In July 2026, Alpöge and Fable exhibited a polynomial map that refutes a conjecture open since 1939. zelph verifies that counterexample over the integers using rules alone. The engine’s C++ core contains no arithmetic at all, so digits, positional arithmetic, signed integers, symbolic differentiation and polynomial normal forms are built up as rules first, and the refutation runs on top of them.

The tutorial: https://acrion.github.io/zelph/math/tutorial-jacobian/