Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing
An arithmetic expression is a word of operations; a motion is a flow on a manifold. For expressions whose evaluation has a single internal spine, the two are one structure: affine for addition and multiplication, projective once the second operand slot is divided, contact-geometric in the order defect. A regular arithmetic expression space (AES) is an oriented Riemannian surface $(M,g)$ with a smooth assignment $a\colon M\to\mathbb R$ satisfying the eikonal equation $|\nabla a|_g^2=\mu^2+\lambda^2a^2$; the four operations then act as motions translating and scaling $a$. In the basic model…
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