ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Flat zero-field data and Minkowski development

Example

Take $M = R^{1, 3}$ with coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , Minkowski metric $g = - (d x^{0})^{2} + (d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2}$ , the standard orientation and future direction, and the hypersurface $Σ = {x^{0} = 0}$ . Set $F = 0$ and $Λ = 0$ , with the fixed positive parameters $κ, μ_{0}$ of Electrovacuum tensor Cauchy system. The induced data are $h = δ$ , $K = 0$ , and $e = b = 0$ . This is a smooth exact solution and its data satisfy all four electrovacuum constraints.

Facts & Assumptions

Given: The source-free tensor system and initial-data conventions of Electrovacuum tensor Cauchy system and Constrained electrovacuum initial data.

[F1]

Electrovacuum tensor Cauchy system: the equations are $G + Λ g = κ μ_{0}^{- 1} Q (g, F)$ , $d F = 0$ , and $d (⋆_{g} F) = 0$ .

[F2]

Constrained electrovacuum initial data: the induced data are tested by the two Maxwell divergences, the Hamiltonian constraint, and the momentum constraint with $K = + \frac{1}{2} L_{n} h$ .

Verification

Given: $g = η$ , $F = 0$ , $Λ = 0$ , and $κ, μ_{0} > 0$ .

1.1F1algebra

In the displayed global coordinates the metric coefficients are constant, so every Christoffel symbol and every curvature component is zero; hence $G_{a b} = 0$ .

2.1F1step 1.1algebra

Since $F = 0$ , one has $d F = 0$ , $d (⋆_{g} F) = 0$ , and $Q_{a b} = T_{a b}^{E M} = 0$ . Thus $G + Λ g = κ T^{E M}$ holds identically, proving that $(g, F)$ is an exact electrovacuum solution.

3.1F2step 1.1algebra∎

The slice has future unit normal $n = \partial_{0}$ , induced metric $h = δ$ , second fundamental form $K = 0$ , and observer fields $e = b = 0$ . Its scalar curvature and all four constraint expressions vanish, so the induced data satisfy Constrained electrovacuum initial data.

Flat zero-field data and Minkowski development

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels