Constructed independent cell Gibbs states

Let $r\ge 1$ be finite. For each cell $i\in \{1,\dots ,r\}$, let ${\mathcal{H}}_i$ be a complex Hilbert space of finite dimension $d_i\ge 1$, let $H_i=H_i^{\ast }$ be a Hermitian proper Hamiltonian in joules, and let $Q_i=Q_i^{\ast }$ be a dimensionless number observable whose eigenvalues lie in ${\mathbb{N}}_0$. Assume $[H_i,Q_i]=0$. Operators $H_i,Q_i$ on the total space $\mathcal{H}:={⨂}_{i=1}^r{\mathcal{H}}_i$ mean their lifts acting on the $i$th tensor factor and as the identity on the other factors; no intercell interaction is included.

Let $(M,g)$ be a smooth time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$, and let $O\subseteq M$ be a connected open set on which a smooth future-directed timelike Killing field $\xi$ is defined. Use length coordinates $x^0=ct$ and the corresponding dimensionless components of $\xi$. Define its dimensionless lapse $N(x):=\sqrt{-g_x({\xi }_x,{\xi }_x)}>0$. Let $c>0$ be the speed of light in $\mathrm{m}{\mathrm{s}}^{-1}$, $k_B>0$ Boltzmann's constant in $\mathrm{J}{\mathrm{K}}^{-1}$, and let $A$ be a global smooth real one-form with SI components in tesla-metres, $F:=dA$ with components in teslas, and ${\mathrm{\Lambda }}_{\xi }:O\to \mathbb{R}$ a global smooth compensator in tesla-metres with ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$. Define ${\mathrm{\Phi }}_{\xi }(x):=-c(({\iota }_{\xi }A{)}_x-{\mathrm{\Lambda }}_{\xi }(x))$ in volts. Let $T:O\to (0,\mathrm{\infty })$ and $\mu :O\to \mathbb{R}$ be smooth fields in kelvins and joules per particle. Fix a common particle rest mass $m>0$ in kilograms and charge $q\in \mathbb{R}$ in coulombs, put $M:=m{c}^2>0$ in joules, and define $u:=\xi \mathrm{/}N$, $\beta :=u\mathrm{/}(k_{B}T)$ in ${\mathrm{J}}^{-1}$, and $\zeta :=\mu \mathrm{/}(k_{B}T)$ dimensionlessly. On the future massive shell ${\mathcal{P}}_M^{+}(O):=\{(x,P)\in TO:g_x(P,P)=-M^2,P\text{ future-directed}\}$, where $P$ has energy-vector units and $g_x(P,P)$ has units ${\mathrm{J}}^2$, let $f(x,P):=\exp (\zeta (x)+g_x(\beta (x),P))$, as defined for the same charged data in Global electrochemical redshift from charged Liouville stationarity, and assume the exact whole-shell stationarity condition ${\mathcal{L}}_{F,q}f(x,P)=0$ for every $(x,P)\in {\mathcal{P}}_M^{+}(O)$, where ${\mathcal{L}}_{F,q}$ is the charged transport field in that supplier's formulation. This is the stationarity premise of Global electrochemical redshift from charged Liouville stationarity. Choose a cell-location map $\chi :\{1,\dots ,r\}\to O$ and set $x_i:=\chi (i)$. Define the lapse, compensated potential, proper temperature, and chemical potential at cell $i$ by evaluation: $N_i:=N(x_i),{\mathrm{\Phi }}_i:={\mathrm{\Phi }}_{\xi }(x_i),T_i:=T(x_i),{\mu }_i:=\mu (x_i)\mathrm{.}$ The same theorem supplies constants $T_K>0$ and ${\mu }_K\in \mathbb{R}$ such that $N_i{T}_i=T_K,N_i{\mu }_i+q{\mathrm{\Phi }}_i={\mu }_K,$ where $q$ is the common particle charge in coulombs. This use of the charged smooth-geometry supplier retains its explicit ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption.

Set ${\beta }_K=(k_B{T}_K{)}^{-1}>0$ in joules${}^{-1}$, ${\beta }_i=N_i{\beta }_K=(k_B{T}_i{)}^{-1}$, $H_K={\sum }_{i=1}^r(N_i{H}_i+q{\mathrm{\Phi }}_i{Q}_i)$, and $Q={\sum }_{i=1}^r{Q}_i$. Then the grand generator $G_K:=H_K-{\mu }_{K}Q$ is Hermitian and $G_K={\sum }_{i=1}^r{N}_i(H_i-{\mu }_i{Q}_i)\mathrm{.}$ The finite partition sums $Z_i:={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)},Z:={\mathrm{Tr}}_{\mathcal{H}}{e}^{-{\beta }_K{G}_K}$ are positive and finite, with $Z={\prod }_{i=1}^r{Z}_i$. The normalized Gibbs density factorizes exactly: $\rho :=Z^{-1}{e}^{-{\beta }_K{G}_K}={⨂}_{i=1}^r{\rho }_i,{\rho }_i:=Z_i^{-1}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)}\mathrm{.}$ It is a ${\beta }_K$-KMS state for the grand-generator dynamics on the full finite matrix algebra. On the neutral subalgebra ${\mathcal{A}}_0:=\{B\in \mathcal{B}(\mathcal{H}):[B,Q]=0\}$, grand-generator and physical $H_K$ dynamics coincide, so the restricted state is ${\beta }_K$-KMS for the physical dynamics there. This constructs only independent finite cells; it does not claim that an interacting system's reduced cell states are Gibbs states.

All deductions are conditional on the actual interfaces in [F1]–[F3]. In particular, the charged redshift relation is used at its stated global-compensator and whole-shell hypotheses; no generic curved-spacetime quantum state is inferred.