The declared finite-mode model and its occupation readouts

(a) Adopted readings. Adopt a declared species sector in the vocabulary of Declared species sectors and their preparations: the framework's reading of P6, the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) in the reading of Assigned sharp observables and the outcome law of the framework, and the quantities and units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). The latter is inherited from the general spectral-PVM construction in P2 and clause (b) of the observable definition used in clause (c) below; the finite joint-eigenspace decomposition in clause (d) requires no additional choice.

(b) The declared finite-mode model. Fix $d\ge 1$, a finite-dimensional complex Hilbert space $K$ of dimension $d$ (Hilbert space) with a declared orthonormal basis $(f_1,\dots ,f_d)$, the declared modes of the model, and an integer $n\ge 1$. The declared finite-mode model of the species with these data is the declared species sector of clause (a) whose one-particle space is $K$, together with its $n$-particle sector space ${\mathfrak{H}}_n^{\sigma }=\mathrm{ran}{P}_{\sigma }^{(n)}\subseteq K^{\otimes n}$ and the declared sector preparations of Declared species sectors and their preparations: the framework's reading of P6 clauses (c)-(d). The declaration is an idealization: it asserts that the model confines the one-particle content to the $d$ declared modes. No claim is made that a species has finitely many modes; whether the declared modes describe a laboratory mode family is a tested identification with uncertainty under the preparation and apparatus postulate P7, not a consequence of this definition.

(c) The occupation readouts. For $j\in \{1,\dots ,d\}$ let $N_j:K^{\otimes n}\to K^{\otimes n},N_{j}x:={\sum }_{i\in \{1,\dots ,d{\}}^n}\mathrm{\mid }i{\mathrm{\mid }}_j⟨x,f_{i_1}\otimes \cdots \otimes f_{i_n}⟩f_{i_1}\otimes \cdots \otimes f_{i_n},\mathrm{\mid }i{\mathrm{\mid }}_j:=\mathrm{\#}\{k:i_k=j\},$ be the occupation operator of Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors clause (d) for the basis $(f_1,\dots ,f_d)$. By that clause $N_j$ is bounded and self-adjoint, the $N_j$ commute pairwise, and $N_j$ preserves the sector ${\mathfrak{H}}_n^{\sigma }$; the sector occupation readout is the restriction $N_j^{\sigma }:=N_j{\mathrm{\mid }}_{{\mathfrak{H}}_n^{\sigma }},$ a bounded self-adjoint endomorphism of ${\mathfrak{H}}_n^{\sigma }$ with spectrum contained in $\{0,1,\dots ,n\}$. The model reads $(N_j^{\sigma },\text{particle count})$ as an assigned sharp observable of the declared sector in the sense of Assigned sharp observables and the outcome law of the framework clause (a): its outcome law in a declared sector preparation $\rho$ is ${\mu }_{\rho }^{N_j^{\sigma }}(B)=\mathrm{tr}(\rho E_{N_j^{\sigma }}(B)),B\subseteq \mathbb{R}\text{ Borel},$ the reading of P2 clause (ii) through the observable definition.

(d) The joint occupation readout. The operators $N_1^{\sigma },\dots ,N_d^{\sigma }$ are simultaneously diagonalised by the sector bases of the cited theorem: the Slater basis in the fermionic case and the family $(m_{\lambda })$ in the bosonic case, with joint eigenvalues ${\mathbf{1}}_J$ and $\lambda$ respectively. Write ${\mathrm{\Lambda }}_n:=\{\lambda \in {\mathbb{N}}^d:{\lambda }_j\le n,{\sum }_j{\lambda }_j=n\}$ and, for $\lambda \in {\mathrm{\Lambda }}_n$, let $P_{\lambda }$ be the orthogonal projection of ${\mathfrak{H}}_n^{\sigma }$ onto the joint eigenspace of $(N_1^{\sigma },\dots ,N_d^{\sigma })$ for the eigenvalue vector $\lambda$; the cited theorem exhibits a line $\mathbb{C}{m}_{\lambda }$ for every bosonic $\lambda \in {\mathrm{\Lambda }}_n$. In the fermionic sector there is a line $\mathbb{C}{f}_J^{\wedge }$ exactly when $\lambda ={\mathbf{1}}_J$ for an $n$-element subset $J$; if any entry of $\lambda$ is at least $2$, the joint eigenspace is $\{0\}$ and $P_{\lambda }=0$. Thus the $P_{\lambda }$, including these zero projections, are mutually orthogonal, their finite sum is the identity of the sector, and $N_j^{\sigma }={\sum }_{\lambda \in {\mathrm{\Lambda }}_n}{\lambda }_j{P}_{\lambda }$. The model reads the joint occupation readout through the finite family $(P_{\lambda }{)}_{\lambda \in {\mathrm{\Lambda }}_n}$: the joint law in a declared sector preparation $\rho$ is ${\mu }_{\rho }^N(\lambda ):=\mathrm{tr}(\rho P_{\lambda })(\lambda \in {\mathrm{\Lambda }}_n),$ and the marginal law of $N_j^{\sigma }$ is recovered by ${\mu }_{\rho }^{N_j^{\sigma }}(B)={\sum }_{\lambda :{\lambda }_j\in B}{\mu }_{\rho }^N(\lambda )$, because the eigenvalue decomposition of $N_j^{\sigma }$ groups the $P_{\lambda }$ by the value ${\lambda }_j$.

(e) Total occupation. By clause (d) of the cited theorem, ${\sum }_{j=1}^d{N}_j^{\sigma }=n{\mathrm{i}\mathrm{d}}_{{\mathfrak{H}}_n^{\sigma }}$; hence the total occupation ${\sum }_j{N}_j^{\sigma }$ has the degenerate outcome law ${\delta }_n$ in every declared sector preparation, and the joint law is supported on ${\mathrm{\Lambda }}_n$: the total number of particles read by the model is $n$ with certainty.

(f) Status and scope. Clauses (a)-(e) fix the vocabulary of the model and read the cited spectral mathematics through P2 under the choice hypotheses of clause (a); they add no physical assumption beyond the sector declaration and the finite-mode idealization of clause (b). The occupation readouts are sector quantities in the sense of clause (e) of Declared species sectors and their preparations: the framework's reading of P6; the exclusion and occupation-count consequences are proved on this page (Pauli exclusion in the declared finite-mode occupation readout, with the bosonic contrast and Finite-mode occupation configurations: fermionic exclusion counts, bosonic counts and truncated dimensions). No measured count, no detector response, no dynamics and no empirical claim is made.

• Mode versus particle. $N_j^{\sigma }$ counts how many of the $n$ particles the model reads in the declared mode $f_j$; it is a collective readout of the sector, not a quantity of a labelled factor, and it is insensitive to which labels carry the occupied modes.
• Choice hypotheses. The tensor construction and the sector projections carry the Axiom of Countable Choice. The reading of the general spectral construction through P2 additionally carries the Axiom of Choice; both hypotheses are declared in clause (a). The finite spectral decomposition of clause (d) uses no further choice.