Prove2Me

The same linearised operator is not identically zero: an RN direction with LE_θθ ≠ 0

Proved
ExactVacuumAtlas.Variation.RN.nonzero_theta_ne

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

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

The same linearised operator is not identically zero: an RN direction with LE_θθ ≠ 0

For all real M, Q and x ∈ domRN: LE_θθ(gRN M (Q²), giRN M (Q²), faradayRN Q; h = tanB, f = 0)(x) ≠ 0, where tanB has the single component h_vv = −1/r² (the b-direction of the RN ansatz at b = Q²).

Scope. This direction is the tangent of the non-solution family b = Q² + ε (cert_iff); it is not a linearised solution. A vanishing first variation elsewhere is never claimed to give a new exact solution.

中文摘要 非零检验：同一 L 在 RN 背景、方向 h_vv=−1/r²、f=0 上 LE_θθ=1/r²≠0。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Certificate
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_RN_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_RN_Frame
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_RN
import Definitions.Def_Atlas_Transport_MatrixCalc
open Real
open scoped ContDiff
open ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.RN
Formal statement
theorem ExactVacuumAtlas.Variation.RN.nonzero_theta_ne (M Q : ℝ) {x : Pt 4} (hx : x ∈ domRN) :
    linEinstein (gRN M (Q ^ 2)) (giRN M (Q ^ 2)) (faradayRN Q) tanB 0 2 2 x ≠ 0 := 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/Variation/RN.lean#L172-L175; SHA-256 7513cd6c91a5e550bb62e1cac2f2c8a8c17fa469087ba3c27bf9c1eab933f66f. 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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