From Number Theory to Neural Networks: The Anchor Edge Framework

Emergent Collective Computation in Constraint-Satisfaction Networks with Applications to Mechanistic Interpretability This paper presents the Anchor Edge Framework, a unified research program discovering that edges in constraint-satisfaction networks carry a measurable structural property — the combination of discrimination, non-redundancy, signal flow, and cascade depth — that predicts which local interactions determine global outcomes. Origin: Applying Michael Levin's TAME (Technological Approach to Mind Everywhere) framework to factorization in imaginary quadratic integer rings (Z[√d]), we show that simple agents with no knowledge of algebraic number theory collectively recover the splitting behavior of primes — a result connected to ideal class groups and the Dedekind zeta function. The converged state is a robust attractor, with 98–100% recovery after complete scrambling. Core discovery: Two local properties — discrimination (asymmetric information flow) and consistency (corroboration from other neighbors) — predict which edges anchor the system's converged state with 89% top-1 and 100% top-2 accuracy. Discrimination transfers universally (91–100%) across hierarchical, grid, core-periphery, and power-law network architectures. Neural network transfer: The Anchor Edge properties translate to neural networks as a mechanistic interpretability tool. The "Cheap Anchor" variant (v5) achieves Spearman ρ = 0.74–0.90 correlation with per-edge ablation ground truth across networks from 60 to 4,128 edges, using only network structure and a single forward pass — no per-edge ablation, loss function, or training data required. It outperforms weight magnitude at all scales above tiny and outperforms gradient-based importance at small scales, while running 3–11× faster than full ablation. Key insight: The three TAME conditions for emergence map directly onto components of edge importance in neural networks, with each condition contributing a measurable, independently validated improvement factor. This mapping was discovered iteratively, not designed in advance.