Prove2Me

Equal-spin ansatz: explicit inverse metric

Proved
ExactVacuumAtlas.EinsteinMaxwell.ES5.esInverse_mul

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

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

Equal-spin ansatz: explicit inverse metric

For all radial functions B D A H W and x : Pt 5 with B(x 1), D(x 1), A(x 1), H(x 1) ≠ 0 and x 2 ≠ 0, 1 − x 2 ≠ 0: metricMatrix (esInverse B D A H W) x * metricMatrix (esMetric B D A H W) x = 1.

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 在 B,D,A,H 非零且 u ≠ 0,1 的点上，esInverse 是 esMetric 的逆矩阵。

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
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
set_option maxHeartbeats 8000000
variable (B D A H W : ℝ → ℝ)
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.esInverse_mul (x : Pt 5) (hB : B (x 1) ≠ 0) (hD : D (x 1) ≠ 0) (hA : A (x 1) ≠ 0)
    (hH : H (x 1) ≠ 0) (hu : x 2 ≠ 0) (hv : 1 - x 2 ≠ 0) :
    metricMatrix (esInverse B D A H W) x * metricMatrix (esMetric B D A H W) x = 1 := 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/Defs.lean#L108-L115; SHA-256 bb9c71fdcb4c409e9bfd1766a9ca9bb3118fe200aee50524ee3c961baaa00f90. 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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