CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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A degenerate BF triad defines no Lorentz metric

Physical scope

Pure mathematics; an explicit classical configuration, with no empirical or quantum claim.

Statement refuted

Every smooth solution of the zero-cosmological-constant three-dimensional BF equations has an invertible triad and determines a nondegenerate Lorentz metric satisfying the vacuum Einstein equation.

Statement references

Chern–Simons and BF action data

Invertible triads identify zero-cosmological-constant three-dimensional BF gravity

Facts & Assumptions

Given: ${A C}_{ω}$ , a nonempty oriented three-manifold $M$ with a global trivialization, and the zero-cosmological-constant BF field equations.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): countable choice, as required by the conditional gravity proposition used for comparison.

[F1]

Chern–Simons and BF action data: the BF curvature is $F_{ω} = d ω + \frac{1}{2} [ω, ω]$ and the covariant derivative of the BF field is $D_{ω} e$ .

[F2]

Invertible triads identify zero-cosmological-constant three-dimensional BF gravity: the BF equations are $F_{ω} = 0$ and $D_{ω} e = 0$ , and only the invertible-coframe sector defines a nondegenerate Lorentz metric equivalent to vacuum gravity.

Counterexample

1.1A1F1F2givenconstruct

Take $M$ to be the closed unit three-ball, choose the trivial $s o (2, 1)$ bundle, and set $e \equiv 0$ and $ω \equiv 0$ . Direct substitution gives $F_{ω} = d 0 + \frac{1}{2} [0, 0] = 0$ and $D_{ω} e = d 0 + [0, 0] = 0$ , so [F2] says this is a BF solution.

2.1F2step 1.1algebra∎

At every $x \in M$ , the map $e_{x} : T_{x} M \to R^{2, 1}$ has rank zero, so it is not invertible. The induced symmetric tensor $g_{e} (X, Y) = η (e X, e Y)$ is identically zero and therefore degenerate; it is not a Lorentz metric. This solution lies outside the metric-gravity sector, refuting the statement.

A degenerate BF triad defines no Lorentz metric

Statement refuted

Statement references

Facts & Assumptions

Counterexample

Depends on

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