Pauli exclusion in the declared finite-mode occupation readout, with the bosonic contrast

[Sector restrictions are self-adjoint] By [F1] the operator $N_j$ preserves ${\mathfrak{H}}_n^{\sigma }$, and both the Slater family and the family $(m_{\lambda })$ consist of eigenvectors of $N_j$ with real eigenvalues and form a basis of the respective sector; a linear endomorphism of a finite-dimensional Hilbert space that is diagonal in an orthogonal basis with real eigenvalues is self-adjoint. Hence each $N_j^{\sigma }=N_j{\mathrm{\mid }}_{{\mathfrak{H}}_n^{\sigma }}$ is a bounded self-adjoint endomorphism of ${\mathfrak{H}}_n^{\sigma }$, which is the readout declared in clauses (c)-(d) of the model.

[Finite spectral measures and laws] Fix $\sigma \in \{-,+\}$ and $j\in \{1,\dots ,d\}$. By [F3], the finite family of projections $F_j(B):={\sum }_{\begin{array}{c}\lambda \in {\mathrm{\Lambda }}_n\\ {\lambda }_j\in B\end{array}}{P}_{\lambda }(B\subseteq \mathbb{R}\text{ Borel})$ is a projection-valued measure: $F_j(\mathrm{\varnothing })=0$ and $F_j(\mathbb{R})=I$ because the $P_{\lambda }$ sum to $I$; each $F_j(B)$ is an orthogonal projection; intersections satisfy $F_j(B\cap C)=F_j(B)F_j(C)$ because the $P_{\lambda }$ are mutually orthogonal; and countable additivity is strong since only finitely many $\lambda$ occur and the Borel sets in a disjoint sequence contain each eigenvalue ${\lambda }_j$ at most once. Its scalar measures ${\mu }_x(B):=⟨F_j(B)x,x⟩$ are finite and supported on the finite set $S_j:=\{{\lambda }_j:\lambda \in {\mathrm{\Lambda }}_n\}$, hence regular: for every Borel $B$, the compact set $B\cap S_j$ and an open set $U$ with $U\cap S_j=B\cap S_j$ give ${\mu }_x(B\cap S_j)={\mu }_x(B)={\mu }_x(U)$. Its spectral integral is the finite sum ${\int }_{\mathbb{R}}rd{F}_j(r)={\sum }_{\lambda \in {\mathrm{\Lambda }}_n}{\lambda }_j{P}_{\lambda }=N_j^{\sigma }\mathrm{.}$ Thus $F_j$ is the spectral PVM of $N_j^{\sigma }$ by uniqueness of the spectral PVM in [F2]. The outcome law in $\rho$ is therefore ${\mu }_{\rho }^{N_j^{\sigma }}(B)=\mathrm{tr}(\rho F_j(B))$. In the fermionic sector [F1] restricts the joint eigenvalue vectors to ${\mathbf{1}}_J\in \{0,1{\}}^d$, so this spectral measure is supported on $\{0,1\}$; in the symmetric sector it groups the lines $\mathbb{C}{m}_{\lambda }$ by the value ${\lambda }_j$. This identifies the readout laws of clauses (c)-(d) of the model.

[Exclusion] Assume $\sigma =-$. By step 2.1 the spectral measure of $N_j^{-}$ is supported on $\{0,1\}$, so $E_{N_j^{-}}([2,\mathrm{\infty }))=0$ and ${\mu }_{\rho }^{N_j^{-}}([2,\mathrm{\infty }))=\mathrm{tr}(0)=0$; moreover $N_j^{-}(N_j^{-}-I)=0$ on the sector because the operator vanishes on the Slater basis by [F1], so its expectation $\mathrm{tr}(\rho N_j^{-}(N_j^{-}-I))$ is zero. For the joint law, [F1] shows that the joint eigenspaces with an entry $\ge 2$ do not occur among the lines $\mathbb{C}{f}_J^{\wedge }$, so the sum of the corresponding projections $P_{\lambda }$ is zero and ${\mu }_{\rho }^N(\{\lambda :{\lambda }_j\ge 2\text{ for some }j\})=0$. This proves (a).

[Bosonic contrast] Assume $\sigma =+$. For every $\lambda \in W(n,d)$ the vector $m_{\lambda }$ is nonzero and spans the joint eigenspace for $\lambda$ by [F1], so $P_{\lambda }\mathrm{\ne }0$; for the pure preparation of the ray of $m_{\lambda }$ the weight is ${\mu }_{\rho }^N(\lambda )=\mathrm{tr}(\rho P_{\lambda })=\mathrm{\parallel }{P}_{\lambda }x{\mathrm{\parallel }}^2\mathrm{/}\mathrm{\parallel }x{\mathrm{\parallel }}^2=1$ for any unit representative $x$ of that ray by [F4]. By [F5] the configurations $\lambda$ with ${\lambda }_j=n$ lie in $W(n,d)$, so they occur; this proves (b).

[Total occupation] By [F1] one has ${\sum }_j{N}_j=nI$ on the whole tensor power, hence on the sector, and therefore ${\sum }_j{N}_j^{\sigma }=n{\mathrm{i}\mathrm{d}}_{{\mathfrak{H}}_n^{\sigma }}$; since the total occupation is the constant operator $nI$, its outcome law assigns probability one to $\{n\}$ in every preparation, and applying the joint law of [F1] gives ${\sum }_j{\lambda }_j=n$ for every configuration of the support; this is (c).

[Assembly and scope] Step 3.1 proves (a), step 3.2 proves (b) and step 3.3 proves (c). The proof is conditional on the declared sector of P6, the declared finite-mode readout and the cited spectral mathematics; no measured occupancy, no apparatus behaviour and no derivation of a species' statistics is asserted.