A Rigorous Proof of θ_tol = 1° from the Axioms of Constraint Network Dynamics: Global Uniqueness, Dynamical Stability, Number-Theoretic Consistency, and Emergent Critical Phenomena in Chain Network Rupture

The Constraint Network dynamical system, defined by three axioms, rigorouslyproves the existence and uniqueness of emergent constants. In prior work,the precise values 1836 and 1837 were derived under the conditionθ_tol = 1°. This paper makes two central contributions. First, we provide arigorous, three-pillar proof that θ_tol = 1° itself is a necessaryconsequence of the axioms. Within the dynamical stability pillar, we derivean explicit lower bound for the sparseification rate using the sealed nodegeometry, strengthening the exclusion argument from a conditional hypothesisto a deterministic proof. Second, we report the discovery of a new criticalphenomenon—chain network rupture—that emerges naturally from the θ_tol = 1°dynamics. When two stable nodes connected by a bi-directional flow channelare gradually separated, the chain ruptures at a finite critical distancewith a universal power-law flux decay exponent β ≈ 0.5. The fluxfluctuations follow Poisson statistics, and the rupture threshold scaleswith node capacity via a power law with exponent γ ≈ 0.3. The convergence ofa rigorous microscopic proof of the fundamental tolerance angle with thediscovery of emergent macroscopic critical phenomena provides a comprehensivedemonstration of the predictive power of the Constraint Network framework.