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\[ da/ds=μcosθ + aλsinθ.\]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 E₀ 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 E₁.The first intrinsic invariant of the theory is arithmetic torsion, arising from the non-commutativity of additiveand multiplicative propagation. To globalize this invariant, we introduce the 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 eM-weighted area integral, and a boundaryintegral.To localize the same phenomenon, we construct a three-dimensional arithmetic expression contact manifoldwith contact form α = da - (μ\,du + λ a\,dv). Its horizontal distribution recovers the arithmetic flow,yields a natural horizontal differential calculus δ, and produces a curvature package proportional toμλ\,du dv. We then define 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.