From Zero-One Tables to Meta-Matrices: An Algebraic and Geometric Framework for Graph Generation — A Holographic Theory of Graphs via Ordered Vertex Patterns

We establish a systematic algebraic and geometric framework generated from the zero–one scanning tables of finite simple graphs. For a graph $G$ and a $k$-vertex set $U$, the table of all $k!$ orderings of $U$ yields, via an adjacency-or-self scanning rule, a family of binary tables. Treating these tables as primary matrices, we construct a canonical hierarchy of mathematical structures: flattened meta-matrices and their row-permutation orbits, support basis spaces and support algebras, normalized operator spaces and probability-functional packages, and derived Gram geometries together with their threshold filtrations and topological shadows.

We prove that every layer of this hierarchy is independent of arbitrary ordering choices and is functorial under graph isomorphism. The framework naturally organizes into three interdependent branches—algebraic, probabilistic, and geometric—all generated from the same table data. At the foundational level, we isolate the precise reconstruction-theoretic consequences: the tower of algebraic and geometric invariants constructed from the tables is faithful if and only if the original graph is determined by its enriched lower-order data. The framework thus provides a unified language in which graph reconstruction, canonization, and generation are simultaneously formulated as questions about the completeness and rigidity of the generated structures.

The theory is deliberately foundational and extensible. By varying the ambient separator datum (real, complex, polynomial, or operator-valued), the same zero–one table machine produces a broad spectrum of enriched graph invariants and derived geometries. This positions the framework as a general method for lifting combinatorial graph data into algebraic, analytic, and topological domains.