Physics theoremStatement: AI-adaptedProof: 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.

Finite bath stabilizes the selected negative-capacity exchange model

Physical scope

Local stability and exponential attraction only for the adopted scalar finite-bath exchange ODE at a matched-temperature point with C_G<0 and 0<C_b<|C_G|, retaining the upstream slow-motion condition for any Newtonian physical-applicability claim; no generic GR-fluid stability, causal transport law, empirical thermalization, or observation.

Statement

Assume the finite-bath preparation Selected finite bath and effective energy-exchange preparation, which supplies $C^{2}$ subsystem and bath entropies on open energy intervals, with subsystem temperature $T_{*} > 0$ , finite capacity $C_{G} < 0$ , bath capacity $C_{b}$ satisfying $0 < C_{b} < ∣ C_{G} ∣$ , conserved total energy, and exchange law $\dot{x} = L S_{t o t}^{'} (x)$ for $L > 0$ . At the matched-temperature point $x_{*}$ , $S_{t o t}^{''} (x_{*}) = - \frac{1}{T_{*}^{2}} (\frac{1}{C_{G}} + \frac{1}{C_{b}}) < 0.$ Consequently $x_{*}$ is a strict local entropy maximum in this energy-exchange direction, the adopted ODE has a forward-invariant local basin with exponential attraction to $x_{*}$ , and along each solution $d S_{t o t} / d t = L (S_{t o t}^{'})^{2} \geq 0$ . These are properties of the selected scalar exchange model only.

Facts & Assumptions

Given: The selected Kepler negative-capacity point, the finite bath entropy and matched temperature, $0 < C_{b} < ∣ C_{G} ∣$ , conserved total energy, and the stipulated scalar exchange ODE.

[F1]

Selected finite bath and effective energy-exchange preparation supplies a subsystem entropy $S_{G} \in C^{2}$ on an open interval with $T_{*} > 0$ and finite $C_{G} < 0$ , the $C^{2}$ bath entropy with constant $C_{b} > 0$ , the matched-temperature point, and $\dot{x} = L S_{t o t}^{'} (x)$ .

[F2]

Finite-bath entropy curvature and specified exchange stability: for $C_{1} < 0 < C_{2}$ , a common-temperature critical point has $S_{t o t}^{''} = - T^{- 2} (1 / C_{1} + 1 / C_{2})$ and is a strict local maximum when $C_{2} < ∣ C_{1} ∣$ ; for the separately adopted $C^{1}$ exchange vector field $L S_{t o t}^{'}$ , the local solution is forward invariant and exponentially attracting, with nondecreasing entropy.

[A1]

The exchange law in [F1] is an adopted effective model assumption, not an empirical transport law.

Proof

1.1givenA1F1F2

The selected point in [F1] has $S_{G}^{'} = 1 / T_{*}$ and the bath has $S_{b}^{'} = 1 / T_{*}$ ; hence the derivative of $S_{t o t} (x) = S_{G} (x) + S_{b} (E_{t o t} - x)$ vanishes at $x_{*}$ . By [F1], both entropies are $C^{2}$ on open energy intervals, and the two finite nonzero heat capacities meet the common-temperature hypotheses of [F2].

2.1F1F2step 1.1algebra

Substituting $C_{1} = C_{G} < 0$ and $C_{2} = C_{b}$ into [F2] gives the stated curvature. Since $C_{b} < ∣ C_{G} ∣$ , $1 / C_{b} > 1 / ∣ C_{G} ∣ = - 1 / C_{G}$ , so the bracket is positive and $S_{t o t}^{''} (x_{*}) < 0$ .

3.1A1F1F2step 2.1∎

The strict curvature and the exchange law in [F1] satisfy the local stability hypotheses of [F2]; therefore $x_{*}$ is a strict local maximum, the local basin is forward invariant and exponentially attracting, and the entropy production is $L (S_{t o t}^{'})^{2} \geq 0$ . This is not a stability statement for a relativistic star or a general fluid evolution.

Depends on

Dependency tree · 2 levels

Sources

P.-H. Chavanis, Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas, arXiv:1908.10806v1 (2019), Sec. III.E