CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

Counterexample: the proper-time integrand of a spacelike curve is not real

Statement refuted

False claim: the formula $τ (b) - τ (a) = c^{- 1} \int_{a}^{b} \sqrt{- η (x^{'} (s), x^{'} (s))} d s$ assigns an elapsed clock time to every regular $C^{1}$ curve of the model.

The claim is refuted by a straight spacelike line: its tangent is spacelike at every point, so $- η (x^{'}, x^{'})$ is negative and the square root is not a real number. A spacelike interval is not elapsed clock time; the geometric spacing along the curve is its spacelike length, a different quantity.

Facts & Assumptions

Given: An event $p \in A$ , a spacelike vector $d$ and the straight curve $x : R \to A$ , $x (s) : = p + s d$ in the oriented time-oriented affine Minkowski model.

[L1]

Causal classification (Timelike, null, spacelike and causal vectors of a Lorentz vector space): $v$ is spacelike when $η (v, v) > 0$ and timelike when $η (v, v) < 0$ .

[L2]

Inertial charts (Inertial frames and inertial charts of the affine Minkowski model): in an inertial chart the matrix of $η$ is $diag (- 1, 1, 1, 1)$ , so for $v = (v^{0}, v^{1}, v^{2}, v^{3})$ one has $η (v, v) = - (v^{0})^{2} + (v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2}$ .

[L3]

Curves (Curves, regular causal curves and future timelike worldlines in affine Minkowski space): the curve $x (s) = p + s d$ has constant tangent $x^{'} (s) = d$ and is a future timelike worldline only when $d$ is future timelike.

[L4]

Proper time (Proper time along a future timelike worldline): the proper-time integrand is $\sqrt{- η (x^{'} (s), x^{'} (s))}$ and the construction is restricted to future timelike worldlines; for a spacelike tangent the quantity $- η (x^{'}, x^{'})$ is negative.

Counterexample

technique · direct

1.1L1L2

In an inertial chart with $η_{0} = diag (- 1, 1, 1, 1)$ let $d : = (1, 2, 0, 0)$ : then $η (d, d) = - 1 + 4 = 3 > 0$ , so $d$ is spacelike by [L1].

2.1step 1.1L3L4

The curve $x (s) : = p + s d$ has constant tangent $x^{'} (s) = d$ by [L3], so $- η (x^{'} (s), x^{'} (s)) = - 3 < 0$ at every point; the integrand $\sqrt{- η (x^{'}, x^{'})}$ of [L4] is the square root of a negative real number and is not defined. The proper-time construction therefore produces no real value on this curve.

3.1step 1.1step 2.1L4∎

The geometric spacing along the curve is governed by the spacelike length per unit parameter, $\sqrt{η (d, d)} = \sqrt{3}$ , not by a clock reading: this curve is not a timelike worldline, has no causal-character tangent that is timelike, and the failure is exactly the hypothesis of the timelike restriction in [L4].

Remarks

Spacelike length and proper time are different constructions. The number $\sqrt{η (d, d)}$ measures a spatial separation in the model; $c$ times it is not an elapsed time along the curve, and no clock is associated with a spacelike displacement by this framework.

Counterexample: the proper-time integrand of a spacelike curve is not real

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels