Scope and empirical status of the massive-particle kinematics items

The items of this pair are conditional consequences of the framework's postulates and of the cited mathematics. The massive spin-s particle model of the framework: state space, operators, units and frame readings founds the spin-$s$ model on the state space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$ with the componentwise position and momentum operators $Q_j,P_k$; Componentwise position and momentum operators on the component-valued wave-function space proves that these operators are self-adjoint with their maximal domains and provides the Fourier representation and the translation identities; Canonical commutation relations on the component-valued core and commutation with the internal factor proves the canonical commutation relations on the Schwartz core, identifies the spectral measures and proves that the internal factor commutes with the spatial measures; The momentum representation of the componentwise kinematics constructs the momentum representation; Position and momentum outcome laws of the massive spin-s model derives the position and momentum outcome laws of the spin-$s$ model, including the Born density $\mathrm{\parallel }\psi (x){\mathrm{\parallel }}_{{\mathbb{C}}^{2s+1}}^2$ and the momentum density $\mathrm{\parallel }{\psi }_p(p){\mathrm{\parallel }}^2$; Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics and Galilean boost covariance of the free massive spin-s model and the moving-frame reading establish the covariance of these laws and of the bounded-potential dynamics under the Galilean translations and boosts of the model's frame class. All of this is conditional on P1, P2, P4, P5 and P6 and on the cited mathematical suppliers; none of it is a statement that any physical system is exactly described by the model.

The mass parameter $m>0$ and the spin label $s$ are model parameters: within the framework they are declared values, and the identical-particle and spin postulate assigns to a species its mass, spin, statistics and couplings as empirical model parameters with uncertainty. This pair derives no value of $m$ or $s$, records no measurement and asserts no species identification; the units in which $m$, $\mathrm{\hslash }$, $Q_j$ and $P_k$ are read are those of Dimensions, SI unit conventions and quantity types of the framework, and the exact defining constants of the current SI recorded there are unit conventions, not measurements of this framework.

The outcome laws are sharp-law statements: an exact position or momentum value has probability zero in every preparation, finite-resolution outcomes are Borel bins, and no detector, efficiency, resolution or sampling model is introduced. The covariance theorems hold for the free model and for real bounded measurable potentials; they assert nothing for singular or unbounded potentials, confined or boundary realizations, massless particles, radiation or relativistic matter. The internal factor is bookkeeping: it is not identified with a spatial rotation, a classical angular momentum or a relativistic spinor, and no empirical consequence of the spin assignment alone is derived. The identification of the operators $Q_j,P_k$ with concrete laboratory procedures remains a tested identification under the preparation and apparatus postulate, external to this pair.