From Binary Randomness to Macroscopic Emergence A Microscopic Derivation of Nonuniformity Emergence, with the Load-Bearing Link Graded in Four Layers

Abstract

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. Where a link cannot be derived it is graded rather than asserted, and in this version the one link that carries the chain is graded in four layers, of which two are rigorous. V10 makes one regrading, two corrections and four completions. They are stated first, because they are what a reader of V9 needs in order to know what is new. The regrading, which is the change that matters. V8 registered the master equation as the single load-bearing phenomenological item in the chain and recorded that its layer propagates to everything downstream; V9 left that registration untouched. It was too coarse in one direction and too pessimistic in another. Too coarse, because the income-minus-expenditure shape is not carried at all: splitting the generator about the passive baseline gives it as an identity, and what is carried is one step inside it, the linearity of the dissipation term. Too pessimistic, because that one step admits a one-sided bound: where the undriven semigroup satisfies a modified logarithmic Sobolev inequality, the true dissipation is not smaller than a linear function of the stock, so the master equation read at that rate is a rigorous upper bound on the stock and the minimum maintenance power acquires a lower bound passing through no phenomenological step at all. Section 2.5 accordingly grades T9 in four layers instead of one, and what propagates downstream is a one-sided inequality rather than an unlabelled reduction (new §5.5). Correction one. Three conclusions were said to rest on one counting fact, and one of the three is incompatible with another theorem of this paper. Section 7 recorded that the vanishing of the diffusion coefficient at an absorbing state forces the multiplicative loss, removes escape from the fossil state, and makes the exchange coefficient diverge there. The third cannot stand beside T6. If the exchange coefficient diverges as the reciprocal of the active fraction, the factor of the active fraction in the income term is cancelled exactly, the income does not vanish at the fossil state, and the fossil state is not a fixed point under positive driving. The two claims are exclusive. This paper keeps T6, which entails that the exchange coefficient is bounded and that its factoring into an efficiency and a thermal energy fails in a neighbourhood of the absorbing state. That is a negative result about one of this paper's own layer-B assumptions, and it carries more information than the limit-ordering repair it replaces (§3.3, §7.1, §9.4). Correction two. The rates of the master equation live on the divergence layer, and conversion to component turnover carries a factor of two. The stock is a Kullback–Leibler functional, and to second order about the passive baseline it is a quadratic form in the departure, so a departure decaying at the principal modal rate gives a stock decaying at twice that rate. The age-transport equation of Section 6, being linear in the density, carries the hazard itself. The two layers were not distinguished; R2 subtracts rates, so the factor does not cancel, and a renewal rate inferred from it is out by a factor of two unless the layer is declared. The correction is stated at the front because §6.3 has now shown what running one of these predictions actually costs, and a protocol defect that moves an inferred quantity by a factor of two is the most expensive kind (new §6.4). Completion one. The stock side acquires its noise, its boundary classification and its stationary law (new §7.2). The third of the three reasons for multiplicative expenditure is a statement about the diffusion coefficient and had been used only on the drift. Carried to the noise, it gives a square-root diffusion on the stock, whose behaviour at zero is settled by a Feller classification rather than by inspection of the drift: the two sign branches are separately invariant exactly when the income rate is at least the stock-side noise intensity. Below that the sign label can flip along a trajectory, which §3.2 says it cannot, so the condition is falsifiable. On the linear segment the stationary law is Gamma and the variance is closed. Completion two. The size coordinate acquires a nonlinear segment, with a fold condition that involves no parameterization (new §7.3). The causal chain of §12 has concentration at both of its ends: spatial gradient surplus raises the effective multiplicative gain, which raises the allocation exponent, which lowers the tail index, which raises concentration. Written into the income term this is a gain factor, and the stock equation ceases to be linear. The folds satisfy a tangency condition holding for any increasing gain factor; on the simplest two-parameter family the bistability condition, the fold positions and a size-side quasi-potential are all closed, and three exact identities follow, one of which contains no parameter at all. Completion three. The type of transition available on the size coordinate is settled, and it is settled by two theorems this paper already had (new T20, §7.4). T6 establishes that the zero of the stock is repelling and not invariant: the expenditures vanish there while the income does not. A coordinate with no invariant branch of fixed points has no structurally stable transcritical bifurcation, so on that coordinate only the saddle-node is of codimension one. Hence a size-coordinate transition that changes the fixed-point set is of jump type and cannot be of continuous type. The result requires no new machinery — it is T6 read against the codimension count — and it is the first statement in this paper that constrains a type rather than a location. Entered as MB-5. Completion four. The quasi-potential of §10 is identified as a minimized action, which supplies one prediction the deterministic reading cannot (§10.3). The quasi-potential this paper computes is the minimum of the Freidlin–Wentzell rate functional of the underlying stochastic dynamics, and the escape rate of §10.2 is its small-noise corollary; the two are one statement rather than two. The identification is a mature borrowed result, but it carries a consequence this paper has not drawn: since the coupled system of §5.4 is not a gradient flow, its most probable escape path is not the time reversal of its relaxation path, and the area the two enclose measures the circulation. Entered as MB-7. What V9 settled, retained here unchanged. V9 made two additions to V8, and one of them added a condition to a prediction. They are described in the two paragraphs that follow, because both remain in force and the first is what §6.4 above now qualifies. One prediction is executed on measured data. Until now every prediction of this set had been argued and none run. Section 6.3 executes P1, the three-sign criterion for the shape of a recovery curve, on measured duration cohorts drawn from seven monthly price series spanning 1990 to 2022, one daily series, and a four-channel electrophysiological recording; Appendix D gives the data ledger. The convex branch is confirmed on two markets and two sampling frequencies and is robust to the at-risk cutoff; the memoryless branch is confirmed within a homogeneous stratum; the concave branch is not resolved at the sample sizes available and is reported as unresolved. One cohort was assigned to the concave branch and came out convex, and the reason is the variance term of (7): pooling series of different volatility adds rate disorder across cohorts, which the framework says convexifies. That repair makes a further prediction carrying no fitted quantity — the pooled slope equals the within-series mean minus the across-series variance of the hazard — and it was checked at three band widths, the prediction falling inside the measured interval in each and the no-heterogeneity null falling outside it in each. The invariance of the criterion under a common rate shift, which is what allows it to be used without shock control, was tested directly and the largest displacement of the statistic was 1.12 standard errors. A methodological condition is added to P1 in consequence: fitting a continuous law to integer-recorded durations manufactures an aging signal, and on one memoryless cohort here it returns a decisive false verdict. The last completeness assertion is audited. Section 2.8 settles whether the layer index of the multi-layer companions is a third coordinate. It is not: no equation of the set transports a unit between layers, so the index carries no generator component and is a structural parameter, the state space of an L-layer system being the L-fold product of the single-layer space together with a coupling matrix. The resolution carries a checkable condition — layers whose membership can change do acquire a third coordinate — and that condition is stated rather than left implicit. What V8 settled, also retained. V8 made twelve corrections and six completions to V7, in four groups. They are described in the paragraphs that follow because a reader who has only this version should know what changed and why, and because two of them are conditions that travel with results Section 6.3 now uses. 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,