Boundary Conditions on Mathematical Emergence: A TAME Framework Analysis of Factorization Domains

Michael Levin's Technological Approach to Mind Everywhere (TAME) framework proposes that cognitive-like properties emerge in systems satisfying three structural conditions: degrees of freedom in component states, cascading consequences of local decisions, and collective resolution of ambiguity. We apply this framework to a novel domain — pure mathematics — by analyzing factorization in algebraic number rings. We prove that the standard integers (ℤ) cannot satisfy TAME's structural prerequisites due to unique factorization, establishing a rigorous boundary condition. We then demonstrate that rings of algebraic integers with non-unique factorization (ℤ[√d] for appropriate d) satisfy all three conditions: ~14% of norms exhibit intrinsic factorization ambiguity, single factorization choices cascade through 90%+ of the network, and agent-based negotiation produces robust collective consensus (Jaccard similarity 0.98–1.00 vs. 0.37–0.42 null). Agents collectively discover the splitting behavior of primes — a fundamental concept in algebraic number theory — through purely local interaction, with split-fine alignment improving +0.10 to +0.31 over null across all tested class numbers (h = 2 through 5). Perturbation experiments reveal that this collective pattern functions as a robust attractor: even complete randomization of agent beliefs results in 98–100% recovery, with the algebraic signal perfectly preserved. These results suggest that the boundary between unique and non-unique factorization domains constitutes a mathematically precise analog to the boundary between systems with and without emergent collective computation.