CounterexampleConstruction: AI-generatedVerification: AI-generated

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.

External forces break the two-body reduction: a two-label model with a static central pair potential and external forces $\pm γ$ has $μ r^{''} = γ \neq 0$ , although its total external force vanishes

Physical scope

A refutation of the unconditional two-body reduction: an explicit two-label force model with the external-plus-pair presentation, a (zero) static central pair potential and antisymmetric constant external forces has an explicit motion whose relative coordinate is not a central-force trajectory, even though the total external force and the centre-of-mass balance are unaffected. The item is a finite check inside the adopted force branch; it asserts no observation and shows that the no-external-force restriction cannot be dropped without replacement, even if the total external force vanishes.

Statement refuted

False statement. Let a two-label model of the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance declare the external-plus-pair presentation of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch with a static central pair potential $u \in C^{2} (J_{0})$ , with no restriction on the external forces. Then the relative coordinate $r = x_{1} - x_{2}$ of every motion is a central-force trajectory for $(μ, u)$ with $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , that is $μ r^{''} = - u^{'} (∣ r ∣) r / ∣ r ∣$ .

The hypothesis of Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection is the stronger statement that the force map be presented through the potential $U (x) = u (∣ x_{1} - x_{2} ∣)$ alone, which excludes external forces; the witness below shows that this exclusion is not decorative. In particular the vanishing of the total external force, which already guarantees the centre-of-mass balance of Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces, is not sufficient for the reduction.

Facts & Assumptions

Given: The model and motion declared in the Counterexample: $m_{0} : = 1 k g$ and $m_{1} = m_{2} = m_{0}$ , time window $R$ , configuration domain $Q = {x \in (R^{3})^{2} : ∣ x_{1} - x_{2} ∣ > 0}$ , pair potential $u \equiv 0$ on $J_{0} = (0, \infty)$ , acceleration scale $a_{0} : = 1 m s^{- 2}$ , speed scale $v_{0} : = 1 m s^{- 1}$ , offset $d_{0} : = 2 m$ , and external forces $F_{1}^{e x t} = γ$ , $F_{2}^{e x t} = - γ$ with $γ : = (1 N) e_{1} = m_{0} a_{0} e_{1}$ ; for an orthonormal basis $(e_{1}, e_{2}, e_{3})$ the curves are $x_{1} (t) = \frac{1}{2} a_{0} t^{2} e_{1}$ and $x_{2} (t) = - \frac{1}{2} a_{0} t^{2} e_{1} + d_{0} e_{1} + v_{0} t e_{2}$ .

[L1]

Model notions (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clauses (i)-(iv), Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clauses (i)-(iv), The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (i), Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): a motion satisfies $m_{i} x_{i}^{''} = F_{i} = F_{i}^{e x t} + \sum_{j \neq i} F_{i j}$ ; a constant force map is continuous and state-Lipschitz with constant $0$ , hence admissible for the force branch; the total external force is $σ = \sum_{i} F_{i}^{e x t}$ ; and $M = m_{1} + m_{2}$ , $μ = m_{1} m_{2} / M$ , $r = x_{1} - x_{2}$ , $∣ r ∣ = ∥ r ∥_{2}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L2]

Curve calculus (Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, $C^{k}$ Euclidean maps and diffeomorphisms, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): sums, differences and scalar multiples of differentiable curves differentiate componentwise, so $r^{''} = x_{1}^{''} - x_{2}^{''}$ for $C^{2}$ curves.

[L3]

The reduction theorem (Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection clauses 1-3, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential): for a model presented through the potential $U (x) = u (∣ x_{1} - x_{2} ∣)$ with $F_{i}^{e x t} \equiv 0$ , the relative coordinate satisfies $μ r^{''} = - u^{'} (∣ r ∣) r / ∣ r ∣$ ; this is the statement whose external-force hypothesis is at issue, and it is used below only to identify what the false statement asserts for the pair potential $u \equiv 0$ , namely $μ r^{''} = 0$ .

Counterexample

technique · direct

1.1givenL1

The model is admissible for the force branch: the pair maps are $F_{12} = F_{21} \equiv 0$ (the negative gradients of $u \equiv 0$ , a $C^{\infty}$ function on $(0, \infty)$ ) and the external maps are the constants $F_{1}^{e x t} = γ$ , $F_{2}^{e x t} = - γ$ ; constant maps on $I_{0} \times Q \times (R^{3})^{2}$ are continuous and state-locally Lipschitz with constant $0$ , so the standing hypotheses of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii) hold, the presentation is the identity $F_{i} = F_{i}^{e x t} + \sum_{j \neq i} F_{i j}$ , and the pair forces are antisymmetric and (trivially) central.

