We study rigidity phenomena for time-scaled intertwining families of dissipative semigroups \(\mathcal {S}_i(t)=e^{-tA_i}\) and prove that a network of bounded injective operators satisfying \(K_{ij}\mathcal {S}_j(t)=\mathcal {S}_i(\lambda _{ij}t)K_{ij}\) and \(K_{ik}=K_{ij}K_{jk}\) necessarily admits a multiplicative gauge representation \(\lambda _{ij}=\tau _i/\tau _j\) , if and only if the renormalized generators \(\{\tau _iA_i\}\) form a common isospectral class with matching eigenspace dimensions; in particular, eigenspaces are transported isomorphically across sectors. The operators \(K_{ij}\) define parallel transport in a flat Hilbert bundle over the index network, with flatness derived from the intertwining constraints rather than assumed. As an application, the mixture observable \(M(t)=\sum _i w_i\mathcal {B}_0K_{0i}\mathcal {S}_i(t)\psi _i\) reduces under finite spectral support to a structured exponential sum. Under spectral separation, the modal parameters are uniquely identifiable, with sector tags determined intrinsically by the operator spectra; under eigenspace observability, active state components are uniquely recovered. Finite-window exact reconstruction holds from 2L samples, and the stability bound \(\Vert \widehat{\Theta }-\Theta _*\Vert _{\mathcal {X}}\le C_{\text {stab}}\kappa _{\text {exp}}\varepsilon \) follows with constants explicitly controlled by the spectral geometry and observability of the network.
