RemarkRemark: AI-adaptedProof: Not applicable

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.

Scope and status of the spherical vacuum model page

Physical scope

Scope and status note for the spherical vacuum model page: it records that the page's items are conditional results of the adopted smooth classical model, that no observation or measured value is asserted on the page, that the empirical interpretation of any redshift, orbit or horizon statement requires separate apparatus and analysis models, and that maximality, global mass, collapse and non-vacuum interior topics are deferred. It asserts no physical fact about nature and reports no measurement.

What the page defines and proves

The model. The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation fixes the datum: the manifold $R \times (s, \infty) \times S^{2}$ , the global smooth metric $g = - f d (x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ with diagonal components $- f, f^{- 1}, r^{2}, r^{2} \sin^{2} ϑ$ in spherical-polar product charts, the time orientation $\partial_{0}$ and the parameters $c, G_{N}, m_{*} > 0$ with $s = 2 G_{N} m_{*} / c^{2}$ . The components carry the unit readings of SI unit and coordinate conventions of the classical gravitational framework; no numerical value of any constant is asserted.
The vacuum calculation. Curvature components and the vanishing of the Ricci and Einstein tensors of the Schwarzschild exterior metric computes the Christoffel symbols, the curvature, the vanishing Ricci and Einstein tensors and the nonvanishing curvature of the datum, and The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter reads the result as a model of the framework's field law with $T = 0$ and $Λ = 0$ , together with the model's static clock readings and the reading of the mass parameter.
Local uniqueness. Local Birkhoff theorem: a vacuum warped product with a round sphere factor is locally Schwarzschild with an extra Killing field shows that a vacuum warped product with spherical orbits is locally of this form with an extra local Killing field, under the explicit hypotheses of that theorem; the conclusion is local and does not assert that every vacuum solution or every spherical vacuum region is an exterior region with positive mass.
Restricted predictions. Restricted redshift prediction of the Schwarzschild vacuum model: the observer-energy ratio between static observers and Restricted geodesic predictions of the Schwarzschild vacuum model: circular free-fall orbits, the stability threshold and the photon sphere are conditional model predictions for ideal clocks, test bodies and light rays, using the adopted coupling rules of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework; they are not measurements and involve no apparatus model. The horizon and singularity distinctions are recorded in Horizon versus singularity in the Schwarzschild model: a chart breakdown, a smooth null surface and a curvature obstruction.

What is not asserted

No observation and no empirical comparison. The page contains no experiment item, no reported observation, no calibration or uncertainty record and no statistical inference. A comparison of the redshift ratio, an orbital frequency or a horizon statement with data would test the model together with an apparatus, preparation and analysis model and would be evidence about the combination, not a proof of the model (Adopted Einstein field equation in the direct-equation presentation and the postulates fix only the model side).
No numerical values. Neither $G_{N}$ , $c$ , $m_{*}$ nor any derived number is evaluated; the statements are identities in the parameters.
Deferred topics. Maximality and global structure of the Kruskal model, causal-boundary and maximal-development theorems, boundary-asymptotic mass, rotating and charged models, collapse and singularity theorems, and the empirical interface are separate modules of the track; nothing on this page supplies or presupposes them. Where an item uses the extension or Kruskal models it says so explicitly, and where it uses the weak-field reading of the mass parameter it carries the comparison warning of the isotropic coordinate lemma.

Scope and status of the spherical vacuum model page

What the page defines and proves

What is not asserted

Depends on

Dependency tree · 2 levels

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