Geometry of Integers

On the classical one-dimensional number line, the structural depth of an integer—its step distance from the origin—coincides identically with its scalar magnitude. Confined to a single degree of freedom, integers can translate only forward or backward along a single flat axis. This paper constructs an infinite hierarchy of higher-degree discrete integer foliations and their reflected integer completions, within which the longitudinal depth of an integer along its structural leaves is decoupled from its ambient scalar magnitude. By partitioning the integers into horizontal power shells aligned along a central median spine, every integer generates its own intrinsic two-dimensional coordinates across three canonical directions. This foliated architecture induces a direction-dependent discrete transport operator that simultaneously generates a global Chebyshev geodesic metric and a native multi-valued, non-commutative, and non-distributive arithmetic governing foliated addition, subtraction, multiplication, division, and primality. Furthermore, whereas global scalar primes scatter across shells, local leaf-depth primality endows every individual leaf with an autonomous Fundamental Theorem of Arithmetic, assigning each integer a coupled triad of unique prime factorizations. In the quadratic foliation, magnitude-coupled self-shift transport exhibits non-vanishing discrete holonomy, positive discrete curvature that converges asymptotically to a constant under quadratic normalization, and an exact holographic conservation law linking prime counts in the initial interval to the corresponding quadratic shell. Finally, under the correspondence principle at degree one, the lateral generators, discrete holonomy, and curvature vanish identically; the integers lose their second degree of freedom, and classical commutative Peano arithmetic is restored on the one-dimensional number line.