Prove2Me

Einstein–Maxwell residual depends only on data on an open set

Proved
ExactVacuumAtlas.EinsteinMaxwell.ES5.einsteinMaxwellD_congr_of_eqOn

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

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

Einstein–Maxwell residual depends only on data on an open set

Componentwise EqOn of g, ginv, F on an open U gives equal einsteinMaxwellD at every x ∈ U (statement as printed).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 开集上数据相同则 Einstein–Maxwell 残差相同。

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_ES5_Defs
open scoped ContDiff
open ExactVacuumAtlas ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.D
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
variable {U : Set (Pt 5)} {f : Pt 5 → ℝ} {F : ℝ → ℝ → ℝ} {x : Pt 5}
variable {g g' ginv ginv' : Metric 5} {Fm Fm' : TwoForm 5}
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.einsteinMaxwellD_congr_of_eqOn (hU : IsOpen U) (hg : ∀ i j, Set.EqOn (g i j) (g' i j) U)
    (hgi : ∀ i j, Set.EqOn (ginv i j) (ginv' i j) U) (hF : ∀ i j, Set.EqOn (Fm i j) (Fm' i j) U)
    (hx : x ∈ U) (a b : Fin 5) :
    einsteinMaxwellD g ginv Fm a b x = einsteinMaxwellD g' ginv' Fm' a b x := 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/ES5/Tools.lean#L69-L78; SHA-256 8eb33cdee8c5956f9a82565c80aa58b049e59e7e5ed43a4f1176098b09966a0a. 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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