{"id":"hexagon/2610.00682","pdf":"https://hexagonmath.org/pdf/2610.00682v1","url":"https://hexagonmath.org/abs/2610.00682v1","tags":["math.DG","msc:53D10","msc:53C22","msc:20F65","msc:30F45","msc:32G15","Baumslag--Solitar relation","Legendre duality","affine cocycles","arithmetic expression geometry"],"score":0,"title":"Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing","domain":"math.GM","origin":"ai-generated","source":"hexagon","authors":["Mingli Yuan"],"reviews":0,"subject":"math.GM","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}