Scope and empirical status of the flat and Newtonian bridge page

This note fixes what the physics items of this page do and do not claim; it proves nothing by itself.

• The flat model is a model. The Minkowski vacuum model with Lambda = 0: field law, free-fall kinematics and clock readings shows that the flat geometry with $\mathrm{\Lambda }=0$ and $T=0$ satisfies the adopted field law and what its ideal free-fall and clock readings are. It is one solution among many: clause (d) also supplies explicit non-flat vacuum and flat quotient comparisons with different global topology. Semiclassical and observational questions attached to "empty space" are outside this page. Statements about the flat model are conditional on the adopted postulates of Adopted Einstein field equation in the direct-equation presentation and Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework, and are not claims about nature.

• The bridge is a controlled comparison of models. The weak-field results of this page — the Einstein and stress hierarchy (The weak-field Einstein hierarchy: linearized Einstein tensor, stress components and the Poisson equation), the geodesic acceleration estimate (Coordinate-time acceleration of weak-field geodesics: the Newtonian leading term with uniform error) and the scaled trajectory convergence (Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data) — are estimates inside an explicitly hypothesized family of spatially factored stationary metrics (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition). Their hypotheses are essential: an existing Einstein family with the stated derivative bounds, the slow-motion condition $\mathrm{\mid }\dot{x}\mathrm{\mid }\le C\sqrt{\epsilon }c$, a common compact spatial region, a comparison motion that exists on the whole finite scaled interval and a bounded second derivative of the potential. The Poisson source match to dust rest-mass density additionally assumes the stated nonnegative dust model, its conservation equations, weak-source bound and slow tilt. The page does not construct the input family from Newtonian initial data, does not prove existence of a global Newtonian limit in any gauge, and does not treat strong fields, fast motion, unbounded scaled times, large-time limits at fixed $\epsilon$, or non-smooth sources. Fixed scaled intervals do permit the stated ${\epsilon }^{-1\mathrm{/}2}$ coordinate-time horizons.

• The correspondence dictionary is adopted, not derived. The Newtonian comparison model and the weak-field correspondence identification fixes the spatial region, time, potential and coupling dictionary. Its Newtonian source ${\rho }_N$ obeys its own exact Poisson equation and is not defined to equal either the relativistic coordinate density ${\rho }_{\epsilon }=T_{00}\mathrm{/}{c}^2$ or the independent dust rest-mass density ${\mu }_{\epsilon }$. The conditional theorem The controlled weak-field slow-motion correspondence with the Newtonian comparison model proves that ${\rho }_N$ and ${\rho }_{\epsilon }$ each match ${\mu }_{\epsilon }$ within explicit errors under the dust hypotheses. No derivation of either framework from the other is claimed, and the identification of the geometric and potential variables is not an identity of theories.

• The displayed spatial form is chart-dependent. The perturbation form $g_{00}=-(1+2\mathrm{\Phi }\mathrm{/}{c}^2)+r_{00}$, $g_{ij}=(1-2\mathrm{\Phi }\mathrm{/}{c}^2){\delta }_{ij}+r_{ij}$ is a statement in the chart of the ansatz. Under a change of chart the components transform by the covariant transformation law (Chart independence and the transformation law of a smooth covariant 2-tensor field), so comparisons of spatial coefficients computed in different charts — for instance the areal-radius forms used in the spherical model — require the explicit coordinate conversion and are not identities; that conversion is deferred to the spherical-model page. Similarly, the gauge transformation of the flat linearization is a change of perturbation inside a fixed chart and is not a change of chart.

• Kinematics versus dynamics. The low-velocity expansions Low-velocity expansions of observer energy and momentum with uniform error bounds and the observer energy and momentum definitions (Observer-relative energy, spatial momentum and rest mass of a tangent vector) are kinematic: they expand the model's energy and momentum readings. The page does not prove conservation of any total energy. The Newtonian source density, coordinate density ${\rho }_{\epsilon }$ and rest-mass density ${\mu }_{\epsilon }$ are distinct objects; the weak-slow theorem gives only the stated conditional error bounds between them.

• No measured input and no observations. Nothing on the page uses a measured value of $c$, $G_N$ or $\mathrm{\Lambda }$; the unit conventions SI unit and coordinate conventions of the classical gravitational framework fix dimensions and normalizations only. A comparison with reported Newtonian, solar-system, clock, redshift or gravitational-wave data would require, in addition: the measured value of $G_N$ with its uncertainty and calibration, the apparatus and sampling model, an identified observable computed from the model, and a statistical account distinguishing prediction, observation and inference. Agreement of such a comparison would be evidence about the combined model and apparatus assumptions, not a proof of the postulates. None of that material is supplied on this page; the empirical interface is developed, if at all, on the track's later experimental pages with source-backed primary reports.