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

The macroscopic phenomenological equations for open systems—the income-and-expenditure master equation, the free-energy gradient flow, and the quasi-potential landscape—have been proposed as phenomenological postulates, without a rigorous connection to the underlying microscopic stochastic dynamics. This paper supplies that connection. Starting from a minimal density-dependent random axiom (A0) and passing through a triple limit (thermodynamic limit, temporal coarse-graining, slow-manifold reduction), we give the microscopic identity of each macroscopic ingredient and verify every link by exact stochastic simulation (the Gillespie algorithm, not tau-leaping). The results organize around one principle: macroscopic nonuniformity arises when the microscopic transition rate depends on the collective state, and this state-dependence has two distinct facets, each with its own microscopic identity. Horizontal axis (size distribution): a single unit's stationary size is negative binomial, whose shape parameter equals the ratio of the additive to the multiplicative birth rate, r = g0/g1—the microscopic identity of the tail exponent. When r < 1 the distribution is a truncated power law (a power-law body with an exponential cutoff), not a genuine heavy tail; its extreme-value type is therefore Gumbel, and a genuine Fréchet extreme requires an untruncated power-law parent. Vertical axis (activity dynamics): the activity equation and the master equation rest on a second density-dependence—an activation rate depending on the active fraction. The ceiling factor (1 − Φ) is not imposed but forced by the law of large numbers; the macroscopic drift equals the difference of microscopic birth and death rates, the fluctuation is of order N^(−1/2), and bistability requires superlinear positive feedback with the explicit condition β > 4γ. If A0 is given space, the screening length has the identity ℓ² = D_hop/|f′|. We are explicit about scope: the independent limit of A0 (no multiplicative coupling) yields only the light-tailed floor; heavy tails and activity dynamics are consequences of the two explicit couplings, not of an independent substrate. Finally, an i.i.d. sequence has two genuine limit operations—summation (Lévy–Khintchine) and taking the extreme (Fisher–Tippett–Gnedenko)—with multiplication reducing to summation via the logarithm (Kesten) and the memoryless exponential as the no-combination case; the extreme-value channel is a second-order axis whose type is set by the parent tail. All quantitative indicators (KS distance, relative error, goodness of fit) are reported throughout.