Prove2Me

Derivative of the primitive H_M

Proved
ExactVacuumAtlas.Schwarzschild.PG.hasDerivAt_HPG

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For real M ≥ 0 and r > 0: u ↦ H_M(u) = u − 2√(2M)√u + 4M log(√u + √(2M)) has derivative 1/(1 + q(M, r)) at r (genuine derivative; r = 2M included).

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_Core_Relation
import Definitions.Def_Atlas_Schwarzschild_Defs
import Definitions.Def_Atlas_Schwarzschild_Domain
import Definitions.Def_Atlas_Schwarzschild_PGDefs
import Definitions.Def_Atlas_Schwarzschild_PGRelation
open Real
open scoped ContDiff
open ExactVacuumAtlas ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas
open ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas.Schwarzschild.PG
set_option maxHeartbeats 4000000
variable (M ε : ℝ)
Formal statement
theorem ExactVacuumAtlas.Schwarzschild.PG.hasDerivAt_HPG {M r : ℝ} (hM : 0 ≤ M) (hr : 0 < r) :
    HasDerivAt (fun u => HPG M u) (1 / (1 + qf M r)) r := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Schwarzschild/PGRelation.lean#L67-L95; SHA-256 15a42669f14c3556d812be0667228c735019b13b898fa9b0552c54b35498feb7. 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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