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. This version rebuilds the classification and corrects four load-bearing arguments. The taxonomy is rebuilt from “four channels” to two coordinates, two readings, and one mixture. The previous version classified channels by the four structural combinations of the affine recursion X₍+₁ = bₙ·Xₙ + aₙ, and two of its four rows do not survive inspection. Setting bₙ ≡ 0 gives X₍+₁ = aₙ, which is the sequence itself and not a sum, so no central limit theorem applies to it; a sum requires bₙ ≡ 1. And setting bₙ constant with aₙ = 0 gives a deterministic trajectory, not a distribution, so it cannot be the generator of the exponential law, which comes from the memoryless waiting time. The recursion is not the classifier; it is the object that carries two coordinates at once, its additive term being a shift in the original coordinate and its multiplicative term a shift in the logarithmic one. Rebuilt on the two genuine degrees of freedom, the coordinate and the reading, the table acquires for the first time a cell for pure Pareto: the constant-hazard zero of the logarithmic coordinate, on exactly the same footing as the exponential in the original coordinate. Gaussian and Lévy become two parameter values of one stable family rather than two parallel channels, and the log-stable cell is named. Correction one: the dividing line is redrawn on dynamical response, not on shape. The previous claim that exogenously variable rates “can only transcribe a heavy tail that is already there” is false, and the counterexample is elementary: mixing an exponential law over a Gamma-distributed rate yields a Lomax law whose tail index equals the Gamma shape parameter, while both the component and the mixing law are light-tailed. Numerically, with shape 1. 5 and four million samples, the survival function matches the closed form across three decades and a Hill estimate returns 1. 501. The line that survives is stated in terms of response to work: an exogenously frozen departure has an identically vanishing derivative with respect to work, and only endogenous state dependence makes the departure a function of work. Three distinct routes to a heavy tail are then counted separately: the multiplier generating one, a heavy-tailed additive term transmitting one, and exogenous mixing producing one from light ingredients. Correction two: the argument for the rarity of power laws is reversed and rewritten. The window in which the tail index falls below one has width exactly equal to the ergodicity gap, so “width exactly the gap” is an identity and not an explanation; and since the gap grows with the volatility of the multiplier, a larger gap makes the window wider, not narrower. Within the whole stationary region power laws are the rule, with tail indices between one and three obtainable everywhere; what is rare is the sub-unit regime, and its exact content is that the stationary law still exists but its first moment diverges. Correction three: the bridging theorem outputs a pure Pareto law with no truncation. Its own algebra gives an exact Pareto law on the half-line above the initial value, since the stopping time has no upper bound. Four places that described the output as a truncated power law are corrected, the truncation is re-attributed to the cutoff scale on the stationary side, and the explanation of a bent head is replaced by a three-candidate discriminant table whose candidates have mutually incompatible signatures. Correction four: completeness is split into two located burdens, and one claimed reason is withdrawn. For readings, a representation theorem for associative aggregation gives exactly two classes, one conjugate to addition and one idempotent, with the idempotent class demoted for a structural reason: taking the maximum commutes with any strictly increasing coordinate transformation, so it produces no new coordinate and has nowhere to sit in a table whose rows are coordinates. For coordinates, completeness is inherited from the coupling dichotomy rather than conjectured separately. The genuine open door is neither a third operation nor a third distribution but the independence premise: replacing classical by free independence, the same addition yields the semicircle law. Separately, the assertion that the stopping-time exponent lies in the interval above one is withdrawn as unsupported, and the disjointness of the two power-law exponents is re-derived by a conversion argument that needs no interval claim at all. Two further items are recorded. A list of generators for power laws is supplied, with a final column giving where each route relaxes to when the work is withdrawn, which turns the guardrail that shape does not adjudicate mechanism from a slogan into a usable instrument; self-organized criticality is re-attributed within that list to critical branching, as a generator parallel to the multiplicative one rather than a special case of it. And the microscopic closure of the heavy-tailed side, previously the leading open task, remains closed by the companion microscopic paper through a random multiplier.
Qinfu Li (Sat,) studied this question.