = (1 − M/(2ρ))/B(ρ); gIso M in (t, ρ, θ, φ): g_tt = −A², g_ρρ = B⁴, g_θθ = B⁴ρ², g_φφ = B⁴ρ² sin²θ, rest 0; giIso = the reciprocal diagonal; r(ρ) = ρ(1 + M/(2ρ))² = (2ρ + M)²/(4ρ); ρ₊(r) = (r − M + √(r(r − 2M)))/2, ρ₋(r) = (r − M − √(r(r − 2M)))/2; toStd(t,ρ,θ,φ) = (t, r(ρ), θ, φ); jacIso = diag(1, 1 − M²/(4ρ²), 1, 1). 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_main : ∀ M : ℝ, 0 ≤ M → VacuumCertificate (Iso.gIso M) (Iso.giIso M) (Iso.domIso 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#L86-L88; 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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