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.V8 makes eleven corrections and five completions to V7. They fall into four groups, and each is argued where it occurs.The classification itself. The mixed coordinate introduced by V7 was admitted without the domain on which it exists, and this is the load-bearing change. A coordinate is a strictly monotone map, and the mixed map is strictly monotone only when the additive part of the rate does not exceed the multiplicative part; above that point the map turns back on itself near the origin and the constant-hazard construction has nothing to deliver. V7 knew this in one place — T4D remarks in passing that for a body-slope parameter above one the prefactor rises rather than falls — but the classification sections, the letter mapping, the transition-path table and the numerical ledger were all written as though the cell covered the whole negative-binomial family, and the ledger verified the cell at a shape of three, which lies inside the regime where the coordinate does not exist. The cell is now read on the open unit interval, and the remainder of the negative-binomial family is assigned to the microscopic route, where it is defined and needs no coordinate (Sections 3.1, 4.1, 4.3, 8, 9, Appendix B). Three further corrections follow from the same repair: the two limits of the mixed row are stated per side, because the classification side reaches pure Pareto while the microscopic side reaches the boundary of stationarity (Sections 4.1 and 9); the two realizations of the cell are named separately, because the lattice guardrail of T6A applies to one of them and not to the other (Sections 4.1 and 7); and the identification of the classification route with the microscopic route drops the word exactly, the two hazard rates agreeing to first order and differing at second (Section 8.1).Labels and registers. The generator list of the guardrail acquires stable labels M1 to M10, because the companion papers order the list differently and an ordinal citation in one resolves to a different row in another. The labels are M and not G: G1, G2 and G3 are the three framework-level guardrails, one of which is the guardrail whose operational form this very list is, so a G-label on the list would give one string two meanings inside one section (Sections 7 and 11.1). The three phenomenological propositions of Section 12 are relabelled PP1 to PP3, the labels P1 to P22 being reserved for the prediction register shared by the five papers; the collision was live rather than latent, since the dragon-king signature is discussed in this paper under its register label. And seven symbol collisions with the companion papers are settled in one place, together with the one theorem-label collision that is not a symbol (Section 4.3).Theorems. T3 restores a fifth condition, which V7 omitted while citing the explicit constant that requires it. T2 is restated as one family with two parameter values rather than as a disjunction corrected one sentence later. And T3B is new: the allocation exponent is put inside the law of the multiplier, so that the chain from allocation to tail index becomes a theorem with an assumption and a domain instead of an asserted link. It was first proved in the cross-scale companion, which states it as Theorem E; T3B is the canonical label across the five papers and the companion records the equivalence, so that one theorem carries one name wherever it is cited. Its domain is stated on the supremum of the chord slope rather than on the support of the fluctuation, the two coinciding only when every exponential moment is finite. Its corollary identifies the diminishing returns reported by the companion treatment of spatial gradients as the convexity of the tail index in the allocation exponent, which is why that result is not an artefact of one parameterization.The axiom system. This paper is one of three that share a classification apparatus, and a reader auditing any one of them should not have to assemble the inputs from the other two. Section 2.3 therefore states the shared axiom system in full — three axioms, two structural hypotheses, one technical hypothesis, two embeddings and three demoted axioms — in the same labels the microscopic companion uses, and marks which of them this paper actually loads.Statements and the ledger. One row of the unified-equation table is withdrawn, a zero-noise ordinary differential equation having no distribution to be the backbone of a cell. The Gamma law is given its two addresses on their two disjoint parameter ranges. And the numerical ledger is corrected in two places where V7’s numbers do not reproduce, and in two further places where the quantity was computed in a way that built the answer in — one of them the very row that verifies this paper’s own constraint against rank regression.
Qinfu Li (Wed,) studied this question.
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