CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck pass

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.

Coordinate velocity is not an inertial-frame-independent quantity

Physical scope

A refutation, on explicit data of the model, of the claim that the coordinate velocity of a worldline is independent of the inertial frame used to describe it: two inertial frames of the worked boost example give distinct coordinate velocities for one worldline. The item asserts no force law and no empirical claim; the correct frame-change law it reports is the velocity law of the cited physics theorem.

Statement refuted

In the model of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure, Inertial frames, stationary observers and the adopted Galilean transformations between frames and Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses, the coordinate velocity of a worldline is independent of the inertial frame in which it is computed: if $x$ and $ξ$ are the coordinate curves of the same worldline in two inertial frames with time coordinates $t$ and $t^{'}$ , then

$ξ^{'} (t^{'}) = x^{'} (t) at corresponding times t^{'} = t + b .$

Facts & Assumptions

Given: The two inertial frames and the worldline of A worked Galilean boost: the same uniform motion in two inertial frames: the standard frame $F$ with coordinates $(t, x)$ of $A = R^{4}$ , $τ = x^{0}$ , $K = {0} \times R^{3}$ , the standard metric; the frame $F^{'}$ with parameters $b = 2$ , $a = (0, 3, 0)$ , $v = (\frac{1}{2}, 0, 0)$ , $R = I_{3}$ ; and the worldline with $F$ -coordinate curve $x (s) = (1 + \frac{1}{4} s, 2, 0)$ .

[L1]

The transformation law between the two frames and the velocity law $η^{'} = R w + v$ for the primed coordinate curve $η (t^{'}) = x^{'} (γ (t^{'} - b))$ , together with the definitions of coordinate velocity and acceleration (Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes, clauses 1-2; Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors).

[L2]

The worked verification of the frame data and of the primed coordinate curve $η (t^{'}) = (\frac{3}{4} t^{'} - \frac{1}{2}, 5, 0)$ of the same worldline (A worked Galilean boost: the same uniform motion in two inertial frames, steps 1.3 and 2.1 of its Verification).

Counterexample

technique · direct; compute the two coordinate velocities of one explicit worldline and exhibit their inequality

1.1L2L1algebra

In the frame $F$ the worldline has coordinate velocity $w = x^{'} = (\frac{1}{4}, 0, 0)$ : differentiating $x (s) = (1 + \frac{1}{4} s, 2, 0)$ gives $w = (\frac{1}{4}, 0, 0)$ , independent of $s$ .

2.1step 1.1L2L1algebra

In the frame $F^{'}$ the primed coordinate curve of the same worldline is $η (t^{'}) = (\frac{3}{4} t^{'} - \frac{1}{2}, 5, 0)$ , whose derivative is $η^{'} (t^{'}) = (\frac{3}{4}, 0, 0)$ ; this agrees with the velocity law $η^{'} = R w + v = (\frac{1}{4}, 0, 0) + (\frac{1}{2}, 0, 0) = (\frac{3}{4}, 0, 0)$ of [L1].

3.1step 1.1step 2.1L1algebra∎

The two coordinate velocities $η^{'} = (\frac{3}{4}, 0, 0)$ and $w = (\frac{1}{4}, 0, 0)$ differ: their first components $\frac{3}{4} \neq \frac{1}{4}$ . The refuted statement therefore fails for this worldline and this pair of inertial frames; the correct law is $w^{'} = R w + v$ of clause 2 of Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes, which is what step 2.1 verified, and the coordinate acceleration $a = 0$ is preserved, in accordance with clause 3 of the same theorem.

Remarks

The refuted statement is the frame-independence of the coordinate velocity. It is distinct from the frame-independence of zero coordinate acceleration proved in Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes: the boost adds the same constant $v$ to the velocity at every time and therefore could not change whether the acceleration vanishes, but it changes the velocity itself.
The counterexample is kinematic and conditional on the frame data displayed; it makes no claim about the velocity of any physical body and uses no force law. A particle at rest in one inertial frame has the boost velocity as its coordinate velocity in the other, which is the content of the displayed inequality in the limiting case $w = 0$ .

Coordinate velocity is not an inertial-frame-independent quantity

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources