Generalized Arithmetic Energy Theory (GTEA) — Revision r03 The Generalized Arithmetic Energy Theory (GTEA) is a mathematical framework that assigns to every arithmetic object a structural invariant — its energetic threshold: the minimum constructive effort required to produce it from first principles, across all admissible operational families. This threshold is independent of any particular algorithm, representation, or computational model. It separates what a number is from what it costs to construct — a distinction that classical arithmetic does not make. "Arithmetic Darwinism": The central innovation of GTEA is the replacement of additive privilege by open competition arising from energetic opportunism. No operational family holds a priori priority: addition, multiplication, exponentiation, duplication, and folding compete on equal footing for every arithmetic object. The GTEA energy is the length of the shortest constructive protocol across all competing families. This competition produces a rich and non-trivial energy landscape in which numbers that appear arithmetically large may be energetically cheap, and numbers that appear simple may be energetically expensive. Crystallization and energetic residues. Certain numbers exhibit sharp local minima in the energy landscape — crystallization events — that reveal deep arithmetic structure without any prior knowledge of what structure to seek. Prime numbers, by contrast, resist reduction in every competing family and emerge as energetic residues: a selection of energetically opportunistic events that no operational shortcut can reach. The Folding Algebra. A new operational family is introduced, founded on rotation by π around axial hyperplanes in a discrete token space. Its mathematical foundations are established via the Huzita-Justin Origami axioms and the theorem of Alperin, which proves that folding strictly exceeds the constructive power of ruler and compass, resolving cubic equations inaccessible to purely additive methods. The folding energy detects rectangular and symmetric structure in numbers through compactness, connecting discrete arithmetic to spatial geometry in a mathematically precise way. Folding is a major contributor to arithmetic efficiency for operations such as: squaring, raising to a power, computing rectangular products, and more generally for any integer admitting a compact rectangular or square decompositions Energetic functions. The framework extends across the integers, rationals, reals, and complex numbers, giving rise to energetic functions that attach a constructive cost to every value of every analytic function at every precision. This opens a new axis of investigation for special functions, the Riemann zeta function, and the distribution of primes — through the lens of energetic profiles rather than algebraic properties alone. Scope and research directions. GTEA is not simply a new way of counting additions. It replaces "arithmetic as linear inscription" with "arithmetic as a competition of geometries" — a unifying framework for complexity, geometry, and energy. With the Competitive Operational Framework and the Folding Algebra, it has the potential to become a new branch of constructive arithmetic, as significant in its time as addition chains in the 1960s or mathematical origami in the 2000s. In analytic number theory, the energetic profile of the Riemann zeta function, the Collatz dynamics, and the prime distribution constitute entirely new objects of study. The theory of energetic-functional equations — constraints of the form H (f, Ef) = 0 — opens a class of problems analogous to functional equations in mathematical physics. In the foundations of computation, GTEA connects naturally to the thermodynamics of computation (Landauer's principle), algebraic complexity theory (Blum-Shub-Smale), and approaches that derive arithmetic structure from physical principles. The notion of constructive cost independent of any primitive offers a machine-independent measure of computational irreducibility. In algorithmic optimization and cryptography, the competitive framework generalizes addition chains to several families simultaneously, with direct applications in compiler design (arithmetic circuit depth reduction), modular exponentiation, and quantum state preparation. The energetic hardness of prime numbers offers a structural perspective on factorization difficulty independent of classical number-theoretic arguments. In constructive geometry, the Folding Algebra creates a direct bridge to the mathematical origami community (Lang, Alperin, Demaine), opening new questions on the minimal energetic cost of geometric configurations and a new chapter in discrete geometry. In number theory, precision-dependent accessibility laws suggest a new classification of transcendental numbers by their energetic profile — a criterion structurally independent of classical irrationality measures and Diophantine approximation exponents. We believe that energetic profiles will become as natural a tool in arithmetic as Kolmogorov complexity is in information theory. The present work lays the foundations. The open space ahead is vast.
Sylvain Geffroy (Mon,) studied this question.
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