Dynamical Attractor Groupoids and Operator Algebras: A Non-Discrete Geometrical Paradigm for Transcendental Continuum Mechanics

This work introduces a novel mathematical framework, termed Dynamical Attractor Groupoids and Wave-Operator Algebras (DAG-WOA), designed to bypass the fundamental limitations of discrete, Turing-complete algorithmic computation when modeling continuous physical systems. Instead of treating the real number line as an agglomeration of discrete symbol-strings or decimal expansions, this framework defines numbers as invariant topodynamical structures emerging from multi-dimensional continuous phase spaces. By shifting the foundational ontological status from numbers to geometric attractors, transcendental quantities such as π and e are naturally represented as stable phase-boundary topologies rather than infinite non-repeating limits. We formalize the structural basis of this framework using topological groupoids for additive-confluent behaviors and non-commutative C∗-algebras for projective-multiplicative modulation. We supply rigorous syntactic operators for execution—namely Confluence (⊕), Interference (⊖), Modulation (⊗), and Amostragem (⊘)—and demonstrate their validity through three foundational physical case studies: classical circular area projection, continuous kinematic gravity fields, and wave-string harmonic resonance. Finally, we map the epistemological advantages and energetic bottlenecks of this continuous paradigm, presenting a rigorous path toward physical, non-algorithmic computing.