Prove2Me

pdchr 2 2 3 3

Proved
ExactVacuumAtlas.Controls.Cylinder.pdchr_2_2_3_3

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real MM (an unused parameter) and x=(t,r,θ,φ)x=(t,r,\theta,\varphi) with r>0r>0 and 0<θ<π0<\theta<\pi, the cylinder metric g=diag⁡(−1,1,1,sin⁡2θ)g=\operatorname{diag}(-1,1,1,\sin^2\theta) and its stated inverse satisfy

∂θΓ332=−(cos⁡2θ−sin⁡2θ).\partial_{\theta}\Gamma^{2}_{33} = -(\cos^2\theta-\sin^2\theta).

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

Preamble
import Mathlib
import Definitions.Def_Atlas_Controls_CylinderTables
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.Cylinder
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 (M : ℝ)
Formal statement
theorem ExactVacuumAtlas.Controls.Cylinder.pdchr_2_2_3_3 (x : Pt 4) (hx : x ∈ domEF)  :
    pd 2 (christoffel (gCyl M) (giCyl M) 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/CylinderTables.lean#L1230-L1235; SHA-256 6853c22124c397d8886c39820e0325b7348d66f0d0a916d41678b132205ebfee. 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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