From Binary Randomness to Macroscopic Emergence Microscopic Derivation and Numerical Verification of Nonuniformity Emergence

The macroscopic phenomenological equations of open systems—an income-minus-expenditure master equation, a free-energy gradient flow, a quasi-potential landscape—were previously posed as postulates, with no rigorous link to the underlying microscopic stochastic dynamics. This paper supplies that link and, relative to the preceding version (V2), tightens the axiomatic structure, the dynamical decomposition, and the honest boundaries. Starting from a minimal density-dependent stochastic axiom (A0), we first supply a necessary premise: A0′ (marked extension)—each binary switch carries a continuous mark. Once A0′ is granted, the “size coupling” and “activation coupling” that V2 stated as two parallel axioms reduce to two species of a single coupling axiom A1′: the transition rate depends either on the unit’s own marked share (the horizontal axis) or on the unmarked global active fraction (the vertical axis). The gain variables of the two projections are not inter-convertible—the branching offspring number satisfies P(b = 0) > 0, so E[ln b] diverges, whereas the multiplicative factor satisfies P(b = 0) = 0—yet both projections issue from the same microscopic object. Through a triple limit (thermodynamic limit, temporal coarse-graining, slow-manifold reduction), every macroscopic ingredient acquires a microscopic identity, verified link by link with exact stochastic simulation (Gillespie algorithm, not tau-leaping). Horizontal axis: the stationary size of a unit follows a negative binomial distribution whose shape parameter r = g₀/g₁ is the microscopic identity of the tail index; r < 1 yields a truncated power law (power-law body plus exponential cutoff), not a genuine heavy tail. Vertical axis: the stationary count of active units is likewise negative binomial—the same distribution appears once on each axis, produced by two different coupling species; the capping factor (1 − Φ) emerges under compulsion from the law of large numbers, and it is itself the existence condition for bistability, whose necessary and sufficient condition is given in purely parametric form.