Algebraic extinction in acyclic compartment models

Unrefereed candidate research note. A positive four-state conservative quadratic model disproves a specified exponential-decay conjecture; the paper proves minimum compartment count, an arbitrary-multiplicity family, an exact phase boundary and a trajectory-dependent spectral criterion with aggregate logarithmic rate. Complete written proofs and finite exact replay are provided. No external human peer review, formal verification, empirical validation or historical-priority certification is claimed.