ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Softened Newtonian particles do not specify the physical collision force

Example

For two Newtonian point masses in $R^{3}$ with positive constants absorbed into $c > 0$ , the unsoftened relative acceleration is $F (r) = - c r / ∣ r ∣^{3}$ for $r \neq 0$ , where $∣ \cdot ∣$ is Euclidean. A softened model uses $F_{ε} (r) = - \frac{c r}{(∣ r ∣^{2} + ε^{2})^{3 / 2}}, ε > 0.$ The softened field is smooth at a collision, but it defines a different dynamical model.

Facts & Assumptions

Given: $c > 0$ , $ε > 0$ , and the two displayed force laws.

[F1]

The norm is Euclidean (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Verification

Given: The unsoftened and softened force formulas in the Example.

1.1givenF1algebra

For $r \neq 0$ , their scalar denominators are $∣ r ∣^{3}$ and $(∣ r ∣^{2} + ε^{2})^{3 / 2}$ , which differ because $ε > 0$ . Thus $F_{ε} (r) \neq F (r)$ for every nonzero $r$ .

2.1givenstep 1.1algebra∎

At $r = 0$ , the unsoftened formula is undefined, whereas the softened formula gives $F_{ε} (0) = 0$ . The denominator $∣ r ∣^{2} + ε^{2}$ is smooth and strictly positive, and $u \mapsto u^{- 3 / 2}$ is smooth for $u > 0$ ; hence the softened field is $C^{\infty}$ on all of $R^{3}$ . Its smooth extension therefore supplies a convention for its own regularized model, not a collision law for the singular Newtonian system.

Depends on

The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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