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 proof that θ_tol = 1° itself isa necessary consequence of the axioms, thereby closing the final link in thederivation chain from the axioms to the fundamental constants. The proofproceeds by exhaustive elimination: the parameter space (0, π) ispartitioned into three mutually exclusive and collectively exhaustiveintervals—θ_tol < 1°, θ_tol = 1°, and θ_tol > 1°—and it is shown thatonly θ_tol = 1° is dynamically viable. The argument is organized into threecomplementary components. The core proof (Section 4) is a dynamicalstability analysis: an upper bound on the accretion rate derived fromphase-space geometry and a lower bound on the sparseification rate derivedfrom escape-cone geometry together exclude all θ_tol < 1°; the conflictbetween the packing and covering constraints excludes all θ_tol > 1°. Alogical bridge (Section 3) connects this uniqueness result to the numericalvalue N_seal = 1836 derived in companion work under θ_tol = 1°. Anumber-theoretic consistency verification (Section 5) confirms that thediscrete orbital parameters obtained in companion work are internallycoherent with θ_tol = 1°. These three components are not three independentproofs; their logical relationship is sequential and complementary. Withθ_tol = 1° rigorously established, the emergent constant N_seal is uniquelydetermined as 1836. The complete derivation chain—axioms → unique θ_tol →unique N_seal—is thus closed without adjustable parameters.