Abstract. The macroscopic phenomenological equations of open systems — an income-minus-expenditure master equation, a gradient-flow relaxation, and a quasi-potential landscape — are usually posed as postulates. This paper supplies the microscopic link. Starting from a minimal density-dependent stochastic axiom A0 together with a marked extension A0′ and a coupling axiom A1′, and through a triple limit, every macroscopic ingredient acquires a microscopic identity, verified link by link against exact stochastic simulation and, where a reversible chain permits it, against exact closed forms. V8 makes twelve corrections and six completions to V7. They fall into four groups, and each is argued where it occurs. The axiom system, which is the completion that matters. V7 stated three axioms and named its hypotheses, but it did not record which of the framework’s earlier axioms had been demoted to theorems, nor list the theorems the derivation chain actually establishes, nor say which maturity layer each sits in. A reader auditing the microscopic inputs therefore had to reconstruct that list from the body. Section 2 now states it: three axioms, two structural hypotheses, one technical hypothesis, two embeddings, three demoted axioms, and a register of sixteen theorems graded by layer. The single load-bearing phenomenological item is identified as the master equation itself, and everything downstream that inherits from it is marked as inheriting its status; in particular the closure of the margin channels in the companion paper is inherited from that reduction and not from the axioms. Two of V7’s three hypothesis labels are split, because one label carried three different things: analyticity is a hypothesis of the dichotomy theorem and not an embedding, and the two coarse-graining windows are taken over two different processes, so a criticism of one does not touch the other (Sections 2. 4 and 2. 5). Two corrections that change what a reader may infer from this paper’s own numbers. The sharp-threshold form of the relaxation kernel is delay-dependent, and V7 printed its value at zero delay as though it were the formula, setting it beside three numbers computed from the delay-dependent expression; the numbers were right and the formula was not, which is the worst of the two arrangements, since a reader who trusts the formula reproduces neither (Section 6. 1). And Section 2. 7 requires every convergence claim to be reported as a fitted exponent with its interval; V7 reported two exponents without intervals and drew a conclusion from one of them, that the terminal-spread exponent is not one half. Computing the interval from V7’s own four sizes gives −0. 810 to −0. 377, which contains one half, so four sizes do not resolve the difference and the conclusion is withdrawn (Section 4. 1). Definition, labels and register. The sign branch is redefined as the sign of the derivative of the concentration reading along the driving branch, rather than as the sign of a difference between two concentration readings. The difference form needed a monotonicity condition to be well behaved, and that condition cannot supply what it was 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 patch that has the divergence and the label vanish together therefore fails. The derivative form is well defined wherever that derivative exists and does not vanish, needs no domain condition, and says directly what the label is for; the difference form is demoted to an estimator with a stated validity condition (Section 3. 2). A lemma is added recording what the label does and does not do, and one consequence is drawn: the surface on which the label changes sign is not a phase boundary, since no structure of the fixed-point set changes there and the law itself, the active fraction, the concentration coordinate and the divergence are all continuous across it (Sections 3. 2 and 11). The sub-additivity of two pushers is separated into two readings, since the framework’s own causal chain routes both through one allocation exponent, where sub-additivity is a convexity and holds for every law of the standardized fluctuation, while the three parameterizations evaluated in Section 12. 2 have the pushers acting on different parameters, where the sign of the mixed second derivative is not universal; the allocation-exponent theorem, its assumption and its domain are stated at the point of use so that a paper which relies on that chain contains the statement it relies on (Section 12. 2). The prediction register shared by the five papers is extended to P22 and three statuses are recorded rather than left to be inferred from an absence (Section 14. 3). Four symbol collisions with the companion papers are settled: the cumulant generating function of the standardized fluctuation is written K_ζ, the inter-layer coupling coefficients are written Aₐb and Aba rather than with the letter reserved for the tail index, the two-layer recovery rates carry the superscript that separates an effective recovery rate from a critical margin, and the generator list is labelled M1 to M10 rather than G1 to G10, because G1, G2 and G3 are the three framework-level guardrails and one of them is the guardrail whose operational form that list is (Sections 14. 1 and 14. 2, Appendix A). Three corrections that a reader recomputing the paper would hit. The simulation check of the quasi-stationary variance was placed immediately after the second parameter set’s table while having been run on the standard set, so that a confirmation appeared, to anyone who recomputed the table, to refute by forty percent the exact figures it confirms; Appendix C now names the set and lists the exact values the runs reproduce, and records that the second set’s table carries no simulation. The domain of the allocation-exponent theorem is stated on the supremum of the chord slope rather than on the support of the standardized fluctuation, the two coinciding only when every exponential moment is finite; written on the support alone the bound admits values of the allocation exponent at which no positive root exists. And one entry of the summary table rounded a fourth-decimal figure the wrong way, disagreeing with the table it summarizes (Appendix C, Sections 12. 2 and 15. 1). Corrections of statement. The mixed cell of the aggregation table acquires the domain on which its coordinate exists, and its identification with the stationary size law of Section 8. 1 is first-order in the hazard rather than exact, a survival function of that algebraic form and a density of that algebraic form not being the same law. The three parameterizations of the mixed second derivative, named without numbers in V7, are given their base points, their tail indices, both first derivatives and the mixed derivative. And two ledger entries are brought into line with the companion papers, the detailed-balance agreement being reported at the same precision in all three and the ratio at the smallest system size being quoted consistently in the summary and in the appendix.
Qinfu Li (Wed,) studied this question.