Unified Principle of Multiplicative Systems: Coordinates, Readings, 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 are mostly modelled independently, so that a unified account is hard to assemble. This paper builds a two-layer source-and-differentiation system, taking the binary random process as the source and the exponential law as the co-located endogenous distribution of that source, and then classifies the classical limit laws by a small number of structural choices rather than by their shapes. V7 makes five corrections and two completions to V6. They are listed here in brief; each is argued where it occurs. One. The table acquires a third coordinate. The cell at the intersection of the mixed coordinate and the non-aggregating reading was declared empty by construction; it is not empty, and what sits in it is the truncated power law that Section 8 derives and Section 11 lists as a generator (Sections 3. 1 and 4. 1). Two. The heading of the mixture row is corrected: the Kesten fixed point is a recursion, not a summation and not a stable-family limit (Section 4. 1). Three. The path count is reconciled at five, the reflected route from the log-normal cell to pure Pareto having been described in prose but omitted from the table (Section 9). Four. A truncated-power-law tail has a moment generating function finite in a neighbourhood of zero, not everywhere; the phrase all exponential moments finite is corrected in five places. Five. Two statements that depend on dimension and on convention: the inner-segment exponent of a screened profile is d − 2, coinciding with the large-distance prefactor exponent only at d = 3 and logarithmic at d = 2 (Table 6) ; and the preferential-attachment limit of one holds for the Simon parameterization read on the complementary cumulative distribution, the network model bottoming out at two (T5). The two completions are: the truncated-power-law cell is given its generating rule, its two parameters and its relation to the other cells, so that the third coordinate is a working part of the classification rather than a hole that has been filled; and a numerical ledger lists every quoted number with the parameter set that produces it. The first correction is the one that changes the classification rather than a statement, so it is worth one sentence here. A coordinate is fixed by a strictly monotone map, and nothing prevents that map from combining the identity and the logarithm. Taking φ (x) = x/sc + (1 − r) ·ln x and requiring a constant hazard gives the survival function x^ (r−1) ·e^ (−x/sc) — the power-law body with an exponential cutoff that is the most frequently used distribution in the framework and that had no cell in the framework’s own table. Refusing that cell on the ground that with no aggregation there is nothing to mix conflates the mixing of coordinates with the mixing of aggregation operations.