Hexagonal Confinement and Stochastic Dynamics of Prime Distribution : A Structural Study via Digital Root Projection

This paper presents a rigorous structural and probabilistic analysis of prime number distribution within the Modulo-9 projection framework, referred to as the Numerical Origin Theory. We establish that all prime numbers greater than 3 are topologically confined to the hexagonal orbit S = 1, 2, 4, 8, 7, 5 ⊂ Z₉, strictly obeying an Exclusion Principle that forbids occupancy of the 36-9 axis. Through exhaustive statistical analysis of N = 100 000 consecutive prime numbers (up to p = 1, 299, 709), we demonstrate that transitions between successive primes within this hexagonal domain follow a structured probabilistic law governed by topological distance. These observations are formalized through the Symmetric Hexagonal Kernel — a Symmetric Circulant Transition Matrix — validated against empirical data. The stationary distribution of this matrix converges exactly to 1/6 per node, in perfect agreement with the observed uniform distribution of primes across the six admissible residue classes. An empirically-corrected version of the kernel is also derived directly from the data.