PostulateStatement: AI-adaptedProof: Not applicablePipeline-generatedprecheck pass

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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.

Selected finite bath and effective energy-exchange preparation

Physical scope

A selected finite Gibbs-volume Kepler subsystem with negative capacity, a finite logarithmic-entropy bath with positive constant capacity, conserved total energy, a matched-temperature point, and the separately adopted scalar exchange ODE, with the upstream slow-motion bound retained for any Newtonian physical-applicability claim; not causal relativistic transport, generic stellar stability, empirical thermalization, or an observed reservoir.

Postulate

Use the finite relative Kepler subsystem and Gibbs-volume entropy convention of Confined relative Kepler shell and volume entropy preparation, with $R > 16 r_{0}$ . In that model, choose $E_{G} = - e_{*}$ with $A / R < e_{*} < A / (16 r_{0})$ , strictly inside the energy branch, and restrict to an open interval $I_{G}$ about $x_{*} = E_{G}$ whose energies stay in that interior. The supplied formula $Ω (- e) = C e^{- 3 / 2} J (e r_{0} / A)$ has $J (δ) = \int_{δ}^{1} s^{1 / 2} (1 - s)^{3 / 2} d s$ and $J^{'} (δ) = - δ^{1 / 2} (1 - δ)^{3 / 2}$ ; because $δ$ stays in $(0, 1 / 16)$ , this derivative is smooth there, so $J$ , $Ω$ , and $S_{G} = k_{B} \log Ω$ are $C^{2}$ on $I_{G}$ . The mathematical result Negative Gibbs-volume capacity in a finite regulated shell gives $T_{G} (x_{*}) = T_{*} > 0$ and finite $C_{G} (x_{*}) < 0$ on this branch; restrict $I_{G}$ if necessary so these strict signs persist. For a physical Newtonian slow-motion interpretation, also retain the upstream condition $2 A / (μ r_{0} c^{2}) \leq δ_{N R} ≪ 1$ for a stated tolerance $δ_{N R}$ ; the scalar exchange calculation itself uses only the explicitly adopted model.

As a separate finite bath, choose $C_{b} > 0$ in $J K^{- 1}$ , reference energy $E_{0}$ in joules, reference temperature $T_{r e f} > 0$ in kelvins, and entropy constant $S_{0}$ in $J K^{- 1}$ . For bath energy $E_{b} > E_{0}$ , adopt $S_{b} (E_{b}) = C_{b} \log ⁣ (\frac{E_{b} - E_{0}}{C_{b} T_{r e f}}) + S_{0} .$ The logarithm argument is dimensionless, and this stipulated bath has $T_{b} = (d S_{b} / d E_{b})^{- 1} = (E_{b} - E_{0}) / C_{b}$ and constant heat capacity $C_{b}$ . Fix total energy $E_{t o t} = x + E_{b}$ , choose $E_{b} = E_{0} + C_{b} T_{*}$ to match the selected subsystem temperature, and restrict $0 < C_{b} < ∣ C_{G} (x_{*}) ∣$ .

Finally adopt the effective scalar energy-exchange law $\dot{x} = L \frac{d}{d x} [S_{G} (x) + S_{b} (E_{t o t} - x)], L > 0 J K s^{- 1} .$ The law is a selected one-dimensional model with time measured in seconds; it does not assert a relativistic transport equation or a mechanism for thermalization. The literature citations give gravitational-thermodynamic context only; the explicit bath and ODE are adopted here, and their mathematical consequences are proved conditionally by the linked local supplier.

Selected finite bath and effective energy-exchange preparation

Postulate

Depends on

Dependency tree · 2 levels

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