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

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 provides a rigorous, three-pillar proof thatθ_tol = 1° itself is a necessary consequence of the axioms, thereby closingthe final link in the derivation chain from the axioms to the fundamentalconstants. Pillar I leverages the global uniqueness of the emergent constantto lock θ_tol at 1°. Pillar II provides an independent dynamical stabilityanalysis, deriving an explicit lower bound for the sparseification rate anda deterministic upper bound for the accretion rate, which together excludeall θ_tol ≠ 1° in three spatial dimensions. Pillar III adds a number-theoretic consistency condition—the gcd(306, 91) = 1 lock—that independentlyconfirms the value. With θ_tol = 1° rigorously established, the emergentconstant N_seal is uniquely determined as 1836, as shown in companion work.The complete derivation chain—axioms → unique θ_tol → unique N_seal—is thusclosed without adjustable parameters.