Prove2Me

pdchr 2 2 3 3

Proved
ExactVacuumAtlas.Controls.Shear.pdchr_2_2_3_3

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real λ\lambda and x=(T,R,α,β)x=(T,R,\alpha,\beta) with R>0R>0 and 0<α<π0<\alpha<\pi, use the sheared control metric gCtl,λg_{\mathrm{Ctl},\lambda} and its stated inverse. Then

∂αΓ332=−(cos⁡2α−sin⁡2α).\partial_{\alpha}\Gamma^{2}_{33} = -(\cos^2\alpha-\sin^2\alpha).

This is the derivative of the indicated Christoffel component, not a claim of zero curvature. The coordinate index 2 denotes alpha.

Preamble
import Mathlib
import Definitions.Def_Atlas_Controls_ShearTables
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_Schwarzschild_Defs
import Definitions.Def_Atlas_Schwarzschild_Domain
open Real
open ExactVacuumAtlas ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas
open ExactVacuumAtlas.Controls
open ExactVacuumAtlas.Controls.Shear
set_option maxHeartbeats 4000000
set_option linter.unusedVariables false
set_option linter.unusedSimpArgs false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false
variable (lam : ℝ)
Formal statement
theorem ExactVacuumAtlas.Controls.Shear.pdchr_2_2_3_3 (x : Pt 4) (hx : x ∈ domEF)  :
    pd 2 (christoffel (gCtl lam) (giCtl lam) 2 3 3) x = (-1) * (cos (x 2) ^ 2 - sin (x 2) ^ 2) := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Controls/ShearTables.lean#L1273-L1278; SHA-256 782cc8c1c151bbdb9e599ba53ea76ee6c2e4203bbca394d8de7a4b95ac66b98e. 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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