Operational Geometry: Foundations of a Geometry of Operations

We construct a formal axiomatization of Operational Geometry: a mathematical foundation in which the primitive entities are operations rather than points, sets, or manifolds. Objects are rigorously defined as equivalence classes of operational sequences, morphisms are generative operations (divide, iterate, rotate, scale, thread), and constants such as φ, π, e, and primes emerge as fixed points or invariants under specific classes of operations. This framework defines a category OpGeom equipped with composition, monoidal product, and primitive invariant functionals. We provide explicit constructions showing how all objects emerge from a unit element via finite or limiting sequences of operations, and we establish a topology on the object space. Crucially, we reveal that constants form an infinite hierarchy indexed by prime numbers, with each prime p generating a constant Ω_p from the fixed point of x = 1 + 1/x^(p−1). This suggests deep connections to physical constants and the Riemann zeta function. The goal is to provide a structural and operational foundation parallel to Euclidean geometry, constructive mathematics, and categorical algebra, but tailored for dynamic generation of form rather than static description.