Abstract: The macroscopic phenomenological apparatus of open flow-through systems — an income-minus-expenditure master equation, a gradient-flow relaxation, and a quasi-potential landscape — is usually posed as a set of postulates. This paper assembles that apparatus into a diagnostic framework and draws its boundary of validity. The organizing proposition is retained as a first principle for open systems, the Principle of Nonuniformity: the state of an open system departs structurally from uniformity along both a cross-sectional and a temporal axis, a departure supplied continuously by work and paid for by non-negative internal entropy production. V13 makes fifteen corrections and five completions to V12. They arrived in two rounds — the first from an internal audit, the second from a cross-check against the phase-transition classification and the cross-scale synchronization treatment — and are listed here in one sequence, since only V12 was ever deposited. Two of the corrections repair an internal inconsistency in the framework’s own statement of what it measures, and two more change a definition and a prediction. Each is argued where it occurs. One. Two different baselines were called the zero of one quantity. Section 2 defined structure as departure from the constrained maximum-entropy baseline, while Section 3 defines the stock as departure from the passive baseline. For a coupled system these are different laws, so the framework had two zeros for the quantity it claims to carry. The two offices are separated: the maximum-entropy baseline defines the domain, that is, when structure exists at all; the passive baseline defines the measured quantity. The two coincide on the uncoupled side and only there (Sections 2. 1 and 3. 4). Two. The mixed coordinate carries a domain, and its identification with the stationary size law is asymptotic rather than exact. The map φ (x) = x/sc + (1 − r) ·ln x is strictly monotone only for r ≤ 1, so the cell exists on r ∈ (0, 1). And the caption of V12’s Table 13 stated that requiring a constant hazard in that coordinate gives a survival function which is exactly the law Section 4. 7 derives — but Section 4. 7 derives a mass function. A law whose survival function has that form and a law whose density has it are different laws; the two hazard rates agree to first order and separate at second (Sections 4. 7 and 10. 2). Three. The closed form of the sharp-replacement kernel was printed at zero delay. The second derivative of the log-recovery curve under a sharp threshold is −λ²e (−λ (s−t) ) / (1−e (−λ (s−t) ) ) ², which depends on the delay; V12 printed the expression with the delay removed and gave no numbers beside it (Section 7. 5). Four. The open-loop gain and its value at full ignition are written apart. V12 defined the gain as a function of the active fraction and then used the same symbol for the pure parameter combination in the same paragraph (Section 4. 4). Five. The count of independent blocks is dimension-dependent. V12 wrote that the ratio of system size to screening length is the number of roughly independent blocks; in d dimensions that number is the d-th power of the ratio. The paper is careful about dimension in Section 8. 2 and was not here (Section 10. 4). Six. The three-coefficient regression is exactly identified, not over-determined. Three regression coefficients determining three physical quantities is a bijection. Over-determination is available, but only if the exchange coefficient is measured elsewhere, and that qualification is now stated (Section 7. 3). Seven. The minimum maintenance power cannot be evaluated from an exchange coefficient measured at the same operating point. Substituting the stationary relation for the coefficient into the expression returns the applied work rate identically, because the two are one equation read twice. V12 recorded only the weaker circularity through the active fraction (Section 4. 5). Eight. A1″ was used without definition. The uniqueness of the coordinate dichotomy was said to rest on A1″, a label this paper never defined. The axiom system is now stated by reference, with every label the body uses (Section 2. 4). Nine. The body carried no citations. V12 listed thirty-four references and cited none of them inline. Every borrowed result is now attributed at the point of use. Ten. Table 2 was called a reconciliation of five scalars inside a section titled three structural quantities. The canonical reconciliation carries six scalars, the sixth being the ergodicity gap of the temporal treatment; the scope of this paper’s table is now stated against it (Section 3. 1). Eleven. The register of predictions and the generator list are cited by the shared labels. P19 enters from the microscopic companion, P17 and P18 are flagged as numbers issued after the shared register was fixed, and the generator list is cited by stable labels rather than by ordinal, since the papers order it differently (Sections 11. 1 and 11. 2). Those labels are M1 to M10 and not G1 to G10: G1, G2 and G3 are the three framework-level guardrails of Section 11. 1, and G1 is precisely the guardrail whose operational form that list is, so a G-label on the list gives one string two meanings inside one section. Twelve. The sign branch is defined on a derivative, not on a difference of two scalars. The branch label had been the sign of the difference between two concentration readings. That definition needs a monotonicity condition to be well behaved, and the condition cannot supply what it is asked to supply: the concentration coordinate is a scalar functional whose level sets are not points, so two equal concentration readings do not imply two equal laws, and the divergence is then non-zero while the label flips. Attaching the condition rescues the definition only by pledging its well-posedness to an unproved empirical property of the branch. The label is redefined as the sign of the derivative of the concentration reading along the driving branch; the difference reading becomes an estimator with a stated validity condition (Section 3. 4). Thirteen. The sign-flip surface is not a transition boundary, and nothing observable crosses it discontinuously. This follows from the redefinition together with the criterion of Section 9. 1, and it corrects a listing that appears downstream, where that surface is carried as one of three boundaries of the phase diagram. No structure of the fixed-point set changes there. The signed quantity jumps because a signed magnitude was packaged as a sign times a modulus; the law itself, the active fraction, the concentration coordinate and the divergence are all continuous (Sections 3. 4 and 9. 1). Fourteen. The sub-additivity of two pushers has two readings, and only the weaker one had been carried. The result had been reported as an evaluation of the mixed second derivative on three parameterizations in which the pushers act on different parameters of the multiplier, with the instruction that the parameterization be named. That is right for that case. But the causal chain of Sections 8. 4 and 8. 5 routes both pushers through the same allocation exponent, and in that case sub-additivity is the convexity of the tail index in that exponent and holds for every law of the standardized fluctuation. Prediction P19 is stated in two clauses, and the theorem the chain needs is stated where Section 4. 7 uses it rather than left to the companions (Sections 4. 7, 8. 4 and 11. 2). Fifteen. The prediction register is extended to twenty-two entries and three statuses are recorded. The phase-transition companion’s engineering criterion, its concentration-to-detection-delay prediction and its closed-form critical tail index enter as P20, P21 and P22; two of that companion’s own labels are word-for-word duplicates of P1 and P16 and are not separately numbered; and the cross-scale companion has merged one prediction into another and recorded a third as an auxiliary, both of which belong in the register rather than being inferred from an absence (Section 11. 2). The five completions are: a compact statement of the axiom system, so that a paper which cites axiom labels defines them; the concentration coordinate’s closed form on a Pareto tail, verified against quadrature and given a ledger entry with values; the Fano sweep stated at the eight coefficients it was measured at, of which V12 reported five and without coefficients; the allocation-exponent theorem, with its assumption and its domain, stated at the point of use in Section 4. 7, so that a paper which relies on the chain from allocation exponent to tail index contains the statement it relies on; and three symbol conventions settled across the five papers (Appendix A).
Qinfu Li (Wed,) studied this question.
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