Holographic Encoding and Prime Infinity in Zeta Function Theory - From Critical Lines to Infinite-Dimensional Information Unification

This paper establishes a complete holographic encoding framework for infinite primes through Riemann zeta functions. We prove that prime distribution emerges as boundary projections of zeta function analytic structures. Extending this holographic perspective, we now detail the mathematical encodings of prime infinity. By treating the critical line $\text{Re}(s) = 1/2$ as an information boundary, we establish precise mathematical connections between the zeta function and AdS/CFT correspondence, black hole entropy, and quantum chaos. Based on Alain Connes' Hilbert space approach and high-dimensional generalizations, we prove that infinite-dimensional Hilbert space, as a mathematical structure with zero Gaussian measure and infinite-dimensional boundary, perfectly encodes all information about the universe. Core discoveries include: (1) zero point distributions on the critical line follow GUE random matrix statistics, embodying the universality of quantum chaos; (2) prime infinity produces apparent randomness through finite computational resource truncation; (3) physical phenomena such as the Casimir effect and critical dimensions in string theory can be precisely predicted through zeta regularization; (4) hidden holographic imprints of the zeta function exist in CMB fine structure. This work establishes a complete bridge from pure mathematics to physical reality, revealing the ultimate unification of information, computation, and existence.