A Reduction Obstruction for Cross-Sector Dynamics on a Shared Quadratic Carrier: Non-Identifiability, Projection Survival, and Apparent Dissipation

This paper studies what can and cannot be inferred about cross-sector dynamics when two heterogeneous systems project onto the same quadratic carrier (A^2+B^2=1). Using a block-operator framework, it shows that norm preservation forces the familiar skew-adjoint exchange structure
[
\mathcal G=
\begin{pmatrix}
\mathcal A & -\mathcal C^\dagger\
\mathcal C & \mathcal D
\end{pmatrix},
]
but leaves the cross-sector map (\mathcal C) otherwise unconstrained. The singular-value spectrum of (\mathcal C) provides the natural mode-resolved description of transmission, with zero singular values defining dark modes and nonzero singular values determining transmitted channels. The paper further characterizes reduced survival through observability of the pair ((\pi,\mathcal G)), gives an exact closure criterion for projected dynamics, and shows how compression of conservative evolution can produce apparent contraction and, after elimination of hidden degrees of freedom, non-Markovian memory.

The central result is a non-identifiability theorem: the shared carrier relation together with the reduced scalar functionals (F_Q=B) and (F_G=2AB) contains no information about the cross-sector operator (\mathcal C). Infinitely many non-unitarily-equivalent cross-operators, with different ranks, kernels, and singular-value spectra, reproduce the same reduced carrier observables. The double-angle relation is shown to be an elementary squaring or Bloch-sphere map and does not impose an intertwining condition between the sectors. An application audit of the Dirac-occupation and relativistic-kinematic reduction concludes that the reduction is too severe to determine, and in its present form too asymmetric in state-space structure even to host, a well-posed cross-sector coupling operator. The paper’s contribution is therefore delineative rather than constructive: shared reduced geometry constrains observables, but dynamical laws, transmission spectra, memory, and cross-sector operators require independent structure beyond the carrier.