proof drop #3 — a cheap router decides whether the locality bet pays

Proof drop #2 showed that searching the right region of a memory beats scanning all of it — but I cheated and handed the search an oracle for which region that was. Production has no oracle. So #3 fires the oracle and bolts on the cheapest router that could work: one centroid per topic (the mean of its items, which the store already has), route the query to the nearest one.

The result splits cleanly in two. When the memory is organized, routing is nearly free and it works: at 128k items a top-3 centroid router recovers 98% of the oracle’s recall at ~60× the speed of a flat scan, and the routing decision itself costs ~0.002 ms — about 4000× cheaper than the scan it replaces. When topics overlap, routing — not retrieval — becomes the bottleneck: realized recall falls along a continuous curve that tracks routing accuracy, down toward chance. The kicker: proof drop #2 ran in exactly that hard regime, so its oracle win was real but uncashable by a trivial router. Routing quality gates the whole locality bet.

→ Full writeup, figures, and repro: /blog/fmm-router/

what’s next

A learned router — to push the crossover rightward and cash the win in harder, more realistic regimes. That’s proof drop #4. The centroid router is the floor; now we measure the ceiling.

— Yan

Research conducted at Éthiqueia Québec inc.