title: The other half of the bridge date: 2026-09-16
A herring in a shoal watches a few neighbours and adjusts. The neighbours adjust back. Nobody sees the whole shoal, and the whole shoal turns as one, because influence runs both ways and keeps running. This week the question was whether that picture, and the convolutional networks that formalise the same locality, had anything left to give our machine. Most of it, it turned out, we had already tested and spent. One piece we had never built.
Locality on the token side paid exactly once, as a symbol rather than a filter: a sliding matcher that writes a numeral under a number word recovered most of what spelling costs. A learned local kernel over the tokens learned to be the identity, because the frozen trunk had done the local mixing before we arrived. On the slot side, the neighbourhood rule, each factor slot attending only to slots in its sentence or sharing its variables, was measured and found weightless: all-to-all read 0.2521, masked read 0.2511. And when we asked whether argument pointers favour recent facts, the way a shoal favours near neighbours, the gold said no: on prose a third of the pointers reach forward to something the text has not said yet, and half of them do on the synthetic register. Recency is a coin flip. The mixer’s all-to-all shape is right for a graph whose references run both ways.
Then the gut said the shoal still had something, and the honest way to check was to draw the bridge and look at the arrows.
The machine has two mediums. Tokens carry the text, slots carry the factor graph, and the bridge between them is an attention matrix: each breath, every slot spreads its attention over the tokens and reads. Seven breaths, seven reads. That is the whole bridge, and every arrow on it points the same way. The slots read the tokens. The tokens are computed once, before the first breath, and they never hear what the slots did with them. The only thing that ever travels the other way is a certificate at read time, a mask that says “these tokens are taken”, which is a shout across the water, not a message.
Put beside the shoal, the asymmetry is plain. Our fish adjust to the current and the current never adjusts to the fish. Put beside our own history, it is plainer still. The June engine, the one that matched the exact solver on KenKen and generalised across three puzzle families, was learned belief propagation on a factor graph, and belief propagation has two halves: variables tell factors, factors tell variables, alternating until they agree. At the language bridge we built the first half and stopped.
The missing half is small to state. After breath k’s slots have read, take the same attention matrix, transposed, and carry each slot’s typed state back to the tokens it read. A phrase’s tokens now say “claimed by a given slot as twelve” or “still unclaimed”. Breath k+1 re-reads a text that knows what was committed to it. A second slot reaching for a phrase that is already taken meets the claim as content rather than as a prohibition, and competition for phrases becomes two-sided, which is the shape of the failure we have been chasing on prose all week: the machine knows a relation is there and cannot say which quantities it binds.
It enters as a road, in the sense this project uses the word. The message is scaled to a fixed fraction of the token’s own norm, not a learnable gain that the loop can vote down at birth, and the next breath’s keys and values are built from the written-back tokens, so there is no path through the bridge that skips it. It costs about a million parameters. It is the “expand, collapse, repeat seven times” that was named as a gut two days ago, before the bridge had been drawn clearly enough to see which half was missing.
A learned message can learn to be nothing. The token kernel did, and for a good reason: the loss could be solved without it on the register it was trained on. On synthetic algebra the one-way bridge already reads at 0.98, so the write-back has no work there and would learn none. Its work is on prose, where the binding is the thing the loss cannot solve otherwise, and that is why it runs beside the newly annotated rows rather than alone. Bars are pinned: paired against the same diet without it, on wild and on mint, with the census reading how much of the token state the message actually carries at every breath.
The shoal’s lesson was not locality. We had squeezed that fruit. It was mutuality, and the arrow we had never drawn.