Generalized Arithmetic Energy Theory (GTEA) — Revision R04: Competitive Operational Framework, Folding Algebra, and Path-Length Arbitrage

This paper introduces the Generalized Arithmetic Energy Theory (GTEA), a foundational mathematical framework that redefines how arithmetic objects are measured and compared. The central idea is simple but far-reaching: instead of asking what a number is, GTEA asks what it costs to construct it. This cost — the energetic threshold — is the minimum number of operational acts required to produce the number from first principles, across all admissible families of operations simultaneously , it can be considered as a kind of "Arithmetic Darwinism" driven by energy . Obviously , 'Energy' isn't used here in a physical sense , but more generally to mean "something that causes change" , that is the case . What is new in this revision (r04) Previous approaches measured arithmetic cost within a single operational family. GTEA r04 introduces a unified competitive framework in which six families — addition, multiplication, exponentiation, duplication, symmetry, and rotation — compete on equal footing via mixed paths. The common currency is the act: one operation, cost 1, regardless of which family it belongs to or how many tokens it moves. This resolves a long-standing conceptual tension between incommensurability and comparability. A new operational family — the Symmetry family — is formally introduced, based on axial reflections of discrete token configurations. Unlike duplication, reflections can cancel tokens by superposition, producing genuine energetic gains verified by exhaustive graph search. The imaginary unit i emerges natively as a quarter-turn rotation, giving GTEA a native foundation for the complex domain. Why it matters GTEA proposes a structural invariant — the energetic signature — that is finer than any single arithmetic measure. It opens new research directions in analytic number theory, algorithmic complexity, constructive geometry, and the thermodynamics of computation. The algebra of transformations,a linear algebra (due to additives property of energy ) announced here as a research program, promises a rich combinatorial theory of how numbers are built from competing geometric operations. Status This is a working preprint deposited on Zenodo for priority and community feedback. The framework is foundational; several directions identified in the Outlook are open research problems.