Prove2Me

KN frame Gram entry (frameP, 33)

Proved
ExactVacuumAtlas.EinsteinMaxwell.KN.frP_3_3

by He Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity

KN frame Gram entry (frameP, 33)

Entry of Eᵀ gKN E for the Kerr frameP at (3, 3) (Carter-type frame; used for the Lorentzian signature).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 KN 标架 Gram 矩阵分量（用于 Lorentz 号差）。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Domain
import Definitions.Def_Atlas_Kerr_Frame
import Definitions.Def_Atlas_Kerr_InAtoms
import Definitions.Def_KerrBL_CoordGeometry
import Definitions.Def_KerrBL_Kerr_Metric
open Real
open ExactVacuumAtlas
open scoped ContDiff
open ExactVacuumAtlas.Kerr.In (Ui Wi)
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.KN
set_option maxHeartbeats 8000000
set_option maxRecDepth 100000
set_option linter.unusedVariables false
set_option linter.unusedSimpArgs false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false
variable (M a Q : ℝ)
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.KN.frP_3_3 (x : Pt 4) (hS : x 1 ^ 2 + a ^ 2 * cos (x 2) ^ 2 ≠ 0) (hD : x 1 ^ 2 - 2 * M * x 1 + a ^ 2 + Q ^ 2 ≠ 0) (hs : sin (x 2) ≠ 0) :
    ((Kerr.frameP a x).transpose * metricMatrix (gKN M a Q) x * Kerr.frameP a x) 3 3 = (x 1 ^ 2 + a ^ 2 * cos (x 2) ^ 2) * sin (x 2) ^ 2 := by sorry
Source
He Wang, Exact Spacetime Atlas IV: Certified Einstein–Maxwell Backgrounds and First Variations. Local source, no public repository; commit 543904bfcfa3bb169f43fc4d9dbd569999b71886; Atlas/EinsteinMaxwell/KN/Parts.lean#L356-L369; SHA-256 27df3365e1b568e37410fd5ea573366b73f33888fe24ccbe6f7bc357788ceccc. Registered definitions and literature: mission/PHYSICS_BOUNDARY.md (§H–§J); Müller–Grave arXiv:0904.4184; Newman et al., J. Math. Phys. 6 (1965) 918.
Human review

  • Endorsed by Shuze Chen · Oct 21, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 21, 2026

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

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