From Binary Randomness to Macroscopic Emergence: A Microscopic Derivation of Nonuniformity Emergence

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.V7 makes six corrections and three completions to V6. They are listed here in brief; each is argued where it occurs.One. The relaxation kernel of an age-structured level is a transport kernel, not a Laplace transform, and the sign of prediction P-A reverses with it (Section 6.1). Two. The open-loop gain carries the squared active fraction and the sign branch, and its denominator is the total decay rate of the stock rather than the effective deactivation rate of the activity equation; the combined bistability parameter carries the sign branch as well (Section 9.3). Three. The exchange coefficient diverges at the fossil state, so the self-check that evaluates the margin there is restated as a limit (Section 9.4). Four. The four count cases and the three phase regions are not on the same logical level; within the capping case, under-dispersion is the signature of saturation and not of the fossil state, and a two-dimensional discrimination protocol is supplied (Section 9.1). Five. A truncated-power-law tail has a moment generating function finite in a neighbourhood of zero, not everywhere; the phrase all exponential moments finite is corrected in two places. Six. The cell at the intersection of the mixed coordinate and the non-aggregating reading is not empty; it holds the stationary law of Section 8.1, and the aggregation table is corrected in line with the classification companion (Section 13.1).The three completions are: the mixed second derivative that decides compounding against diminishing returns is given with the three parameterizations on which it was evaluated; the convergence order of the central-limit check is reported as a fitted exponent; and the numerical appendix states parameters, replicate counts and seeds for every entry.The first correction is the one that changes a conclusion rather than a statement, so it is worth one sentence here.A Laplace transform of a positive measure is log-convex, so the V6 kernel returned a positive second derivative of the log-recovery curve for every age profile whatever, and the prediction it produced was a property of the transform rather than of the mechanism. With the transport kernel, heterogeneity in age-independent rates convexifies while age-directed replacement concavifies, so one perturbation-recovery experiment now separates three mechanisms by three signs instead of distinguishing a positive value from zero.