A Rigorous Proof of θₜol = 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 θₜol = 1°. This paper completes the final step of the entire theoretical edifice: the rigorous proof that θₜol = 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 Nₛeal without any assumption on the numerical value of θₜol. Second, an exhaustive dynamical stability analysis demonstrates that only θₜol = 1° permits a stable equilibrium between the competing forces of densification and sparseification. Third, a purely number-theoretic consistency condition—θₜol must divide gcd (306, 91) = 1—independently locks the value at 1°. The convergence of these three independent lines of reasoning transforms θₜol = 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, θₜol 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 θₜol = 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, θₜol = 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 θₜol = 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 Nₛeal, established by Theorem 7 independently of θₜol, combined with the precise calculation of Nₛeal = 1836 from Reference 2, directly forces θₜol = 1°. Pillar II (Section 4): An exhaustive dynamical stability analysis proves that among all possible values of θₜol, only 1° permits a stable equilibrium between the antagonistic forces of densification and sparseification. Pillar III (Section 5): A purely number-theoretic condition—θₜol 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 θₜol = 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 θₜol. Theorem 5 (Existence, Uniqueness, and Parity of the Emergent Constant). The aggregate number Nₛeal of a sealed node is even and uniquely determine