Primitive and derived notions of the spacetime-primitives page

This remark records the primitive and derived status of the notions introduced on the spacetime-primitives page, as required before the page's derived claims. It is bookkeeping: it asserts no physical law and no empirical claim.

Primitive mathematical structure of the model. The quadruple $(A,V,\eta ,C^{+})$ of Oriented time-oriented affine Minkowski space (the spacetime primitives): the affine space $A$ with its action of $V$ and the displacement $q-p$, the Lorentz form $\eta$ of signature $(-,+,+,+)$, the orientation of $V$ fixed as a positive ray of ${\mathrm{\Lambda }}^{4}V$, and the time-orientation component $C^{+}$ of the timelike set.

Primitive physical interpretation and parameters adopted on this page. The physical event interpretation together with the covariance assumption (Adopted inertial covariance of the model equations); the ideal-clock and ideal-vacuum-light interpretation (Adopted ideal-clock and ideal-vacuum-light interpretation); and the conversion constant $c$, whose SI numerical value is exact by unit definition and whose physical interpretation as the vacuum ray speed is adopted, as recorded in SI units and coordinate conventions of the special-relativity framework. These are adopted assumptions of the framework, not deductions.

Derived (constructed, not adopted) on this page. Displacements and intervals; the causal classification of displacements; inertial frames and their inertial charts (Inertial frames and inertial charts of the affine Minkowski model); the smooth structure of $A$ making it a smooth $4$-manifold and the canonical identification $T_{p}A\cong V$; the oriented volume form and its components (The oriented volume form epsilon and the permutation symbols of an inertial frame); and the spatial Euclidean structure and vector product of a named inertial frame (The Euclidean spatial structure and cross product of an inertial frame). These are mathematical constructions from the primitive structure, with the stated frame-dependence where it occurs.

Reserved for later development. The general proper-time parametrization theory, physical particle momentum and energy, charge, and force or matter laws. The units item gives explicit kinematic definitions only for bookkeeping; it supplies no dynamics or empirical particle identification. Later pages introduce their own adopted premises and must repeat this status audit for them; a definition on a later page is not made primitive by this page's conventions.

Mathematical versus empirical status. The mathematical statements of this page concern only the stipulated geometry. They do not assert that the model describes nature; the physical interpretation enters through the two adopted postulates, whose empirical support would be evidence about the specified model rather than a proof.