A Rigorous Proof of θ_tol = 1° from the Axioms of Constraint Network Dynamics: Global Uniqueness, Dynamical Stability, and Number-Theoretic Consistency Menggang Yu Independent Researcher Honggutan District, Nanchang, Jiangxi Province, China, 330100 Email: ymg198702@163.com ORCID: 0009-0004-1943-6776 Abstract The Constraint Network dynamical system, defined by three axioms, rigorously proves the existence and uniqueness of emergent constants. In prior work, the precise values 1836 and 1837 were derived under the constraint θ_tol = 1°. This paper completes the final step of the entire theoretical edifice: the rigorous proof that θ_tol = 1° itself is a necessary consequence of the axioms. The proof rests on three mutually independent pillars. First, Theorem 7 (Global Attractor) establishes the strict uniqueness of the emergent constant N_seal without any assumption on the numerical value of θ_tol. Second, an exhaustive dynamical stability analysis demonstrates that only θ_tol = 1° permits a stable equilibrium between the competing forces of densification and sparseification. Third, a purely number-theoretic consistency condition—θ_tol must divide gcd(306, 91) = 1—independently locks the value at 1°. The convergence of these three independent lines of reasoning transforms θ_tol = 1° from an input parameter into a mathematically inevitable output, thereby closing the complete derivation chain from the three axioms to the fundamental constants. Keywords: Constraint Network, emergent constant, global attractor, dynamical stability, number theory, θ_tol 1. Introduction The Constraint Network dynamical system is founded upon three axioms [1]: the universe consists of energy possessing direction and moving at light speed; encounters result in collision, with accretion occurring when post-collision directions are steered into a common main lobe by the local accretion zone; and the total aggregate number is strictly conserved. From these axioms, eight foundational theorems were rigorously proved in [1], establishing the existence, uniqueness, and parity of an emergent constant. Building on this foundation, the precise values 1836 and 1837 were analytically derived in [2] under the specific condition θ_tol = 1°. This derivation proceeded along two independent pathways—static geometric optimization and dynamic accretion overshoot—providing a powerful cross- verification of the result. However, in all prior work, θ_tol = 1° appeared as an externally imposed parameter. The ultimate promise of the theory—to derive all fundamental constants from the three axioms without any adjustable inputs—remained incomplete. This paper fulfills that promise. We present a rigorous, multi-faceted proof that θ_tol = 1° is not an input but an inevitable consequence of the axioms. The proof is structured into three mutually reinforcing pillars: Pillar I (Section 3): The global uniqueness of N_seal, established by Theorem 7 independently of θ_tol, combined with the precise calculation of N_seal = 1836 from Reference [2], directly forces θ_tol = 1°. Pillar II (Section 4): An exhaustive dynamical stability analysis proves that among all possible values of θ_tol, only 1° permits a stable equilibrium between the antagonistic forces of densification and sparseification. Pillar III (Section 5): A purely number-theoretic condition—θ_tol must divide gcd(306, 91) = 1—provides an independent mathematical lock, confirming the consistency of the entire discrete framework. The convergence of these three independent arguments establishes θ_tol = 1° with a degree of certainty that no single line of reasoning could achieve alone. 2. Axioms and Foundational Theorems Axiom 1 (Ontology and Motion). The universe consists of energy units, each characterized by a direction vector d ∈ S² and an aggregate number N ∈ ℤ⁺. All units move at constant speed c = 1. Axiom 2 (Interaction). When two units encounter, collision occurs. Equal portions rebound specularly. In unequal encounters, the residual portion of the larger unit continues along its original direction. Post-collision directions are steered by the local accretion zone toward its main lobe. If both are steered into the same main lobe, accretion occurs—the units merge and their aggregate numbers sum. Axiom 3 (Conservation). The total aggregate number is strictly conserved. From these axioms, Reference [1] proved the following theorems: Theorem 2 (Irreversibility of Accretion and Monotonicity). Accretion is irreversible. The maximum aggregate number is non-decreasing. Theorem 3 (Convergence to Steady State). The system converges to a steady state in finite time. Theorem 4 (Geometric Constraint). In steady state, the interior directions of a sealed node must cover the unit sphere S², with pairwise angular separation strictly greater than θ_tol. Theorem 5 (Existence, Uniqueness, and Parity of the Emergent Constant). The aggregate number N_seal of a sealed node is even and uniquely determine