Soft equality — making what you learned transfer

Parent: recursive-harness

Hard equality makes learning nearly worthless in an agent: a no-good learned in context c applies only to a LITERALLY identical context — and nothing ever recurs exactly. The softening of the equality relation is what lets learning compound.

The softening: equality → distance

Behavioral/bisimulation metrics (quantitative bisimulation): instead of "equal or not", states and contexts get a distance d(c, c′), with the guarantee that things at distance $\varepsilon$ behave $\varepsilon$-similarly. Identity relaxed into geometry — the same relaxation move as everywhere else, applied to the equality predicate itself.

The payoff: knowledge transfers with confidence decay

A no-good, a subsumption claim, or a price learned at context c applies at c′ with confidence discounted by $d(c, c′)$. The coordinator's no-good store upgrades from exact lookup to metric retrieval (nearest-neighbor): everything ever learned is evidence for every sufficiently-similar situation. This is the difference between a cache and an experience.

Also: memoization before generation

SOLVE-or-SPLIT (root-protocol) gains a cheaper first option: RETRIEVE — "have I solved an $\varepsilon$-equal leaf before?" A hit costs a lookup plus a re-verification against the current spec. The retrieved artifact still passes the local gate — trust the metric for retrieval, never for acceptance.

The catch: the metric is the hard part again

honest-caveats #1 applies with full force: a learned embedding gives a cheap d but is Goodhartable and opaque; mechanical features (same file, same API signature, same failing-test fingerprint) give a coarse but trustworthy d. The two-tier rule again: retrieve with the cheap metric, ACCEPT only through the gate — the metric proposes, the gate disposes.

Soften equality into distance and the no-good store becomes an experience — everything learned radiates to its neighborhood, discounted by distance, and the gate still has the last word.