Primitive, adopted and derived status of the objects of the framework page

This remark records the status of the notions introduced on the framework-and-geometry page. It is bookkeeping: it asserts no physical law and no empirical claim, and every mathematical statement it repeats is proved in the item cited.

Primitive mathematical structure of a model. The triple $(M,g,N)$ of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: the smooth boundaryless four-manifold $M$ with its smooth structure, the Lorentzian metric $g$ of signature $(-,+,+,+)$, and the time orientation $N$, a smooth nowhere-zero timelike vector field. These are the primitive geometric data; the definition names no equation that $g$ must satisfy.

Adopted physical assumptions on this page. The reading of $(M,g,N)$ as the spacetime of a classical continuum model, with the stated smooth-branch and four-dimensional restrictions, is the postulate Adopted spacetime model and classical domain of the gravitational framework; the statement that a model's nongravitational matter content, its equations and its stress-energy tensor are model input, together with the optional adoption of energy conditions, is the postulate Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions. The unit and coordinate conventions of SI unit and coordinate conventions of the classical gravitational framework are part of the formulation; they fix dimensions and contain no measured value. The stress-energy slot Stress-energy tensor field of a model and the energy-condition inequalities The pointwise energy conditions WEC, NEC, DEC and SEC are formulation-level definitions available to a model. When a model adopts an energy condition independently, it is an extra assumption. When the condition is proved from that model's equations and hypotheses, it is a derived result under those hypotheses, as allowed by Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions clause (iv); using that result requires its proof and prerequisites, not a separate adoption of the condition. The general framework imposes no energy condition automatically.

Derived (constructed, not adopted) on this page. The inverse metric, the musical isomorphisms ${\mathrm{♭}}_p,{\mathrm{♯}}_p$, the raised and lowered component conventions and the metric trace (The inverse metric, musical isomorphisms and index raising), with the smoothness and transformation law of the inverse-metric components (Smoothness and transformation of the inverse-metric components); the pointwise causal classification of tangent vectors; the future/past classification of nonzero causal vectors by $N$, the identification of the future-directed causal vectors with $\mathrm{cl}(C_p^{+})\setminus \{0\}$, and the fact that every nonzero causal vector has nonzero pairing with the specified time-orientation vector $N_p$ (Classification of future and past causal vectors by a timelike field); the agreement or opposition of two time orientations on connected sets (Two time orientations agree or are opposite on each connected set); the component form of torsion freedom and metric compatibility (Torsion freedom and metric compatibility in chart components). These are mathematical constructions and theorems from the primitive structure; they carry no empirical content.

Early constructions and richer continuation. The metric-selected class of Metric compatibility, torsion freedom and the metric-selected connection class is defined by its two conditions and its unique member is constructed by the complete local nondegenerate Koszul proof; curvature and its Ricci and scalar contractions are then defined from that connection. The smooth-case proper-time and four-velocity constructions of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model use a supplied smooth future timelike curve and mathematical parameter $c>0$, before their physical unit reading. The later pages retain their fuller worldline regularity classes, geodesic and tensor-derivative constructions, and curvature symmetries. No motion law or clock coupling follows from these mathematical constructions.

Explicitly not claimed on this page. No field law, no matter branch or matter equation, no conservation identity for $T$, no field-equation consistency condition, no clock, test-body or light-ray coupling, no empirical statement. Those are adopted or proved on later pages of the framework, each with its own hypotheses.

Time-orientation caveat. Existence of the smooth timelike field $N$ is adopted as part of the model data. It is an additional global condition: this page does not prove that every smooth Lorentzian four-manifold carries a smooth nowhere-zero timelike vector field, and the framework restricts to models for which one is specified. Where a physical or mathematical development needs time orientability, it states that hypothesis explicitly.