Prove2Me

Fréchet derivative of a map with genuine slice derivatives

Proved
ExactVacuumAtlas.fderiv_eq_mulVec

by He Wang · Oct 15, 2026 · Mathlib 0df444a (Lean v4.33.1)

exact-vacuum-atlasgeneral-relativitylean4mathlib

Dimension 4. If U is open, Φ is C^∞ on U, x ∈ U and the slices of Φ have genuine derivatives J x k i at x, then fderiv ℝ Φ x v = (J x) ·ᵥ v for every vector v.

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Certificate
import Definitions.Def_Atlas_Core_Relation
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_Curvature
import Definitions.Def_Atlas_Transport_MatrixCalc
open scoped ContDiff
open Matrix Filter Topology
Formal statement
theorem ExactVacuumAtlas.fderiv_eq_mulVec {U : Set (Pt 4)} (hU : IsOpen U) {Φ : Pt 4 → Pt 4}
    (hΦ : ContDiffOn ℝ ∞ Φ U) {J : Pt 4 → Matrix (Fin 4) (Fin 4) ℝ} {x : Pt 4} (hx : x ∈ U)
    (hJ : ∀ k i : Fin 4, HasDerivAt (fun u : ℝ => Φ (Function.update x i u) k) (J x k i) (x i))
    (v : Pt 4) : fderiv ℝ Φ x v = J x *ᵥ v := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Transport/Inverse.lean#L19-L40; SHA-256 42307986732fed628bcabd660036bfa0c301394389ffa58632d43a90d35f262c. Mathematical background: See mission/PHYSICS_BOUNDARY.md §A-F for registered definitions and literature references (Wald, Müller-Grave arXiv:0904.4184, Visser arXiv:0706.0622, Martel-Poisson gr-qc/0001069).
Human review

  • Endorsed by Shuze Chen · Oct 18, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 18, 2026

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

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