Prove2Me

giPG is a genuine left inverse of gPG on domEF

Proved
ExactVacuumAtlas.Schwarzschild.PG.giPG_mul_gPG

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For all real M ≥ 0, ε with ε² = 1, and x ∈ domEF: giPG M ε (x) · gPG M ε (x) = I (uses q² = 2M/r and ε² = 1 in the (T, r) block).

Preamble
import Mathlib
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_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
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.giPG_mul_gPG (hM : 0 ≤ M) (hε : ε ^ 2 = 1) {x : Pt 4} (hx : x ∈ domEF) :
    metricMatrix (giPG M ε) x * metricMatrix (gPG M ε) x = 1 := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Schwarzschild/PGTransport.lean#L28-L43; SHA-256 21d7802bc903aa00fee3b30ad8bd8c392d214d0c016091aad5299c0d904b982f. 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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