Zero-energy bosonic mode trace obstruction

Given: Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the Fock basis and Lebesgue-null singleton suppliers. Let $I_0=({\mathbb{Z}}^3\setminus \{0\})\times \{1,2\}$ be the enumerated base mode set of Actual photon box trace and Planck energy/entropy flux, with its stated max-norm shell and lexicographic ordering.

Form the ordered disjoint union $I^{\mathrm{\prime }}=\{0_{\star }\}\bigsqcup I_0$, placing the new label $0_{\star }$ first. Give $0_{\star }$ energy ${ϵ}_{0_{\star }}=0$ and each $i=(n,s)\in I_0$ its energy from Actual photon box trace and Planck energy/entropy flux. Let ${\mathrm{\Omega }}_B^{\mathrm{\prime }}$ be the set of finite-support maps $m:I^{\mathrm{\prime }}\to {\mathbb{N}}_0$, let $E^{\mathrm{\prime }}(m)={\sum }_{i\in I^{\mathrm{\prime }}}{ϵ}_i{m}_i$, and set ${\mathcal{H}}^{\mathrm{\prime }}={\mathrm{\ell }}^2({\mathrm{\Omega }}_B^{\mathrm{\prime }},\mathbb{C})$ with its occupation basis. Define $H^{\mathrm{\prime }}$ by $(H^{\mathrm{\prime }}a{)}_m=E^{\mathrm{\prime }}(m)a_m$ on $D(H^{\mathrm{\prime }})=\{a\in {\mathcal{H}}^{\mathrm{\prime }}:{\sum }_m{E}^{\mathrm{\prime }}(m{)}^2\mathrm{\mid }{a}_m{\mathrm{\mid }}^2<\mathrm{\infty }\}$, and define the bounded positive diagonal multiplier $G_{\beta }$ by $G_{\beta }\mathrm{\mid }m⟩=e^{-\beta E^{\mathrm{\prime }}(m)}\mathrm{\mid }m⟩$. Its diagonal partition sum is $Z_{\beta }^{\mathrm{\prime }}:={\sum }_{m\in {\mathrm{\Omega }}_B^{\mathrm{\prime }}}{e}^{-\beta E^{\mathrm{\prime }}(m)}\in [0,+\mathrm{\infty }]$. This is the stipulated extension of the finite-box diagonal model by one zero-energy bosonic mode, without a zero-point term.

Occupation bases and actual CCR/CAR domains, Statement: for a specified enumerated mode set, finite-support bosonic occupation configurations give an orthonormal basis of the bosonic Fock space. The current batch 45 evidence records this draft carrier as Step-1-ready; that is not independent review or proof acceptance, so the use remains conditional on its inspected interface.

Trace class operator, Definition: a trace-class operator is compact and its singular values have finite sum. The compactness condition uses the exact definition in def-compact-linear-operator: the image of every bounded set must have compact closure; compactness of that closure is the finite-open-subcover property in def-metric-compactness.

Actual photon box trace and Planck energy/entropy flux, Statement: the base model has nonzero wavevectors only and its corresponding diagonal Gibbs trace is finite under the explicitly stated summability conditions.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the Fock-basis supplier assumes ${\mathrm{A}\mathrm{C}}_{\omega }$.

Every affine hyperplane of ${\mathbb{R}}^n$, and hence every proper linear subspace, is Lebesgue null, Statement: assuming ${\mathrm{A}\mathrm{C}}_{\omega }$, every proper linear subspace of ${\mathbb{R}}^3$ is Lebesgue measurable and null; in particular, $\{0\}$ is null.