Unified GCD Modular Framework

Mathematical Framework

Core Principle: For each modulus m ≥ 1 and residue r ∈ {0, 1, ..., m-1}, we define the angular location θ_m,r = 2πr/m on the unit circle.

GCD Classification: Each point is assigned to a GCD class based on gcd(r, m):

• gcd(r, m) = 1: Coprime (open channels) - φ(m) points representing primitive directions
• gcd(r, m) > 1: Composite (closed channels) - reduced fractions from smaller moduli

Gaussian Integer Connection: Primitive lattice vectors (a,b) with gcd(a,b) = 1 map to visible angles on the unit circle. When a² + b² = p (prime), these correspond to Gaussian primes with unique representations up to symmetry.

Density Result: The coprime channel density converges to 6/π² (Basel constant), demonstrating the fundamental connection between modular arithmetic and analytic number theory.