Time-Scaled Intertwining Cocycles and Identifiability of Multi-Semigroup Mixtures on Hilbert Operator Networks

We prove that a network of dissipative semigroups ${\mathcal{S}}_i(t)=e^{-t{A}_i}$ admits time-scaled cocycles $K_{ij}{\mathcal{S}}_j(t)={\mathcal{S}}_i({\lambda }_{ij}t)K_{ij}$, $K_{ik}=K_{ij}{K}_{jk}$, if and only if the renormalized generators $\{{\tau }_i{A}_i\}$ form a common isospectral class with matching eigenspace dimensions; the scaling factors are then rigid, ${\lambda }_{ij}={\tau }_i\mathrm{/}{\tau }_j$, and eigenspaces transport isomorphically across sectors. The operators $K_{ij}$ constitute parallel transport in a flat Hilbert bundle over the index network; flatness follows from the intertwining constraints, not assumed. The mixture observable $M(t)={\sum }_i{w}_i{\mathcal{B}}_0{K}_{0i}{\mathcal{S}}_i(t){\psi }_i$ reduces under finite spectral support to a structured exponential sum. Under spectral separation, sector tags are uniquely recoverable; under eigenspace observability, active state components are determined. Finite-window exact reconstruction holds from $2L$ samples. The stability bound $\mathrm{\parallel }\widehat{\mathrm{\Theta }}-{\mathrm{\Theta }}_{\ast }{\mathrm{\parallel }}_{\mathcal{X}}\le C_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}{\kappa }_{\mathrm{e}\mathrm{x}\mathrm{p}}\epsilon$ holds with constants explicit in the spectral geometry and observability of the network.