Prove2Me

Union of two certified domains for the same metric data

Proved
ExactVacuumAtlas.VacuumCertificate.union

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

Total dimension m + 1. If (g, ĝ) carries a vacuum certificate on U₁ and on U₂, then it carries one on U₁ ∪ U₂ (openness, nonemptiness and the pointwise obligations combine; smoothness on a union of open sets).

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Certificate
import Definitions.Def_Atlas_Core_Relation
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_Curvature
import Definitions.Def_Atlas_Transport_MatrixCalc
open scoped ContDiff
open Matrix Filter Topology
Formal statement
theorem ExactVacuumAtlas.VacuumCertificate.union {m : ℕ} {g gi : Metric (m + 1)} {U₁ U₂ : Set (Pt (m + 1))}
    (h₁ : VacuumCertificate g gi U₁) (h₂ : VacuumCertificate g gi U₂) :
    VacuumCertificate g gi (U₁ ∪ U₂) := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Transport/Inverse.lean#L121-L133; SHA-256 42307986732fed628bcabd660036bfa0c301394389ffa58632d43a90d35f262c. 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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