Unified Principle of Multiplicative Systems The First Principle, the Complete Axiomatic System, and the Second-Order Extreme-Value Readout

Abstract Complex-systems science has long lacked an axiomatic micro-generation framework: the generation mechanisms of the classical distributions (exponential, Gaussian, power-law, Lévy-stable, and so on) 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 central-limit channel, generating the Gaussian by additive light-tailed aggregation), Channel B (the generalised central-limit channel, generating the Lévy-stable family by additive heavy-tailed aggregation), and Channel C (the Kesten multiplicative channel, generating the log-normal, the power law, and the Reed–Hughes truncated power law by state-dependent multiplicative aggregation). With core theorems and phenomenological propositions, a dual pillar of micro-generation and macro-steady-state is formed. The whole is organized around a single dividing line: whether the transition rate of the binary switch depends on the system’s own state. The independent side yields the reference structures—memoryless waiting times give the exponential, independent aggregation gives the Gaussian, and light-tailed parents give the corresponding extreme-value type; only the coupled side grows heavy tails. The four channels are accordingly cut 3 + 2 rather than laid out as four equal tiles. The horizontal (size-shape) axis is characterised by two parameters—a body slope and a cutoff scale—corresponding respectively to the tail-index side and the stopping-rate side. Taking the extreme is not a fifth channel but a second-order readout mounted on the horizontal axis and determined by the parent tail; it requires no measure of its own, since the shape parameter of the generalized extreme-value distribution is itself a scalar readout. Every assertion here carries an explicit boundary. The bridging theorem is given two failure paths (a shallow barrier makes the stopping time non-exponential; parameter drift through a fold likewise does). The exponential form of the stopping rate is made decidable by two criteria. The extreme-value readout carries three guardrails: the Fréchet criterion is regular variation of the tail, not unboundedness of support; the maximum of a lattice parent converges to no non-degenerate limit; and for dependent sequences the type is unchanged while the effective sample size is discounted. A guardrail—shape does not adjudicate mechanism—runs through the whole. One honest negative result is stated in the body: a deterministic multiplicative rate yields a light-tailed negative binomial, not a genuine Kesten power-law tail; closing the heavy-tailed side of Channel C requires the multiplicative rate itself to fluctuate in time.