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

The first version sorted phase transitions in complex systems into two classes by the dynamic scaling law followed during the transition: first-order exponential rapid transitions and second-order power-law slow transitions. The second version retracted that naming and supplied a three-layer classification. The present version makes four corrections and three completions to the second version’s skeleton; two of the corrections invert a verdict.The reason for retraction is retained and made more precise. The word order was carrying three distinct jobs — how the state switches, how far influence propagates, and which coordinate the threshold sits on — while the fourth thing the first version was actually about, namely whether one can react in time, had no name at all and could only be expressed by borrowing the word. On a class of real systems the four give mutually opposite verdicts.Correction one, the relation between the two potentials. The second version stated that a quartic truncation and the rate function agree to second order near a fixed point and that well depth lies beyond the Taylor radius. The quartic is the antiderivative of the deterministic drift with its sign reversed; for polynomial drift it is exact rather than a truncation of anything, and its quadratic coefficient is (c−B)/2, not c/2. The quasi-potential is the WKB potential of the jump process. The two share every critical point, but their curvatures at a fixed point differ by the exact factor 1/(Φ★·c) — 1.654 at the maintaining state and 4.464 at the barrier for the standard parameter set — so the claim of second-order agreement does not hold. The well depth is corrected accordingly to +0.0954, the two barrier quantities are separately named as an escape barrier of 0.0646 and a depth difference of 0.0954, their ratio is 2.21, and the Maxwell point is corrected to 0.747.Correction two, the three layers are not three orthogonal axes. By the two-coordinate decomposition of the empirical measure together with the generic classification of codimension-one bifurcations, exactly two axes are genuinely orthogonal: the coordinate and the thermodynamic. The spatial decay type is locked to the thermodynamic layer at a transition point by the closed form ℓ = √(D/V_det″), so it is a conditional reading rather than a third axis; the long-range Ising model is the genuine exception, and its reason is that the gradient expansion presupposes a short-range kernel. The engineering layer, meanwhile, is not a third property of the system: it is a joint property of system and observer. This restatement makes the engineering layer stand more firmly, since it explains why that layer cannot be absorbed into the other two.Correction three, the denominator of the screening length and the shape of the profile. On the activity field the denominator is the critical margin rather than the spontaneous loss rate, and that slot is a role variable to be identified per application. Inside the screening length the profile is dominated by the algebraic denominator, so it is literally a truncated power law rather than an exponential everywhere; its inner-segment exponent depends on dimension, and in two dimensions the inner segment is logarithmic and not a power law at all.Correction four, the two layers have two intersections rather than one. The first is a vanishing second derivative of the potential, equivalent to a diverging screening length. The second is the Maxwell point, where the travelling-front velocity of a bistable reaction term vanishes and changes sign; it predicts a sign rather than a correlation.Completion one, the engineering criterion acquires a closed-form critical tail index. From the lower bound on identification delay together with the closed form of distinguishability, the engineering-jump criterion reads: transition time less than the logarithm of the false-alarm tolerance divided by distinguishability, plus response time; and distinguishability equals the product of drift magnitude and tail index. Solving for the tail index yields a threshold in closed form. This preserves the old intuition that systems nearer the power-law end are more dangerous while replacing its grounding by the identification-delay law, so that it no longer depends on the retracted equation of power law with second order.Completion two, the engineering layer has two criteria rather than one. In time the question is whether one can react; in space it is whether failure can be isolated, given by the ratio of system size to screening length. The two are independent under passive diffusion and degrade together under multiplicative driving.Completion three, an absent slowing signal has five possible causes rather than three. Besides the supercritical case, the beyond-spinodal case, and genuine distance from criticality, a limit cycle and a resident critical state produce misjudgements in the opposite direction: both read a vanishing recovery rate permanently, and neither is approaching a transition. The dynamical script must therefore be confirmed before slowing is discussed at all.