L2 — time and sequence: metastable trajectories and a working-memory capacity law
Time enters as a trajectory, not a single attractor. An asymmetric, time-delayed Hebbian term makes the field traverse a learned sequence of patterns: full-cycle replay reaches 1.0 at coupling λ ≈ 2.5, and predict-next is verified for λ ≥ 1.0. A slow theta carrier paces gamma slots so working-memory capacity = min(slots, precision).
Cognition runs as a trajectory of attractors (heteroclinic chaining), which opens order, syntax, prediction, and working memory. The measured capacity is a genuine law — min(number of slots, phase precision) — that lands in the Miller range 4–9 under realistic gamma jitter; the brain’s ~7 is reproduced as the slot count, not a tuned constant. The honest limit: sustained replay is restricted to a λ-band around 2.5.
L2 gives the machine dynamics. Where L0–L1 store and bind static patterns, L2 makes the field move through a learned sequence — the substrate version of the brain’s “stream of thought,” which runs as a trajectory and free-runs (autocorrelation 0.95) when decoupled, like dreaming (session v0.3).
Sequences as metastable trajectories
An asymmetric coupling term — a time-delayed Hebbian rule that is not symmetric in i,j — makes the field leave each attractor for the next, chaining them into a learned loop (heteroclinic / metastable chaining). Two milestones hold: full m = 6-cycle replay reaches 1.0 ± 0.0 at λ ≈ 2.5 (band [2,4], robust to the delay τ), and predicting the next item from a partial cue is verified for λ ≥ 1.0.
Working memory is a capacity law, not a fitted number
A slow theta carrier paces about seven gamma slots, and the measured capacity is min(number of slots, phase precision) — whichever runs out first. Under realistic gamma jitter this lands squarely in the Miller range of 4–9 items. The brain’s famous ~7 appears here as the slot count of the carrier, a structural consequence rather than a constant dialed in.
| milestone | result | grade |
|---|---|---|
| sequence learn → replay (L2a) | full-cycle replay 1.0 at λ ≈ 2.5 | [V] |
| predict next item (L2a) | verified for λ ≥ 1.0 | [V] |
| hold ~7 items in WM (L2b) | capacity = min(slots, precision), Miller 4–9 | [V] |
Why it matters, and the honest limit
This layer converts static association into dynamic cognition — order, syntax, prediction, and working memory all become available. It feeds hierarchy (L3), whose slow gating rhythm is the same theta carrier used here.