Deduction and Reduction of the Ternary‑System Equations: From Three Meta‑Axioms to Dissipative Structures, Allometric Scaling and Autopoiesis Theory (V24‑rev01)《三元方程的推演还原:从三公理到耗散结构、异速生长律与自创生理论》(V24‑rev01)

Abstract

The ternary‑system equations serve as the unified standardized mathematical carrier for three indigenous foundational meta‑axioms: the Closed‑Loop Axiom, the Scale‑Matching Axiom, and the Physical‑Bottom‑Line Axiom. The logical self‑consistency and global universality of a novel foundational axiom‑system can be rigorously assessed via the deduction‑reduction method: if the globally‑applicable ternary‑system equations can be purely mathematically reduced to well‑established classical foundational theories under prescribed mathematical boundary constraints without empirical fitting parameters, the axiom‑system is demonstrated to be compatible with existing frameworks of physics, biology and systems science. Conversely, failure to achieve such fitting‑free mathematical reduction indicates irreconcilable logical flaws within the axioms. Based on two standard models — the global ternary time‑delay reaction‑diffusion equation suite and the discrete Euler iteration equation suite (employing the revised steady‑state solution and the axiom‑layer self‑repair term from Paper V23) — this paper hierarchically imposes boundary constraints and carries out formal deduction‑reduction proofs for eight classical models. Beyond the original three interdisciplinary landmark theories (Prigogine’s non‑equilibrium dissipative‑structure theory, Kleiber’s 3/4 allometric‑scaling power law, and the Maturana‑Varela autopoiesis theory), the v2 iterative edition further performs formal reductions for five additional classical models of physics and systems science: Newtonian classical mechanics (including Hamiltonian canonical equations), the Einstein field‑equations of general relativity, the thermodynamic law of entropy increase for isolated systems, the Logistic population‑growth equation, and the Lotka‑Volterra predator‑prey equations. A complete deduction‑reduction spectrum spanning physics‑biology‑ecology‑cosmology is thereby constructed. Drawing on the Closed‑Loop Axiom, this work reconstructs Prigogine’s non‑equilibrium dissipative‑structure theory and derives approximate analytical solutions for steady‑state order‑degree as well as approximate Hopf‑bifurcation critical criteria, complementing the missing necessary‑and‑sufficient mathematical conditions for long‑term steady‑state persistence in the classical theory. Using the Scale‑Matching Axiom, a purely mathematical derivation of Kleiber’s 3/4 allometric‑scaling power law is presented. Topological analysis shows that fractal transport networks constitute the only evolutionary configuration for living systems satisfying the rigid scale‑stress boundary condition, and quantitative dynamics for ecosystem carrying‑capacity are further developed. Grounded in the Physical‑Bottom‑Line Axiom, autopoiesis theory is reformulated by means of closed survival manifolds in phase space. Drawing on Noether’s theorem, a candidate survival conservation quantity is defined, and quantitative criteria for critical disintegration across multi‑hierarchy living systems are established. On this basis, this paper implements projections of the underlying unified logic of the ternary meta‑axioms onto the frameworks of thermodynamics, quantum mechanics and relativistic cosmology. The covariant ternary‑system equations are used to interpret three long‑standing foundational puzzles: the anthropic window, quantum‑measurement fidelity, and local negative entropy in living systems. All newly‑added derivations in the v2 iterative edition strictly obey the three validity criteria of the deduction‑reduction method, with no empirical fitting parameters or ad‑hoc artificial assumptions. All algebraic and differential simplification steps are reproducible. Closed‑loop verification for the self‑consistency of the axiom‑system is accomplished at the level of formal deduction. system.