ricci 2 2

Proved

ExactVacuumAtlas.Controls.Shear.ricci_2_2

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real $\lambda$ and $x=(T,R,\alpha,\beta)$ with $R>0$ and $0<\alpha<\pi$ , the coordinate Ricci tensor of $g_{\mathrm{Ctl},\lambda}$ with its stated inverse satisfies

R_{22} = 1+\lambda^2\cos^2\alpha\sin^2\alpha.

This component is strictly positive throughout this domain.

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.ricci_2_2 (x : Pt 4) (hx : x ∈ domEF)   :
    ricciOf (gCtl lam) (giCtl lam) 2 2 x = 1 + lam ^ 2 * (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#L1600-L1605; 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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