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ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)
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.

Two SPH particles receive equal and opposite pressure forces

Example

For N=2 in Fixed-width regularized conservative SPH particle system, write F_12 for the contribution of particle 2 to m_1v̇_1 and F_21 for the contribution of particle 1 to m_2v̇_2. These pair contributions are equal and opposite and act along the particle separation line.

Facts & Assumptions

Given: Two positions q_1,q_2 in R^d with positive masses and positive SPH densities.

[F1]

Fixed-width regularized conservative SPH particle system: F_12=−m_1m_2(A_1+A_2)∇W_ε(q_1−q_2).

[F2]

SPH pressure forces are equal-opposite and central: F_21=−F_12 and F_12 is parallel to q_1−q_2.

Verification

1.1F1given

The two-particle acceleration equations contain precisely F_12 in the first momentum equation and F_21 in the second, by [F1].

2.1F2step 1.1∎

Applying [F2] gives F_12+F_21=0 and makes each force a scalar multiple of q_1−q_2; hence the pair exerts no net force or torque.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources