ExampleConstruction: 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.

Worked example: the clock at rest in an inertial chart

Example

Worked computation. In an inertial chart of the model with $η_{0} = diag (- 1, 1, 1, 1)$ let $c > 0$ and $(a_{1}, a_{2}, a_{3}) \in R^{3}$ . For $t \in R$ consider the coordinate worldline $x (t) = (c t, a_{1}, a_{2}, a_{3})$ of a clock at rest at the spatial position $(a_{1}, a_{2}, a_{3})$ , parametrized by the coordinate time $t$ . Then $x^{'} (t) = (c, 0, 0, 0)$ , so $- η (x^{'}, x^{'}) = c^{2}$ and the proper-time function is $τ (t) - τ (t_{0}) = t - t_{0}$ ; the four-velocity is $U = x^{'} / τ^{'} = (c, 0, 0, 0)$ , with $η (U, U) = - c^{2}$ , and the four-acceleration vanishes, $A = 0$ . Relative to the chart's observer $e_{0}$ the dimensionless four-velocity is $u = U / c = (1, 0, 0, 0) = e_{0}$ , so $γ = - η (e_{0}, u) = 1$ and the relative velocity is $w = c h (u) / γ = 0$ : the clock is at rest in the chart.

Facts & Assumptions

Given: An inertial chart with $η_{0} = diag (- 1, 1, 1, 1)$ , a spatial position $(a_{1}, a_{2}, a_{3}) \in R^{3}$ , a real $c > 0$ , and the worldline $x : R \to A$ with coordinates $x (t) = (c t, a_{1}, a_{2}, a_{3})$ .

[L1]

Inertial charts (Inertial frames and inertial charts of the affine Minkowski model): in an inertial chart the matrix of $η$ is $diag (- 1, 1, 1, 1)$ and the chart's observer is $e_{0}$ .

[L2]

Curves (Curves, regular causal curves and future timelike worldlines in affine Minkowski space): the difference quotients of $x$ are identically $(c, 0, 0, 0)$ , so $x^{'} (t) = (c, 0, 0, 0)$ and $x$ is a $C^{1}$ future timelike worldline (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L3]

Proper time (Proper time along a future timelike worldline): for a constant tangent $x^{'}$ the proper-time increment is $c^{- 1} (t - t_{0}) \sqrt{- η (x^{'}, x^{'})}$ .

[L4]

Four-velocity and four-acceleration (Four-velocity and four-acceleration of a future timelike worldline): $U = x^{'} / τ^{'}$ and $A = U^{'} / τ^{'}$ when $x$ is $C^{2}$ .

[L5]

Decomposition (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition): clause (d) with observer $n$ gives $γ = - η (n, u)$ , relative velocity $w = c h (u) / γ$ and $u = γ (n + w / c)$ ; clause (c) gives $h (u) = u + η (n, u) n$ .

[L6]

Bilinearity (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space): $η (u, u)$ is computed from the diagonal matrix by $η (u, u) = - (u^{0})^{2} + (u^{1})^{2} + (u^{2})^{2} + (u^{3})^{2}$ in the chart.

Verification

technique · direct

1.1givenL1L2L6

The difference quotients of $x$ are $(x (t + h) - x (t)) / h = (c, 0, 0, 0)$ for every $h \neq 0$ , so $x^{'} (t) = (c, 0, 0, 0)$ ; by [L1] and [L6], $- η (x^{'}, x^{'}) = c^{2} > 0$ , so $x$ is a $C^{1}$ future timelike worldline by [L2].

2.1step 1.1L3

By [L3] the proper-time increment is $τ (t) - τ (t_{0}) = c^{- 1} \cdot (t - t_{0}) \cdot \sqrt{c^{2}} = t - t_{0}$ , so the clock's proper time is coordinate time up to an additive constant.

3.1step 1.1step 2.1L4L6

The four-velocity is $U = x^{'} / τ^{'} = (c, 0, 0, 0) / 1 = (c, 0, 0, 0)$ by [L4]; the constant map $U$ has zero derivative, so $A = U^{'} / τ^{'} = 0$ ; and $η (U, U) = - c^{2}$ from the diagonal form [L6], in agreement with clause (a) of [L5]'s theorem.

4.1step 3.1L1L5∎

The dimensionless four-velocity is $u = U / c = (1, 0, 0, 0) = e_{0}$ ; with $n : = e_{0}$ this gives $γ = - η (e_{0}, u) = 1$ and $h (u) = u + η (e_{0}, u) e_{0} = u - e_{0} = 0$ by clause (c) of [L5], so the relative velocity is $w = c h (u) / γ = 0$ : the clock is at rest relative to the chart's observer, and $γ = 1$ matches the general criterion $γ = 1 ⟺ u = n$ .

Remarks

The computation is the model's notion of rest. "At rest in the chart" here means exactly that the relative velocity $w$ of clause (d) vanishes; it is a statement about the ideal observer $e_{0}$ of the chart and the worldline, not about any apparatus.

Worked example: the clock at rest in an inertial chart

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels