Complex-systems science has long lacked an axiomatic micro-generation framework: the generation mechanisms of the classical distributions (exponential, Gaussian, power-law, Lévy-stable, etc.) are mostly modelled independently, making a unified theory hard to assemble. The unified principle of multiplicative systems proposed here builds a two-layer "source–differentiation" axiomatic system, taking the binary random process as the source (the smallest unit of information theory and the simplest prototype of probability algebra) and the exponential law as the co-located endogenous distribution of that source. Four independent generation channels realise a unified traceability of the classical distributions: Channel E (the fundamental exponential channel, the real-world expansion of the exponential/geometric family), Channel A (the CLT channel, generating the Gaussian by additive light-tailed aggregation), Channel B (the GCLT channel, generating the Lévy-stable family by additive heavy-tailed aggregation), and Channel C (the Kesten multiplicative channel, generating log-normal, power-law and Reed–Hughes truncated power-law by state-dependent multiplicative aggregation). With seven core theorems and three phenomenological propositions, a dual pillar of "micro-generation and macro-steady-state" is formed. This third version makes two additions. First, it makes explicit that the four tail-shape channels are complemented by a second, orthogonal state coordinate — the dispersion axis (Fano factor F ≡ σ²/μ) — on which the binomial, Poisson and negative-binomial families occupy definite positions (under-dispersed, the F = 1 zero point, and over-dispersed, respectively). A single binary source thus generates not only four tail types but an entire dispersion coordinate, strengthening the source-layer first principle; and the Poisson zero point is correctly characterised by independent-increment (Poisson-process) structure rather than by maximum entropy, since the discrete maximum-entropy law at fixed mean is geometric, not Poisson. Second, it identifies the zero-noise backbone of the unified evolution equation as the cleanest expression of the exponential law as the neutral zero point between the additive and multiplicative directions. Version 3 adds a discrete dispersion axis orthogonal to the four-channel tail-type axis, which solves the problem that binomial and Poisson distributions had no matching coordinates in Version 2. It proposes the equivalence of dual zero-point limits: exponential interval observations and Poisson count observations are dual projections of the same memoryless process.
Qinfu Li (Sun,) studied this question.