Prove2Me

PG → EF coordinate relation on all of domEF

Proved
ExactVacuumAtlas.Schwarzschild.PG.relation_toEF

by He Wang · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

exact-vacuum-atlasgeneral-relativitylean4mathlib

For all real M ≥ 0 and ε with ε² = 1: CoordRelation (gPG M ε) (gEF M ε) domEF domEF (pgToEF M ε) (pgFromEF M ε) (jac M ε): pgToEF(T,r,θ,φ) = (T + ε H_M(r), r, θ, φ) and pgFromEF are mutually inverse, pgToEF maps domEF bijectively onto domEF, both are C^∞ on domEF, jac (radial mixed entry ε/(1+q), det 1) is the genuine Jacobian, and jacᵀ gEF(pgToEF x) jac = gPG x on domEF.

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
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_Schwarzschild_PGDefs
import Definitions.Def_Atlas_Schwarzschild_PGRelation
open Real
open scoped ContDiff
open ExactVacuumAtlas ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas
open ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas.Schwarzschild.PG
set_option maxHeartbeats 4000000
variable (M ε : ℝ)
Formal statement
theorem ExactVacuumAtlas.Schwarzschild.PG.relation_toEF (hM : 0 ≤ M) (hε : ε ^ 2 = 1) :
    CoordRelation (gPG M ε) (gEF M ε) domEF domEF (pgToEF M ε) (pgFromEF M ε) (jac M ε) := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Schwarzschild/PGRelation.lean#L229-L240; SHA-256 15a42669f14c3556d812be0667228c735019b13b898fa9b0552c54b35498feb7. 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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