Probing Algebraic Structure in Grokked Networks: Cycle Closure Reveals Quotient Learning as a Coordinate Artifact

## Abstract Mechanistic interpretability of grokking often relies on label accuracy or geometric/topological proxies to claim that networks learn algebraic structure. We argue these indicators are insufficient, and introduce **Cycle Closure Count (CCC)**, a functional probe grounded in group axioms: a network respects group G if Net(Net(a, b), b⁻¹) = a for all pairs. CCC measures algebraic consistency directly, independent of element-level accuracy. Applying CCC to dihedral group D₁₂ multiplication, we report three findings: 1. **CCC tracks generalization more reliably than persistent homology** (r = 0.996 vs r = 0.301 at step 15k; r = 0.997 vs r = 0.042 at step 40k)2. **Apparent "quotient-first learning" is a coordinate artifact**—under randomized indexing, early parity accuracy collapses from 90% to chance while CCC drops from 54% to 5%3. **Weight decay is a critical driver of grokking**—without it, networks plateau at 15% validation accuracy despite developing partial functional consistency (CCC 63%), a regime invisible to standard metrics We also document a cautionary negative result: a synchronized "collapse" at step 22,000 observed under Float32 vanishes under Float64, confirming it as numerical artifact rather than learning dynamics. Float32's Softmax overflow limit (logit range ~88.7) triggers spurious gradient spikes, creating illusory phase transitions. These results recommend CCC as a reliable probe for algebraic generalization and caution that coordinate alignment can produce illusory evidence of structural learning. Grokking is compression-driven structure extraction where regularization is the engine, label encoding creates shortcuts, and numerical precision is a critical confound. **Keywords**: grokking, mechanistic interpretability, group theory, weight decay, generalization, functional probes