The Galilean regime of the framework: frame-relative protocols, the absence of an intrinsic slow-motion parameter, and the empirical status of a validity claim

The spacetime of the framework is the affine event space with its surjective absolute-time covector, its three-dimensional Euclidean spatial displacement space and its fixed orientation (Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure); it contains no inner product on the full displacement space, no light cones, no null displacements and no universal speed bound. The frames are the adapted affine charts of these data, and the allowed changes between them are the Galilean transformations $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Rx+vt+a$ with $R\in SO(3)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames); the resulting kinematic laws, the frame-independence of vanishing coordinate acceleration and the group structure of the changes are the content of Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes. The force model transforms by its own declared covariance rule (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance), and the existence and uniqueness statements for maximal motions are stated in the chosen frame (Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder).

An operational observable model (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error) is specified in one inertial frame: its observation times are times of that frame, its readout is a function of the coordinate states, and its error random element carries the units of the readout coordinates. Changing the frame shifts the coordinate times, transforms positions affinely and rotates and boosts coordinate velocities. The prediction map in the new frame must be computed using the transformed protocol and its declared readout law; it need not differ numerically from the original map. For example, the allowed constant dimensionless readout $R\equiv 0$ gives $\mathrm{\Phi }\equiv 0$ in both frames. A measurement model must therefore name its frame, and a claim that two frames "agree" is a claim about the declared transformation law of the model, not a separate empirical discovery.

The dimension group of the framework is generated by length, duration and mass (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework clause (a)); the framework contains no quantity of dimension $\mathrm{\ell }{\tau }^{-1}$ that is a constant of nature. Consequently the framework cannot by itself form a dimensionless parameter that measures how close a system is to some other kinematic regime, and it states no numerical cutoff: every regime statement such as "the motion is adequately described by this model at the required accuracy" is an external, empirically qualified premise of a use of the model, not a theorem of it. A deductive approximation bound relative to a distinct physical model requires explicit hypotheses and a proved comparison estimate; a speed scale is needed only if that comparison uses one. Calibrated observations can instead bound discrepancies between specified predictions and readouts under declared apparatus, calibration and sampling assumptions, carrying their uncertainty and tested conditions. Such empirical bounds require neither another physical model nor a universal speed scale, and do not establish a universal validity boundary. No comparison theorem or empirical bound is asserted here.

The framework's kinematic data distinguish no inertial frame (Status audit and model limits of the framework page: primitive data, adopted assumptions, derived objects and what is not claimed); "at rest" is a frame-relative description, not an absolute state. No measured speed, no numerical validity boundary, no comparison with any other kinematic framework, and no claim that a physical system is exactly Galilean is made. The Galilean transformations relate the adopted inertial frames and give the velocity and acceleration laws of Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes. Covariance of the equation of motion also uses the transformed force map on the transferred domain prescribed by Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (vi). A symmetry of a particular force model means a transformation carrying its solution curves to solutions of the same prescribed force law on the same domain; this requires the force law and its domain to be invariant under that transformation and does not follow from the kinematic theorem alone. Whether a physical situation is adequately represented by the model is an empirical question that this item leaves open.