Arithmetic Expression Geometry: Foundations

We introduce Arithmetic Expression Geometry (AEG), a framework in which arithmetic expressions are studied throughthe geometry of their evaluation histories rather than only through their final numerical values. For a distinguishedclass of threadlike expressions, additive and multiplicative updates determine two generating directions of propagation,and their infinitesimal interplay yields the arithmetic flow equation\[ \frac{da}{ds}=\mu\cos\theta + a\lambda\sin\theta.\]This leads to the notion of an Arithmetic Expression Space: a Riemannian surface equipped with an assignment fieldsatisfying the corresponding first-order law. We construct the basic model $\mathfrak{E}_0$ on the upper half-plane,where $a=-x/y$ satisfies both the flow equation and a Laplace eigenvalue equation, and we record the first singularmodel $\mathfrak{E}_1$.The first intrinsic invariant of the theory is \emph{arithmetic torsion}, arising from the non-commutativity of additiveand multiplicative propagation. To globalize this invariant, we introduce the \emph{Accumulative Commutative Space}(ACS), a plane of total additive and logarithmic multiplicative charges, and prove a Stokes--type triple identity thatexpresses global torsion equivalently as an evaluation difference, an $e^M$-weighted area integral, and a boundaryintegral.To localize the same phenomenon, we construct a three-dimensional \emph{arithmetic expression contact manifold}with contact form $\alpha = da - (\mu\,du + \lambda a\,dv)$. Its horizontal distribution recovers the arithmetic flow,yields a natural horizontal differential calculus $\delta$, and produces a curvature package proportional to$\mu\lambda\,du\wedge dv$. We then define \emph{arithmetic holomorphic fields} by an arithmetic Cauchy--Riemannsystem and show that they satisfy a twisted second-order equation, providing the first analytic structure of the theory.Taken together, these results establish the first foundational layer of AEG: arithmetic evaluation admits a pathlanguage, intrinsic invariants, local and global geometric structures, and the beginnings of a function theory.