2.1step 1.1L1L2algebra

General relative acceleration: by [L2], any motion of a model with this structure has $r^{''} = x_{1}^{''} - x_{2}^{''} = \frac{F_{1}}{m_{1}} - \frac{F_{2}}{m_{2}} = \frac{F_{1}^{e x t}}{m_{1}} - \frac{F_{2}^{e x t}}{m_{2}} + \frac{F_{12}}{m_{1}} - \frac{F_{21}}{m_{2}}$ ; with $F_{21} = - F_{12}$ and $μ^{- 1} = \frac{1}{m_{1}} + \frac{1}{m_{2}}$ this reads $μ r^{''} = F_{12} + μ (\frac{F_{1}^{e x t}}{m_{1}} - \frac{F_{2}^{e x t}}{m_{2}})$ ; for the model at hand $F_{12} = 0$ , $F_{1}^{e x t} = γ$ , $F_{2}^{e x t} = - γ$ and $m_{1} = m_{2} = m_{0}$ , so $μ r^{''} = μ (2 γ / m_{0}) = γ$ .

3.1step 1.1step 2.1L1L2algebra

The displayed curves are a motion: $x_{1}^{''} = a_{0} e_{1}$ and $x_{2}^{''} = - a_{0} e_{1}$ , so $m_{0} x_{1}^{''} = γ = F_{1}$ and $m_{0} x_{2}^{''} = - γ = F_{2}$ ; the curves are polynomials hence $C^{2}$ . Their relative coordinate is $r = (a_{0} t^{2} - d_{0}) e_{1} - v_{0} t e_{2}$ , and $∣ x_{1} - x_{2} ∣^{2} = (a_{0} t^{2} - d_{0})^{2} + (v_{0} t)^{2} > 0$ for every $t$ : if the second square vanishes then $t = 0$ , when the first square is $d_{0}^{2} > 0$ . Thus $x (t) \in Q$ throughout.

4.1step 2.1step 3.1L1L3algebra∎

Failure of the reduction: for this motion $r^{''} = 2 a_{0} e_{1}$ , so $μ r^{''} = m_{0} a_{0} e_{1} = γ \neq 0$ , while the false statement asserts for the pair potential $u \equiv 0$ that $μ r^{''} = - u^{'} (∣ r ∣) r / ∣ r ∣ = 0$ ; the two disagree, so $r$ is not a central-force trajectory for $(μ, 0)$ . Moreover the total external force is $σ = F_{1}^{e x t} + F_{2}^{e x t} = γ - γ = 0$ , and the centre of mass $X = \frac{1}{2} (x_{1} + x_{2}) = \frac{d_{0}}{2} e_{1} + \frac{v_{0} t}{2} e_{2}$ is affine, consistent with the balance of Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces; hence a vanishing total external force alone does not imply the relative central-force equation. Thus the no-external-force restriction cannot be dropped without an appropriate replacement; the witness does not prove that each external force must vanish in every motion satisfying the relative equation.

Remarks

What the witness isolates. The failure is produced by the antisymmetric external pair $\pm γ$ , which cancels in the total momentum but not in the relative equation; by contrast a common external acceleration $F_{i}^{e x t} = m_{i} a$ cancels in $μ r^{''}$ because $a - a = 0$ , so by step 2.1 the relative central-force equation holds exactly when the evaluated external accelerations $F_{1}^{e x t} / m_{1}$ and $F_{2}^{e x t} / m_{2}$ agree along the motion. Nonzero common external acceleration preserves that relative equation but gives $X^{''} = a$ , so it need not preserve the affine centre-of-mass conclusion of the full reduction theorem.
Scope. The computation is a finite check inside the adopted force branch with declared data; it reports no observation and identifies only the unproved strengthening, not any defect of Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection, whose sufficient hypotheses the witness does not satisfy.

External forces break the two-body reduction: a two-label model with a static central pair potential and external forces $\pm γ$ has $μ r^{''} = γ \neq 0$ , although its total external force vanishes

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources

External forces break the two-body reduction: a two-label model with a static central pair potential and external forces ±γ±γ has μr′′=γ≠0μr′′=γ=0, although its total external force vanishes

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources

External forces break the two-body reduction: a two-label model with a static central pair potential and external forces $\pm γ$ has $μ r^{''} = γ \neq 0$ , although its total external force vanishes