Generalized Arithmetic Energy Theory (GTEA): Threshold Crossing, Canonical States, and Energetic Functions

Most mathematical frameworks describe objects by what they are or by how they can be optimally constructed. GTEA takes a different position: what matters is the least energetic threshold at which an object, a state, a variable, or a function first becomes realizable through some admissible process. The system does not wait for the most elegant construction. It crosses the threshold as soon as one admissible realization exists. This single principle, called threshold crossing, drives the entire theory and separates it from both optimization-based and complexity-based formulations. From this starting point, GTEA builds a unified constructive framework covering the natural numbers, the integers, the rationals, the reals, and the complex numbers, organizing each domain through its own class of admissible instantiations and its corresponding threshold law. In the discrete additive regime, the energetic threshold is proved to coincide exactly with the classical minimal addition-chain length, providing a first rigorous external calibration. On the rational domain, a sub-additivity result is established, and the relationship between arithmetic normalization and energetic threshold is shown to be non-trivial, with a central open problem identified. On the real and complex domains, the threshold becomes a precision-dependent accessibility law, encoding the arithmetic nature of a number through the behavior of its constructive cost as approximation improves. The framework extends naturally to functions. To any function defined on a subset of the complex numbers, GTEA associates an energetic function measuring the threshold cost of approximating its values. Seven fundamental properties of this construction are established, including monotonicity, sub-additivity, behavior under composition, and a striking characterization of zeros: the zeros of a function are precisely the points where the energetic threshold collapses to zero. This result opens a potential bridge between threshold dynamics and the study of zero sets in analytic contexts. A discrete dynamical illustration is developed for the accelerated Collatz map, showing how the threshold-crossing decomposition organizes the local balance between constructive injection and structural dissipation at each step of a trajectory. The broader ambition of GTEA is to provide a uniform energetic language for arithmetic, functional, and dynamical mathematics, grounded not in the best way to construct an object, but in the first level at which its construction becomes possible.