Elnamaki Coding (EC) redefines the foundations of arithmetic[1] by replacing scalar value representation with recursive structural emergence. In this framework, natural numbers[1] are not static entities but semantic trajectories—dynamic paths through a topologically recursive Fibonacci manifold. Each numerical identity arises from morphic decompositions, Zeckendorf[3] style expansions, and invertible modular transforms. At the heart of EC lies the Sequanization Theorem, which induces a non-Euclidean metric on Z based on recursive path existence, redefining proximity and arithmetic continuity. Two reversible operators—Lowe and Elevate—generate an algebra of additive evolution, enabling a complete grammar for morphic arithmetic. This generative system encodes identity as structure, not magnitude, establishing a symbolic substrate for logic, computation, and complexity. Unlike compression or classical encoding schemes, EC constructs an intrinsic arithmetic language grounded in recursive algebra, path redundancy, and topological invariants. It supports high-entropy, non-linear mappings with direct implications for post-quantum cryptography, symbolic AI architectures, and structural modeling of recursive growth. EC’s semantic lattice allows encoding of entangled state graphs, recursive tensor webs, and morphogenetic trajectories in symbolic and quantum domains. The result is a universal grammar for number theory—self-referential, reversible, and structurally exact.

Fibonacci lattice Zeckendorf decomposition Lowe and Elevate maps Sequanization Theorem parametric recursion morphic encoding recursive number systems modular arithmetic nested seed expansion generative arithmetic language topological number identity Elnamaki Coding