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Vacuum condition within the EF family: Ricci-flat certificate iff β = 0

Proved
ExactVacuumAtlas.EinsteinMaxwell.EF5.vacuumCertificate_iff

by He Wang · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-viexact-vacuum-atlasfive-dimensional-gravitygeneral-relativity

Vacuum condition within the EF family: Ricci-flat certificate iff β = 0

For all real μ, β: VacuumCertificate (gEF5 μ β) (giEF5 μ β) domU5 ⇔ β = 0, where VacuumCertificate (Atlas I) asks for an open nonempty domain, symmetric C^∞ metric with C^∞ inverse, Lorentz signature and ricciOf = 0.

Scope. All real μ, β (no field strength appears); domain domU5 = {x | 0 < x 1 ∧ 0 < x 2 ∧ x 2 < 1}.

中文摘要 对所有实数 μ, β：EF 拟设在 domU5 上 Ricci 平坦证书 VacuumCertificate ⇔ β = 0。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_D_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_EF5_CertificateEF5
import Definitions.Def_Atlas_EinsteinMaxwell_EF5_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_EF5_EinsteinMaxwellEF5
import Definitions.Def_Atlas_EinsteinMaxwell_EF5_Frame
import Definitions.Def_Atlas_EinsteinMaxwell_EF5_RicciOffDiag
open scoped ContDiff
open ExactVacuumAtlas ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.D
open ExactVacuumAtlas.EinsteinMaxwell.EF5
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.EF5
set_option maxHeartbeats 4000000
set_option linter.unusedVariables false
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.EF5.vacuumCertificate_iff (mu beta : ℝ) :
    VacuumCertificate (gEF5 mu beta) (giEF5 mu beta) domU5 ↔ beta = 0 := by sorry
Source
He Wang, Exact Spacetime Atlas VI: Five-Dimensional Einstein–Maxwell Certificates and Equal-Spin Radial Reduction. Local source, no public repository; commit 4a4048a; Atlas/EinsteinMaxwell/EF5/CertificateEF5.lean#L123-L145; SHA-256 cfead95b85a714aef90d980c00d0cf62ef54959a25cf5716e7e611261a95d8d3. Registered definitions: mission/PHYSICS_BOUNDARY.md (§N–§O); literature: R. C. Myers, M. J. Perry, Ann. Phys. 172 (1986) 304; F. R. Tangherlini, Nuovo Cimento 27 (1963) 636.
Human review

  • Endorsed by Shuze Chen · Oct 29, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 29, 2026

    Confirmed by the mission captain (proposal self-audit).

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