Daopological Measures and Entropies

In Daopology: An Axiomatic System for the Speakable Dao, we established a complete system based on two axioms (the Dao-Name Axiom and the Six-Act Axiom), deriving 9 primitives, 6 Heng families, 6 Jiao families, 8 Cha families, 7 Que families, 10 Suan families, 21 He families, a 9-layer framework, and 4 transfer types—79 minimal expressive parameters in total. This paper is the quantitative companion to that system. We answer a question that was not systematically addressed before: How can we quantify the "importance," "dispersion," "structural influence," and "emergence potential" of a Daopological node (instrument, kernel, combination, or the whole system)? Starting from the act of "measurement" within the six acts, we construct a measure family and an entropy family, presenting five base measures, two entropies, one fused measure, nine signal dimensions, and a four-quadrant health diagnosis based on measure-entropy distributions. All quantitative indices are derived from the axiomatic system without additional assumptions. The core results are: Five base measures: μ_count, μ_structure, μ_compress, μ_emerge, μ_transfer; Two entropies: transferability entropy H_transfer, Daopological entropy H_D; Fused measure: μ_total(x) = μ_structure(x) · exp(H_transfer(x)); Nine signals: novelty, rarity, heat, burst, isolation, cross-domain, potential, structural influence, entropy contribution; Four-quadrant health diagnosis: classifying nodes into enriched, concentrated, dispersed, and impoverished types. The paper provides Daopology with a computable, comparable, diagnosable quantitative layer, upgrading the system from "structural description" to "structural measurement."