Finite response and spectral clock redshift

Fix a cell label $i\in \{1,\dots ,r\}$ in Constructed independent cell Gibbs states, let $x_i=\chi (i)$ be its assigned point, and write $N:=N_i=N(x_i)>0$, $H_{\mathrm{l}\mathrm{o}\mathrm{c}}:=H_i$, $Q:=Q_i$, and ${\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}:={\mu }_i$. Let ${\beta }_K>0$ in joules${}^{-1}$ and ${\mu }_K$ in joules be the global parameters supplied there. Put $G_{\mathrm{l}\mathrm{o}\mathrm{c}}:=H_{\mathrm{l}\mathrm{o}\mathrm{c}}-{\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}Q$ and $G_K:=N{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}$. Set ${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}:=N{\beta }_K$ and define the cell normalizer $Z_{\mathrm{l}\mathrm{o}\mathrm{c}}:=Z_i={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}}={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_K{G}_K}>0$. Let $\sigma$ be the cell Gibbs density supplied by that theorem, so $\sigma =Z_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}{e}^{-{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}}=Z_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}{e}^{-{\beta }_K{G}_K}$. For neutral Hermitian matrices, define ${\alpha }_{\tau }^{\mathrm{l}\mathrm{o}\mathrm{c}}(C):=e^{i{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}\tau \mathrm{/}\mathrm{\hslash }}C{e}^{-i{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}\tau \mathrm{/}\mathrm{\hslash }},{\alpha }_t^K(C):=e^{i{G}_{K}t\mathrm{/}\mathrm{\hslash }}C{e}^{-i{G}_{K}t\mathrm{/}\mathrm{\hslash }}\mathrm{.}$ For this selected cell, let $q$ be the common charge in coulombs and ${\mathrm{\Phi }}_i:={\mathrm{\Phi }}_{\xi }(x_i)$ its compensated potential in volts from the supplier. Define $H_{K,i}:=N{H}_{\mathrm{l}\mathrm{o}\mathrm{c}}+q{\mathrm{\Phi }}_{i}Q$ in joules, and write $H_K:=H_{K,i}$ in the physical control below. The supplier normalization gives $N{\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}+q{\mathrm{\Phi }}_i={\mu }_K$, hence $H_K=G_K+{\mu }_{K}Q$.

Let $A=A^{\ast }$ and $B=B^{\ast }$ satisfy $[A,Q]=[B,Q]=0$. Then their local proper-time and Killing-time dynamics agree after $\tau =Nt$.

Let $h:[0,S]\to \mathbb{R}$ be continuous, with $S>0$ seconds, and let $h(\tau )B$ have units joules. For dimensionless $ϵ\in \mathbb{R}$, the proper-time grand-generator control $G_{\mathrm{l}\mathrm{o}\mathrm{c}}-ϵh(\tau )B$ corresponds in Killing time to $G_K-ϵNh(Nt)B$. Define $E_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )$ and $E_K(t)$ as the finite-time expectations of $A$ from the same initial density $\sigma$ under these controls. Because $A$ and $B$ commute with $Q$, replacing each grand generator by its physical Hamiltonian changes the propagator only by a unitary generated by a constant multiple of $Q$ and leaves the expectation of $A$ unchanged. Thus these expectations also equal those obtained from the respective physical controls $H_{\mathrm{l}\mathrm{o}\mathrm{c}}-ϵh(\tau )B$ and $H_K-ϵNh(Nt)B$. In particular, $E_K(t)=E_{\mathrm{l}\mathrm{o}\mathrm{c}}(Nt)(0\le t\le S\mathrm{/}N)\mathrm{.}$ The finite-time Kubo expansions have identical first-order terms under this rescaling and the same quadratic remainder bound. With $R_{\mathrm{l}\mathrm{o}\mathrm{c}}(u):=\frac{i}{\mathrm{\hslash }}\mathrm{Tr}\text{\hspace{0.17em}⁣}(\sigma [{\alpha }_u^{\mathrm{l}\mathrm{o}\mathrm{c}}(A),B]),C_S:=\frac{2}{\mathrm{\hslash }}{\int }_0^S\mathrm{\mid }h(s)\mathrm{\mid }\mathrm{\parallel }B\mathrm{\parallel }ds,$ the local expansion is $E_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )=\mathrm{Tr}(\sigma {\alpha }_{\tau }^{\mathrm{l}\mathrm{o}\mathrm{c}}(A))+ϵ{\int }_0^{\tau }h(s)R_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau -s)ds+r_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau ),\mathrm{\mid }{r}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }\frac{(\mathrm{\mid }ϵ\mathrm{\mid }{C}_S{)}^2{e}^{\mathrm{\mid }ϵ\mathrm{\mid }{C}_S}}{2}\mathrm{.}$ Use $\mathrm{\hslash }>0$ in joule-seconds, so $C_S$ is dimensionless and $R_{\mathrm{l}\mathrm{o}\mathrm{c}}$ has units $[A][B]\mathrm{/}(\mathrm{J}\mathrm{s})$. The Killing-time response kernel is $R_K(t)=R_{\mathrm{l}\mathrm{o}\mathrm{c}}(Nt)$ and its control amplitude is $Nh(Nt)$.

For the equilibrium correlation spectral measures, use the Fourier convention of Finite spectral balance and distributional fluctuation response. If $D_N(\omega ):=N\omega$, then $S_K=(D_N{)}_{\ast }{S}_{\mathrm{l}\mathrm{o}\mathrm{c}},S_K(\omega )=N^{-1}{S}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\omega \mathrm{/}N)$ in distribution notation; the measure form preserves the weights of delta-function spectral lines. Since ${\beta }_K\mathrm{\hslash }(N\omega )={\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}\mathrm{\hslash }\omega$, detailed balance is unchanged. Any zero-frequency atom remains an atom at zero and is not assigned a value by the coth formula, which remains restricted to nonzero frequencies.

For a static local perturbation $G_{\mathrm{l}\mathrm{o}\mathrm{c}}-\lambda B$, where $\lambda B$ has units joules, the re-prepared Gibbs family satisfies ${\frac{d}{d\lambda }\mathrm{Tr}({\sigma }_{\lambda }B)\mid }_{\lambda =0}={\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B)=N{\beta }_K{\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B)\ge 0.$ Here $\delta B:=B-\mathrm{Tr}(\sigma B)I$ and ${\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B):={\int }_0^1\mathrm{Tr}\text{\hspace{0.17em}⁣}\left({\sigma }^{1-u}\delta B{\sigma }^u\delta B\right)du\mathrm{.}$ It reduces to the ordinary variance when $B$ commutes with $G_{\mathrm{l}\mathrm{o}\mathrm{c}}$; no such reduction is asserted otherwise. These are exact finite-dimensional deductions for the constructed neutral cell. They imply no infinite-time relaxation, dc transport coefficient, or response law for arbitrary interacting reduced states.

The response variables are finite matrices and finite-time controls, not measured apparatus data. The exact Kubo expansion and the static derivative concern different preparations: the former fixes $\sigma$, while the latter re-normalizes the Gibbs family for each $\lambda$.