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 (thermodynamic limit, temporal coarse-graining, slow-manifold reduction), 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. This version makes one retraction and four structural corrections. The retraction concerns the reference measure of the load-bearing variable. Earlier versions took the ordered free energy to be the Kullback–Leibler divergence from an exponential baseline, converted to energy units. Section 8. 1 of the present paper proves that the stationary size law of a unit with a positive multiplicative rate is not exponential; under the linear birth–death specification used there it is negative binomial, but the load-bearing step is the negative statement alone. Hence a quantity referenced to the exponential is not a rate function on any coupled system this framework is about, and is not zero on such a system’s own stationary law. On the uncoupled special case it remains mathematically consistent, and that is the whole of its surviving domain. The paper contained the refutation of its own choice. The correct construction splits one quantity into three, distinguished only by which measure sits in the reference slot: a structural stock measured from the passive baseline (the law to which the system relaxes when driving is withdrawn), which is what the master equation carries; a displacement, which is the quasi-potential of the stationary distribution and enters the escape exponent; and a structure reading measured from the memoryless baseline, which locates the stationary law on the form spectrum and does not enter any equation. Correction one: the framework is made dimensionless, and the fluctuation temperature is confined to the appendix. The structural stock is a pure number. Work is converted by a single coefficient η̂ ≡ η/ (kB·T_*), of dimension inverse energy, which is directly measurable without knowing the temperature: η̂ = (λₜot + σ·fₛhock) ·|Uₛtr|/ (Φ★·W), all four factors on the right being independently observable. This yields a proposition worth stating in the abstract: every falsifiable prediction of this framework is free of the effective fluctuation temperature. The escape exponent is a logarithmic integral and is dimensionless by construction, so the temperature enters only the Kramers prefactor and the absolute calibration of the minimum maintenance power, and neither carries falsifiable content. Correction two: all three expenditures multiply the existing stock. Earlier versions of the present paper wrote the exogenous shock as an additive term, while the companion phenomenological paper had already corrected it to multiplicative form. Three independent reasons force the multiplicative form: non-negativity of a divergence, isomorphism with the activity equation, and the exactly vanishing diffusion coefficient at an absorbing state given by the microscopic reduction. The shock rate changes accordingly from a power to a rate. The two companion papers are thereby aligned on this point. Correction three: the potential is two objects, not one object at three truncation levels. The quartic is the antiderivative of the drift with its sign reversed, and for polynomial drift it is exact rather than a Taylor truncation; its quadratic coefficient is (c−B) /2, not c/2. The quasi-potential is the WKB potential of the jump process. The two share every critical point but their curvatures at a fixed point differ by the exact factor 1/ (Φ★·c), so they do not “agree to second order, ” and only the quasi-potential may be used for well depths, the Maxwell point, and escape rates. A useful identity falls out: the critical margin equals the second derivative of the deterministic potential at the maintaining state. Correction four: age structure is added, and with it the lowest-threshold prediction in the framework. Giving the stock one extra dimension of component age separates the two maintenance classes, which are indistinguishable on the activity coordinate alone. It also removes the principal obstacle to the relaxation-rate prediction: an exogenous shock imposes a common rate shift on every age mode, so the second derivative of the logarithm of the recovery curve is exactly invariant under that shift. The falsifiable statement therefore becomes “the log-recovery curve is convex for a high-turnover system and straight for a low-turnover one, ” which requires no control over the shock. Two further items are recorded. The heavy-tailed side of the multiplicative coordinate remains closed by a random multiplier, with the non-lattice condition now stated and the alternative route through a heavy-tailed additive term named. And the previously open task “quantitative map from potential to tail index” is withdrawn rather than carried forward: the structure reading is V-shaped in the tail index and therefore non-invertible, so the map does not exist; it is replaced by the implicit-function sensitivity.