Attention as a Force: What a Conservative Transformer Keeps, Remembers, and Pays

Softmax attention is a convex average of values; it conserves nothing from layer to layer. We make attention a force: every token carries a momentum next to its state, the attention layer adds the gradient of a log-sum-exp energy to the momentum, and a layer is one step of a symplectic integrator. Such layers preserve phase-space volume and are exactly invertible. We prove three obstructions that delimit the class, the sharpest being that a conservative attention delivers from each key the very projection it matched on. We then add a read-only memory of Casimir coordinates, which every layer reads and none can write, and prove that matching on it decouples whom a token attends to from what it receives. With every criterion fixed before its measurement: on associative recall, matching on Casimirs binds where the same model matching on its moving state does not (1.000 vs 0.078, three seeds), keeps 0.884 binding at 32x the training length (Transformer with the same input: 0.408; capacity-matched control: 0.217), and is carried by one head whose matching table reads "key = predecessor". The price on language modeling is +6.7% to +16.4% perplexity and is unchanged by Casimirs. Where conservation is the task, a 15k-parameter conservative attention keeps the orbits of a planetary system with an unknown force law for 10^4 orbits (41 of 48 systems over three seeds, all 16 on two seeds) while Transformers with 1x and 10x its parameters lose all 48; on NASA ephemerides it keeps every orbit without energy drift, but Transformers predict longitudes more accurately. Files: the preprint (PDF) and the complete code, tests, validation gate, pre-registration log and measurement files from which every number in the paper is generated (python paper/make_assets.py).