Geometric Irreducibility: Prime Numbers Under the Icosahedral Action on S³/2I

We introduce the concept of geometric irreducibility — irreducibility under the three-dimensional icosahedral action on the Poincaré homology sphere S³/2I — and show that it is strictly stronger than arithmetic primality. Every geometrically irreducible integer is arithmetically prime; the converse is false. The integers 2, 3, and 5 are arithmetically prime but geometrically reducible: under the icosahedral action, they decompose into the construction operators of S³/2I — the edge-flip (χ = 2), the face-rotation (D = 3), and the vertex symmetry (the golden prime, 5). They are the manifold's structure, not positions on it. The integer 7 = 2^D − 1 is both arithmetically prime and geometrically irreducible: it…