Prove2Me

C^∞ of the adjugate

Proved
ExactVacuumAtlas.Variation.contDiffOn_adjugate_of

by He Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity

C^∞ of the adjugate

Entrywise C^∞ of a matrix field implies entrywise C^∞ of its adjugate (any normed real domain).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 伴随矩阵的 C^∞。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Transport_MatrixCalc
open Filter Topology
open scoped ContDiff
open ExactVacuumAtlas
variable {n : ℕ}
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
Formal statement
theorem ExactVacuumAtlas.Variation.contDiffOn_adjugate_of {s : Set E} {A : E → Matrix (Fin n) (Fin n) ℝ}
    (hA : ∀ a b, ContDiffOn ℝ ∞ (fun p => A p a b) s) (a b : Fin n) :
    ContDiffOn ℝ ∞ (fun p => (A p).adjugate a b) s := by sorry
Source
He Wang, Exact Spacetime Atlas IV: Certified Einstein–Maxwell Backgrounds and First Variations. Local source, no public repository; commit 543904bfcfa3bb169f43fc4d9dbd569999b71886; Atlas/EinsteinMaxwell/Variation/Joint.lean#L159-L169; SHA-256 4b63c47f27fd80bedab505cf38c5c13014839c88f3159852417904171ca49a55. Registered definitions and literature: mission/PHYSICS_BOUNDARY.md (§H–§J); Müller–Grave arXiv:0904.4184; Newman et al., J. Math. Phys. 6 (1965) 918.
Human review

  • Endorsed by Shuze Chen · Oct 21, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 21, 2026

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

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