Three Layers of Phase-Transition Classification in Complex Systems Thermodynamic, Spatial, and Engineering Criteria

Abstract The first version sorted phase transitions in complex systems into two classes by the dynamic scaling law followed during the transition. The second version retracted that naming and supplied a three-layer classification. The third version corrects five items and completes two; one of the corrections rebuilds the argument on which the paper’s headline claim rests, and one reverses the sign of a prediction. The five corrections, in brief; each is argued where it occurs. One. Sections 2. 2 and 3. 5 of V2 asserted opposite things. §2. 2 held that at a transition point the continuous type necessarily carries a power-law spatial profile and the jump type an exponential one, and used that locking to demote the spatial layer to a conditional reading; §3. 5 then observed that the second half fails on the spinodal. §3. 5 is right: the screening length diverges wherever the margin vanishes, and the margin vanishes at a saddle-node exactly as at a transcritical. The demotion survives but its argument is rebuilt — the spatial layer is not a third axis because it is a reading of the margin, which is itself read from the thermodynamic layer, and not because the two bifurcation types carry different profiles (§2. 2, §3. 4, §3. 5). Two. The transition criterion is phrased on the structure of the fixed-point set rather than on losing a fixed point, since a transcritical bifurcation loses none; and the invariant-branch hypothesis is written into the completeness theorem, which in V2 was stated for generic families and immediately contradicted by its own footnote (§3. 1, §3. 2). Three. Two adjacent sentences of §3. 4 contradict each other. The exponent of the algebraic denominator is (d−1) /2 and belongs to the large-distance prefactor; the inner-segment exponent is d − 2. They coincide only at d = 3, so the profile inside the screening length is not “dominated by the denominator” except in three dimensions (§3. 4, Table 5). Four. The sign of prediction E4. The relaxation kernel of an age-structured stock is a transport kernel and not a Laplace transform, so age-directed replacement gives a concave log-recovery curve while heterogeneity in age-independent rates gives a convex one. V2 had both on the convex side (§6, E4). Five. Notation. V2 declared that distinguishability is written Id rather than D because diffusion and hopping coefficients each claim a D elsewhere, and then used a bare D for the diffusion coefficient throughout. The spatial coefficient is written Dₕop here, the screening length ℓₛcr, and a notation appendix is supplied, which V2 lacked (Appendix A). Two completions. The non-emptiness of the jump-type size cell is not established by the theorems this paper cites, and is recorded as an external input rather than as a consequence (Table 2). And a numerical ledger states the parameter set for every quoted number, including two figures V2 gave without one (Appendix B). What V2 got right is kept without change: the engineering layer as a joint property of system and observer, its two criteria, the closed-form critical tail index, the five causes of an absent slowing signal, and the corrections to the two potentials and to the denominator of the screening length.