Structural Descent and Continuous Realization of Return Dynamics

This technical report studies the transport, continuous realization, and observation of a selected return law, together with obstructions to its realization in a restricted linear representation class. The principal result is an exact exclusion: no real four-dimensional matrix acting by congruence on symmetric matrices has characteristic polynomial λ¹⁰ − λ⁹ − 1. Any finite composition within the same representation class is also excluded. An explicit coefficient reduction and a polynomial identity certify this obstruction. For the constructive interface, a smooth four-coordinate chart expresses a logarithmic closure functional as a constant plus an independent residual coordinate, preserving full-state transport, finite composition, and equivariance. A concrete reciprocal-affine instruction with gains 2 and 1/2 satisfies all hypotheses of an attributed shear-flow realization theorem. An independent scalar factor supplies residual contraction while retaining physical height. A separate endpoint-matched analytic flow illustrates the construction without being identified with the upstream fluid trajectory. Established quotient and predictive-state methods characterize when observations support autonomous reduced dynamics. A supporting spectral argument constrains bounded observations of the canonical operator’s expanding Perron mode. The accompanying package includes analytical proofs, exact symbolic certificates, reproducible numerical examples, negative controls, and selected Lean-checked results with pinned dependencies and explicit evidence boundaries. The report distinguishes its target-specific exclusion and selected constructions from established mathematical tools. It does not derive a unique residual contraction constant, identify an unspecified physical system, or establish physical universality.