Three Layers in the Classification of Complex-System Phase Transitions Thermodynamic, Spatial, and Engineering Criteria

Abstract

Abstract The first version classified complex-system transitions into two kinds by the dynamical scaling law the transition itself follows. The second withdrew that naming and gave a three-layer classification. The third made five corrections and two completions, one of which rebuilt the argument under this paper's headline claim and another of which reversed the sign of a prediction. V5 makes one withdrawal, two completions and one recorded distinction. The withdrawal is the one this paper has been waiting for since V2, and it is the only entry that changes a verdict rather than a statement. The withdrawal. The upper-right cell is no longer an external input. V3 and V4 recorded that the two dichotomies do not establish that every cell of Table 2 is occupied, and that the jump-type cell on the size coordinate in particular is supplied from outside rather than generated by the theorems cited here, because the size-side evolution equation is linear in the ordered stock and therefore has exactly one fixed point for every parameter value. That ground has been removed. The causal chain this paper itself states in Section 3.4 — spatial gradient surplus raising the effective multiplicative gain, raising the allocation exponent, lowering the tail index, raising concentration — has concentration at both of its ends, so within the size coordinate it is a feedback and not a one-way link; written into the income term it is a gain factor, and the equation is no longer linear in the stock. The folds are then located by a tangency condition that contains no parameterization, and the cell is generated by T12 and T16 rather than imported. Section 3.2 is rewritten accordingly, and the note that this paper could not fill the cell is withdrawn. Completion one. The two interlayer intersections both acquire a second landing place, and the dissipation slot acquires its last identification (Sections 3.4 and 3.5). The correction V2 made to the denominator of the screening length — that it is the margin and not the spontaneous loss rate — was stated for the activity field. On the stock field the slot was the total decay rate, and only because the stock equation was linear; with the gain factor it is the curvature of a size-side potential, so the two fields now take the same form and the V2 correction becomes a rule across coordinates rather than a fix on one of them. The Maxwell point likewise: a bistable reaction term on the stock field carries a front whose velocity reverses where the two wells are equally deep, at a size-side value of 0.415424, so the front-reversal prediction has a second coordinate on which it can be run. It is recorded as an extension of the register entry it already owns, not as a new number. Completion two. The phase diagram carries two more boundaries, and they are inside T16 rather than outside it (Section 3.6). V4 removed the sign-flip surface from the boundary list, leaving two on the activity coordinate. V5 adds two on the size coordinate, the folds of the newly generated cell. Both are saddle-nodes, so the completeness of T16 is not touched, and this must be said explicitly, because two boundaries appearing after a completeness theorem invites the reading that the theorem has been refuted. The recorded distinction. The two size-side cells are occupied by objects of different kinds, and V5 records the question rather than resolving it (Section 3.2). The upper-right cell is now a bifurcation of the fixed-point set of a scalar equation. The lower-right cell — the form transition, in which the cutoff scale diverges and the tail passes from exponential to power law — is a change in the shape of a stationary law at a fixed point that does not itself collide with anything. Whether T12 classifies the second at all is a question this paper is the right place to ask and is not the place to answer alone; both readings and what would settle them are stated where the table is. What V4 settled, retained here unchanged. V4 made five corrections and three completions to V3. All five come from reading the five papers of this set together rather than one at a time, and the first is the one that would have cost the most, because it is a defect in the instrument the whole set uses to avoid double counting. One. The prediction labels collided with the shared register, and two of them were duplicates. V3 numbered its six predictions E1 to E6 and stated that they did not occupy the register shared with the companion papers. They did: the label range E1 to E6 is already used in that register as the old numbering of the cross-scale companion's six predictions, so a reader following the old numbers maps E4 to two different predictions. Worse, two of V3's entries are word-for-word duplicates of register entries — its recovery-curve shape criterion is P1 and its Maxwell-point front reversal is P16 — so the register was double-counting exactly the two predictions it most needs to keep single. The six are folded into the shared register (Section 8). Two. The sign-flip surface of the ordered stock is not a transition boundary, and this paper is where that has to be said. The framework carries a phase diagram on which the surface where the sign branch of the ordered stock changes sign is listed as one of three boundaries. By the criterion of Section 3.1 it is not one: no structure of the fixed-point set changes there, and nothing observable crosses it discontinuously. What jumps is a signed magnitude packaged as a sign times a modulus. A paper whose subject is the classification of transitions cannot leave a non-transition on the boundary list (Section 3.6). *Three. The mixed-coordinate row of the generator list said equivalent to where only first-order agreement holds, and carried no domain.* The classification companion delivers a survival function of the form of a truncated power law; the density-dependent birth–death chain delivers a mass function of the same algebraic form. Those are different laws whose hazard rates agree to first order and separate at second. And the mixed coordinate is a strictly monotone map only for a body-slope parameter at or below one, so the cell exists on the open unit interval (Section 6, Table 5). Four. The generator list is cited by stable labels, and its first row is a destination rather than a generator. The five papers order that list differently, so an ordinal citation in one resolves to a different row in another. The rows carry the labels M1 to M10 in all five, and the letter is M and not G because G1, G2 and G3 are the three framework-level guardrails of the companion papers, one of which is the guardrail whose operational form this very list is; a G-label on the list would give one string two meanings inside one section. M1, the constant-hazard zero of the logarithmic coordinate, does not itself produce a power law; what delivers a system to it is M3, M4 or M5, so its relaxation behaviour is whatever its anchor's is (Section 6). Five. The five causes of a missing slowing signal had no common cause stated, and the one carried downstream is wrong. The downstream text gives the common cause as the system is already past the disappearance of a fixed point, which does not hold for the supercritical side of a transcritical bifurcation, where the fixed point has not disappeared but exchanged stability — which is precisely why that case is reversible. The common cause is that the system is not in the neighbourhood of a bifurcation point (Section 7). The four completions are these. The Maxwell point is quoted unrounded, because the prediction attached to it is a sign change located at that point and rounding erases the precision the prediction is about. Symbol and label conventions are settled across the five papers: the generator labels above, the superscript that separates an effective recovery rate from a critical margin, the subscripted screening length, and the canonical name of the allocation-exponent theorem, which this paper uses in Section 4.4 and which is cited as T3B throughout the set. The numerical ledger's screening-length sweep is put on the same parameter values as the companion papers' so that the two can be cross-checked without a reader suspecting a discrepancy, and the sharp-replacement kernel is given its closed form together with the delay dependence that closed form carries. And a reading convention is added to the ledger: where an entry reports a distance, a relative deviation or a fitted exponent rather than a p-value, the figure is an effect size and not the outcome of a test. What V3 got right is kept unchanged: the engineering layer as a joint property of system and observer, its two criteria, the closed-form critical tail index, the five causes of a missing slowing signal, and the corrections to the two potentials and to the denominator of the screening length.