L3 — hierarchy and abstraction: nested phase coupling breaks the flat ceiling
Hierarchy is nested phase coupling — a slow field’s phase gates which fast sub-field is active (theta-gamma generalized to many levels). It does four things: selects the level (correct-vs-wrong routing margin +0.92), recognizes an abstract category from never-stored instances (up to 32 categories), composes held-out combinations with zero train/test gap (to 576 combos, depth 3), and multiplies effective capacity ~6×.
Concepts become attractors of attractors: upper levels carry categories, lower levels carry instances, and a category attractor binds its instances. Under strict full clean-up the flat field collapses at T ≈ 12 items while gated sub-fields hold to T = 72 — the measured remedy for L1’s shallow depth. The honest negative: that capacity win is criterion-dependent (it vanishes under a forgiving read-out), and hierarchical recovery assumes the correct gate, which L4 must derive.
L3 is the measured answer to L1’s shallow-depth ceiling and L2’s slot-bounded memory. The brain stacks rhythms — slow ones gating fast ones — and represents concepts as attractors of attractors. The wave machine does the same with nested phase coupling, and all four claims pass under stress (session v0.4, digest ea4c6723…).
Four verified capabilities
| capability | result | reach |
|---|---|---|
| H1 select the level | routing margin +0.92 | sharp gates reject, broad gates leak |
| H2 abstraction | category from novel instances 1.0 | to ρ = 0.4, up to 32 categories |
| H3 compositional generalization | train/test gap 0.00 | to 576 combos, depth-3 via permute-extraction |
| H4 break the flat ceiling | ~6× capacity | flat dies at T ≈ 12, hierarchy holds to T = 72 |
H2 is true abstraction: the category is recognized from instances that were never stored, and instance and category dissociate cleanly. H3 is genuine composition: held-out combinations — never built during training — decode identically to enumerated ones, and generated instances are valid and novel.
Hierarchy multiplies capacity
Storing instances in gated sub-fields, rather than one flat field, multiplies effective capacity about sixfold. Under the strictest read-out (full clean-up, overlap ≥ 0.95) the flat field collapses at roughly twelve items while the gated sub-fields still hold seventy-two. This is the direct, measured remedy for the d* ≈ 2–4 shallow depth that L1 recorded.