Mirror-neuron computation in a Kuramoto reservoir: emergent sparse coding, the single-attractor theorem, and template-conditioned recall in symbolic reasoning

▎ RIAN is an untrained Kuramoto oscillator reservoir paired with a k=1 nearest-neighbour retrieval readout and a one-line ▎ family-aware stop policy. On a strict 77-query reasoning battery the system reaches 98.7% EXACT without a single gradient ▎ step, versus 17.0% for the ridge-regression baseline on the same substrate (+81.7 pp without backpropagation). ▎ ▎ This version reframes the system as a computational mirror-neuron substrate and adds six new sections of measurements: (i) ▎ emergent sparse coding matching the canonical 6.25% Kenyon-cell statistics of the Drosophila mushroom body (1012/1024 broad ▎ participation, no neuron above 50%, overlap structure 60% above chance) without explicit lateral inhibition; (ii) a ▎ single-attractor theorem characterising the geometric ceiling — top-K masking during integration preserves accuracy but ▎ collapses pairwise prompt cosines from 0.50 to 0.95, confirming Kuramoto's all-to-all coupling as the cause; (iii) a ▎ held-out battery showing 0% out-of-distribution accuracy across six families (add, cond_x, pipe_x, solve_x, copy>30, is>30) ▎ — a clean characterisation of recall vs. reasoning; (iv) two ensemble experiments showing that neither corpus diversity ▎ (1.0/6 unique answers) nor geometric diversity (2.18/6) alone crosses the OOD ceiling; (v) a principles brain trained on ▎ Peano axioms, algebraic properties and 2×2 linear systems that solves a held-out (x+y=20, x−y=6) procedurally from first ▎ principles, computing 20+6=26, 26/2=13, 20−13=7 internally; (vi) the operand-identity diagnosis: the system executes ▎ mathematical procedures correctly on substituted operands when the prompt's operands are out of distribution — the ▎ architectural fix needed is precisely an operand-identity slot, not new computation. ▎ ▎ Bilingual (English + Spanish). All scripts and corpora are in the GitHub repository for reproducibility. Code: ▎ github.com/ian333/RIAN.