math.GM — General Mathematics

Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing

Contributed by Mingli Yuan

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 $\mathfrak E_0$, the upper half-plane with the left-invariant metric of $\mathrm{Aff}^+(1,\mathbb R)$ (curvature $-\lambda^2$), addition and positive multiplication are affine translations and dilations obeying a Baumslag--Solitar law, and $a$ is a Laplace--Beltrami eigenfunction of eigenvalue $2\lambda^2$. Left-expanded words stay in the affine Borel cell fixing $\infty$; the right-expanded language, once second-slot division is admitted, generates all of $\mathrm{PGL}_2(\mathbb R)$ through the rank-one Bruhat decomposition, and pulling standard horocycles back along one operator word gives the dual reading, ripple geometry. Commutativizing the motions by total charge yields the accumulative commutative space, where the order defect of two histories is an exact weighted area, the tearing. The same defect is the constant $\mu\lambda$ in the bracket $[D_u,D_v]=\mu\lambda\partial_a$ of the horizontal fields of $\alpha=da-\mu\,du-\lambda a\,dv$, a contact form whenever $\mu\lambda\ne0$, and it is the curvature of that horizontal connection; for $\mathfrak E_0$ the basic model is proved to be the Reeb quotient of the contact model, and fixed-charge tearing and fixed-state Reeb drift are Legendre dual readings of the same curvature. The generators $\partial_a$, $a\partial_a$, and $a\log a\,\partial_a$ span an infinite-dimensional Lie algebra, so no finite-dimensional Lie group carries all three ranks, and no observable of the charge endpoint detects tearing. For every finite puncture count a conformally compactifiable punctured AES is constructed, with an explicit four-circle zero locus and exact reconstruction from three-ended pants blocks. All results are elementary, planar, or local to a three-dimensional contact model.