The rule in motionThe learning rule as a landscape of fates — birth, death, and the memory in between

In the learning-rule chapter you met K's equation of motion as a formula: K̇ = η[R₀(K)cos(Δθ) − 2K/r] — grow when you see more alignment than your coupling predicts, shrink when you see less. A formula tells you the rule. It does not show you the fates. For that, physicists draw a phase portrait: take everything a single bond can be — its phase gap Δθ, its coupling K — as a plane, and at every point draw an arrow showing where the rule pushes next. The fates appear as the shapes of the flow: where arrows converge, a destiny; where they part, a decision.

1. A bond's whole life, on one plane

Below is that portrait for one pair of oscillators with a frequency gap you control. Two things move at once — the phases (horizontally: the gap drifts unless the coupling is strong enough to hold it) and the coupling (vertically: the learning rule). Click anywhere to seed a bond at that condition and watch it live out its fate. Read the furniture first:

• The shaded band is the dead zone — alignments below the threshold cos(Δθ) = 4/r from the learning-rule chapter. While a bond is passing through it, its coupling can only shrink.
• The green curve is where the learning rule rests (K̇ = 0): the equilibrium K* for each alignment — the same curve as the bifurcation figure in foundations, now embedded in the flow that visits it.
• The blue curve is where the drift rests (Δθ̇ = 0): couplings strong enough to hold the phase gap still against the frequency gap.
• The dot where they cross is the locked state — alive at K*, holding a fixed small angle. The flow spirals into it from a whole basin of starting conditions, and to K = 0 from everywhere else.

frequency gap 1.20

Warm start (K = 1.5) Cold start (K ≈ 0) Clear trails

One honest subtlety the portrait makes visible. From a strictly cold start — K genuinely near zero — an isolated pair almost never bootstraps, at any gap: with no grip, the phase gap drifts through alignment and misalignment alike, and what little coupling grows in the friendly half of each lap is crushed in the hostile half. Yet foundations promised that lattices do light up from nothing. Both are true, and the difference is the network: in a lattice, a bond's endpoints are held nearly still by their other neighbors, so alignment lingers long enough for the rule to catch. Bootstrap is a collective act. Hold that thought — it is exactly why the healing in the next section works.

2. Sixty bonds at once: the field goes binary

Now stop watching one bond and release the whole population. Sixty oscillators on a ring, every bond starting at the same middling coupling, every bond running the same rule on nothing but its own endpoints. The histogram shows the coupling field sculpting itself: within a minute the middle empties out, and every bond has committed — dead at zero, or alive up near its own K*. This is the binary K-field foundations §03 predicted: a sharp dead-or-alive decision on every bond, produced by purely local dynamics. The network has decided which of its connections are worth keeping, and the shape of that decision is its learned structure.

Restart New natural frequencies