DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

Finite softened gravitational Hamiltonian and domain

Definition

Use the phase space and dimensionless reference measure of Finite gravitational phase and entropy reference. Fix an integer $N \geq 1$ , a bounded Lebesgue measurable $B \subseteq R^{3}$ with positive finite volume, a common mass $m > 0$ , a gravitational coupling $G > 0$ , and a softening length $a > 0$ . On $Γ_{N} = B^{N} \times R^{3 N}$ define $H_{N, a} (q, p) = \sum_{i = 1}^{N} \frac{∣ p_{i} ∣^{2}}{2 m} + U_{a} (q), U_{a} (q) = - \sum_{1 \leq i < j \leq N} \frac{G m^{2}}{\sqrt{∣ q_{i} - q_{j} ∣^{2} + a^{2}}} .$ The positions and momenta are mathematical phase variables; the softening is a fixed regulator parameter. The Hamiltonian is measurable and its potential is bounded because each pair denominator is at least $a$ . A point-particle model is a different Hamiltonian and is not included at $a = 0$ .

Depends on

Finite gravitational phase and entropy reference

Finite softened gravitational Hamiltonian and domain

Definition

Depends on

Dependency tree · 1 level

Sources