Generalysed Arithmetic Energy Theory (GTEA) begins from the idea that a mathematical object should not be understood only through its numerical value, but also through the constructive process by which it can be generated, represented, or accessed. In its initial form, GTEA is based on the natural numbers, where the primitive constant 111 is taken as freely available and all other numbers are obtained through admissible operations, most fundamentally addition. The energy of an object is then defined from the minimal constructive effort needed to instantiate it. A generalized TEA extends this principle beyond the discrete additive setting. Its purpose is not merely to assign scalar costs to numbers, but to organize mathematical objects through their minimal constructive structures. In this broader perspective, energies are derived invariants, while structures are primary. The theory is therefore not limited to exact finite constructions on N, but applies to increasingly rich domains such as Z, Q, R,and C, as well as potentially to functions, operators, and higher-order generative systems. In the generalized framework, an object is associated with a family of admissible instances. These instances may be finite DAGs, quotient-type constructions, recursive generators, approximation schemes, or other finitely specified constructive mechanisms. A cost is assigned not directly to the object itself, but to its admissible realizations. The energy of the object is then obtained by minimizing over all such realizations, or, in continuous settings, by minimizing over all admissible approximations within a prescribed tolerance. This generalized viewpoint leads to a major conceptual shift. Canonicality is no longer sought primarily at the level of symbolic representatives, such as reduced fractions, but at the level of minimal constructive structures. Thus a generalized TEA distinguishes clearly between value, representation, and structure. It asks not only how expensive an object is, but what its minimal mode of generation is. Such a theory naturally separates into two regimes. In the exact regime, one studies objects that admit finite exact instantiation. In the accessibility regime, one studies objects that can only be reached through approximation, and energy becomes a law depending on precision. This opens the way to energetic signatures of irrational numbers, constructive profiles of functions, and coupled accessibility laws in the complex plane. Generalized TEA may therefore be viewed as a structural theory of minimal generation. Its ambition is to unify discrete arithmetic complexity, quotient-based representation, approximation theory, and constructive access to continuous domains within a single conceptual language.
Sylvain Geffroy (Thu,) studied this question.