CANOPY: Closed-Form Neurosymbolic Composition with a Small State-Space Reducer

Compositional reasoning benchmarks such as long-form LRA-ListOps and the DeepMind Mathematics arithmetic suite are tree-evaluation problems written as linear token streams. A model reading only the tokens must recover the tree structure, a language problem, and execute the operator rule at each node, a symbolic one. These carry different inductive biases, and end-to-end sequence models continue to trail systems that route inputs through an explicit composition rule. CANOPY separates the two at inference time. A parser supplies the tree, a small state-space reducer approximates each atomic operator call as a probability-mass function (PMF) over the answer, and a fixed closed-form composer propagates those distributions up the tree. Reducer and composer exchange nothing but PMFs, so either can be swapped or retrained without touching the other. A sub-1M-parameter reducer reaches 0.9535 on long-form LRA-ListOps, above the tree-induction band and above end-to-end baselines carrying up to 90x more parameters. On DM Math addition chains it reaches 0.989 against a 30M-parameter reference at  0.94. Mambino, the reducer instantiated here, carries an internal predictor branch that feeds its own one-step prediction error back into the recurrence. We partition every evaluation by whether an operator call was seen in training. Ablating the branch costs 2.8 pp or less on seen calls, and 25.0, 11.3 and 5.1 pp on unseen multiplication, addition chains and single-operation addition, respectively. On LRA-ListOps, whose operators are saturating, ablating it instead improves composition by 4.45 pp. The branch earns its parameters only where the reducer must compute rather than recall.