Generalized Arithmetic Energy Theory (GTEA): Canonical States, Accessibility Laws, and Superstructural Rules

This paper develops Generalized Arithmetic Energy Theory (GTEA), a constructive framework in which arithmetic objects are studied through the minimal cost of their admissible realizations rather than through their numerical or positional properties alone. The foundational layer, Arithmetic Energy Theory (TEA) on the positive integers, is built on arithmetic directed acyclic graphs over the purely additive language. The standard additive energy of an integer is defined as the minimal number of addition steps required to construct it from the primitive constant 1. The central result of the foundational layer is a proof that this invariant coincides exactly with the classical addition chain length, providing a rigorous external anchor connecting GTEA to the established theory of arithmetic complexity. The framework is then extended to integers, rationals, reals, and complex numbers through successive constructive regimes. On the integers, energy is derived by minimization over signed difference representations. On the rationals, it is derived by minimization over quotient-type instances, with an explicit separation maintained throughout between arithmetic normalization and energetic minimality — two notions that do not coincide in general, as a consequence of the non-monotonicity of addition chain length under multiplication. On the reals and complex numbers, exact constructive realization is replaced by precision-dependent accessibility laws, and the primary energetic object becomes a function of the approximation tolerance rather than a scalar minimum. Three structural observations on the rational domain are examined in detail. Bounds on the inverse instantiation cost are established, with the exact value determined for dyadic denominators. The multiplication simulation cost is shown to be non-negative by construction, and a conjecture on its upper bound in terms of the addition chain length of the common divisor is stated, with an explicit identification of the prior open problem — the cost of multiplication in the additive language — that any proof must resolve. A composition-reduction decomposition identity is introduced as a definitional organizational tool rather than an independent theorem. Canonical value-structure states, accessibility laws, asymptotic accessibility classes, and structural transport under energetic operators are developed as the superstructural layer of the theory. The article concludes with a preliminary discrete illustration using the accelerated Collatz map, and a formulation of six superstructural rules governing admissible realization, minimality, canonical quotienting, value-structure separation, the exact-to-accessibility transition, and structural transport. The article is presented as a structured framework at an early stage of development. Established results and open problems are explicitly separated throughout.