A Geometric Origin of Prime Numbers: The Prime-Property Generator as a Geometry-First Model of Primality

A Geometric Origin of Prime Numbers: The Prime-Property Generator as a Geometry-First Model of Primality This paper presents a geometry-first framework for interpreting prime numbers as arithmetic representations of canonical states generated by an underlying geometric symmetry structure. The central object of the framework is the Prime-Property Generator (PPG). Rather than treating prime numbers as primitive numerical objects and subsequently studying their properties, the model reverses the explanatory direction: a finite geometric state space is equipped with local symmetries, orbit structures, canonicalization rules, and global geometric relations. Under an arithmetic projection, the resulting canonical states correspond to the objects conventionally called prime numbers. The paper therefore distinguishes between the generative geometric structure and its conventional arithmetic representation. In this interpretation, primality is not introduced as the fundamental source of the structure; rather, the numbers conventionally labeled as primes arise as arithmetic labels of canonical geometric states. A central feature of the construction is the interaction between local mirror symmetries, finite group actions, hypercube-like orbit structures, and global antipodal geometry. These structures provide a common geometric framework in which several familiar prime-number properties can be represented as arithmetic projections of geometric relations. The paper deliberately makes no claim to prove classical open problems such as Goldbach's conjecture, the twin-prime conjecture, or Polignac's conjecture. Such problems are discussed only as examples of arithmetic questions that may admit a geometric interpretation within the same underlying structure. The primary objective is therefore not to introduce another prime-number sieve or a more efficient primality-testing algorithm. Instead, the paper proposes a different explanatory perspective: geometric structure ⟶ canonical states ⟶ prime properties ⟶ arithmetic representation. The resulting framework suggests that the familiar sequence of prime numbers may be viewed not as the fundamental object itself, but as the numerical projection—or arithmetic ``shadow''—of a richer geometric structure. The paper develops this interpretation formally and identifies the mathematical correspondence between the geometric construction and the conventional prime-number representation, while keeping the distinction between established mathematical results and the proposed geometric interpretation explicit.