Equal-spin inverse components are C^∞
ProvedExactVacuumAtlas.EinsteinMaxwell.ES5.esInverse_smootheinstein-maxwellexact-spacetime-atlas-viexact-vacuum-atlasfive-dimensional-gravitygeneral-relativity
Equal-spin inverse components are C^∞
If R is open, B, D, A, H, W are C^∞ on R and B, D, A, H ≠ 0 there, every esInverse component is C^∞ on esDomain R (statement as printed).
Scope. Exactly the hypotheses printed in the formal statement.
中文摘要 esInverse 各分量在 esDomain R 上 C^∞。
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_smooth {R : Set ℝ} (hB : ContDiffOn ℝ ∞ B R) (hD : ContDiffOn ℝ ∞ D R)
(hA : ContDiffOn ℝ ∞ A R) (hH : ContDiffOn ℝ ∞ H R) (hW : ContDiffOn ℝ ∞ W R)
(hB0 : ∀ r ∈ R, B r ≠ 0) (hD0 : ∀ r ∈ R, D r ≠ 0) (hA0 : ∀ r ∈ R, A r ≠ 0)
(hH0 : ∀ r ∈ R, H r ≠ 0) (i j : Fin 5) :
ContDiffOn ℝ ∞ (esInverse B D A H W i j) (esDomain R) := by sorrySource
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#L209-L235; 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
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.