Classical full-phase interacting ensemble preparations

Fix the model, physical reference conventions, mass, potential and energy window specified by Interacting marked phase-space and reference measures and Total-energy concentration and controlled canonical windows. This is a single-species model on the flat torus of side ${\mathrm{\ell }}_{0}L$, represented by ${\mathrm{\Lambda }}_L=[0,L{)}^d$. Identify the action reference $a_0$ of Classical particle counting and physical observables with $h_{\mathrm{r}\mathrm{e}\mathrm{f}}={\mathrm{\ell }}_0{p}_{\ast }$ from the marked model, and take the controls $\lambda =(L,bc)$ in Canonical thermal preparation. Its physical Hamiltonian is $H_{N,\lambda }=H_{L,bc}^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=E_0{\mathcal{H}}_{L,bc}$ from Interacting marked phase-space and reference measures, so ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{N,\lambda }=b{\mathcal{H}}_{L,bc}$. In the corresponding canonical coordinates, the exact-$N$ reference $R_{L,N}$ of that mathematical model is $e^{-V_L}{\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, where ${\bar{m}}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is the one-species quotient reference of Classical particle counting and physical observables. The factor $e^{-V_L}$ is independent of $N$ and cancels when normalizing a fixed-$N$ canonical law, a fixed-$N$ band restriction, or the full grand-sector law. Adopt the following alternative preparations whenever their normalizations are finite and positive:

1. At fixed exact particle number $N$, controls and $T>0$, use the canonical density proportional to $e^{-\beta H}$ relative to the declared phase reference, as in Canonical thermal preparation.
2. For a sequence satisfying all hypotheses of Total-energy concentration and controlled canonical windows—including AC, the ultrafilter extension principle, DC, HB, an interior attainable density, the periodic exact-particle log-rate hypothesis near $b$, and differentiability at $b$—at fixed exact $N_n$ use the uniform full-phase reference law restricted to its constructed band $A_n$. “Uniform” means normalized restriction of that finite reference measure; it is an energy-window preparation, not a zero-width surface measure. This postulate makes no existence claim for that specific band when those theorem hypotheses fail.
3. For particle exchange, use the countable-sector grand preparation of Grand canonical particle-exchange preparation. Its full dimensionless activity is the fugacity $z_{\mathrm{p}\mathrm{h}}:=e^{{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mu }>0$, where ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=1\mathrm{/}(k_{B}T)$ in J${}^{-1}$ and the chemical potential $\mu$ is in J per particle. When matching positional activity $z=e^{\alpha }$ at inverse temperature $b$, choose $z_{\mathrm{p}\mathrm{h}}=z\mathrm{/}{\kappa }_b$, as in Interacting marked phase-space and reference measures.

These laws are adopted statistical preparations. They do not follow from Hamiltonian energy conservation, do not assert that any isolated dynamics reaches these distributions, and do not encode an experimental observation. The band sequence is the particular controlled shrinking window constructed conditionally by Total-energy concentration and controlled canonical windows.