Adopted finite canonical and band preparation rules

For the finite $q$-level tensor model of Additive spectral Hamiltonian and normalized ensembles, adopt the following preparation rules in addition to the state and outcome assumptions P1 and P2 of State and composition postulate P1: Hilbert spaces, preparations and composite systems and Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes. The quantities in this item are model assignments; they are not inferred from the cited texts or from data.

1. For declared temperature $T>0$ kelvin and Boltzmann constant $k_B=1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, the fixed exact SI Boltzmann constant, use $\beta =(k_{B}T{)}^{-1}>0$ joules${}^{-1}$. In this formulation $k_B$ is the primitive SI conversion constant between temperature and energy; it is not derived from this finite-level model or fitted from its data. The real scalar levels $e_j$ in Additive spectral Hamiltonian and normalized ensembles are primitive model energy parameters in joules. The total label energy $E(j_1,\dots ,j_N)={\sum }_{a=1}^N{e}_{j_a}$ and the one-site canonical mean $E={\sum }_{j=1}^q{p}_j{e}_j$ defined there are derived scalar energies in joules. Adopt the canonical finite density ${\gamma }_N$ of that definition.
2. For declared band width ${\delta }_N\ge 0$ in joules, if its label set ${\mathrm{\Gamma }}_N$ is nonempty, the uniform band preparation is the density ${\sigma }_N={\mathrm{\Pi }}_N\mathrm{/}\mathrm{\mid }{\mathrm{\Gamma }}_N\mathrm{\mid }$ from that definition. If the set is empty, this normalized preparation is undefined and no band state is assigned.
3. For a stated finite POVM $(M_a)$ on the register, the outcome law is the adopted Born law $p_a=\mathrm{Tr}(\rho M_a)$ of P2. A prediction is the resulting model probability, not a reported detector frequency.

The positive-temperature interpretation uses SI units as fixed in Dimensions, SI unit conventions and quantity types of the framework. No preparation mechanism, thermalization process, sampling law, apparatus error, or observed data is asserted. A statement that a physical preparation realizes one of these densities requires a separate empirical identification and uncertainty analysis.