From the largest to the smallest — from celestial bodies down to particles — all things are discrete. Order, or law, connects all things.

                             ———SuHang Zhang

Within DOG, this connection is precisely encoded as a three‑layer mathematical structure:

B(C,n) \;\stackrel{\text{recursive generation}}{\longrightarrow}\; r_n(C)=\frac{1}{C+\dfrac{1}{C+\cdots}}
\;\stackrel{\text{generating function}}{\longrightarrow}\;
F_C(x)=\sum_{n=0}^\infty N_n x^n=\frac{1}{1-Cx-x^2}.

where:

- B(C,n) denotes the discrete geometric primitive at level n;
- r_n(C) stands for the continued‑fraction convergent, which defines scales and nesting relations;
- F_C(x) is the generating function that elevates the discrete counts N_n into an analytic function, which further projects onto modular forms, L-functions, and propagators in quantum field theory.

Mappings among discrete geometric primitives, continued‑fraction coefficient sequences, and the generating function together form a complete logical chain from discrete order to continuous physics.

The global ontology is discrete, manifested in the discrete coupling between geometric primitives. Inside each lattice primitive cell, a local Riemannian manifold is permitted as an internal degree of freedom. Global continuous spacetime is merely a macroscopic emergent special case obtained by taking the continuum limit of the discrete system.

DOG lattice can accommodate arbitrary dimensions, geometries and topological configurations.

In DOG (Discrete‑Order Geometry), lattice primitives possess no inherent scale. Coarse‑graining and refinement are intrinsic properties of the geometric ontology, rather than externally imposed approximation operations. Large‑scale objects may be directly treated as‑order‑one cells; a cell can be decomposed downward into sub‑lattices, and may also embed Riemannian manifolds internally as local continuous degrees of freedom. Physical statements derived at distinct levels belong to separate domains of discourse and cannot be directly compared.