Triadic Probabilistic Balance Theory: Unifying Prime Distribution through Euler's Constant

This paper introduces a comprehensive probabilistic framework that connects prime number distribution with Euler's constant (gamma, approximately 0.57721). Building upon ternary balance representations of even integers N=p+q+2k, we develop the Triadic Probabilistic Balance Theory, which establishes gamma as the fundamental constant governing equilibrium in prime distributions. The theory reveals: Exponential decay of the neutral pole k: Expressed as P(k > sqrt(N)) <= exp(-gamma * ln(N)). Gaussian convergence of balanced representations. Phase transitions at beta_c = gamma / (ln(ln(N))). Potential applications include quantum-resistant cryptography and advancements in particle system simulations. This work bridges analytic number theory, probability theory, and statistical physics through the unifying lens of Euler's constant.