Small Transformers Learn Kepler's Equation with a Number Line and an Eccentricity Register

We train a family of 210 small transformers to solve Kepler's equation $M=E-e\sin E$, then reverse-engineer the algorithm they learn. To our knowledge this is the first circuit-level account of a continuous, transcendental equation. The task is trigonometric, yet the digit embeddings form a number line, not a circle. Attention writes $e$ into a single residual-stream direction largely hidden from the readout, the e-register. The MLP reads it and writes the $e\sin (nM)$ corrections. The output carries harmonics of $M$ to $n\approx 30$. Scale the register and the corrections scale with it; set it to zero and the model tracks its own circular orbit. The line and the register hold across the 210-model family. We release the models and the analysis code as a testbed: the register's variable is known to be $e$, so a method that claims to find hidden computation can be scored against it.