From Primitive Root Cycles, Absolute Zero, and π to the Riemann Hypothesis: A Derivational Approach Under the Information-Potential Conversion Framework

The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie precisely on the critical line Re(s)=1/2 in the complex plane. This paper provides a first-principles derivational approach to this conjecture under the Information-Potential Conversion Framework. The mathematical starting point is a rigorous group-theoretic fact: the 124875 cycle (modulo-9 multiplicative group) derived from cell division and the 142857 cycle (modulo-7 multiplicative group) derived from 1/7 are isomorphic groups. This isomorphism expresses the Information-Potential Conservation Principle. We construct a complete derivation chain: (1) Prime numbers are redefined as markers of information processing events, and the zeta function is interpreted as the potential generating function of these events; (2) Nested primitive root scheduling generalizes discrete traversal to multi-scale hierarchies, with pi emerging as the geometric invariant in the continuum limit; (3) Absolute zero is the benchmark for information processing cost, with a precise numerical link to the fine-structure constant and modulo-13 structure; (4) Under traversal completeness and information-potential conservation, the principle of least action yields Re(s)=1/2 as the unique optimal solution. The Mpemba effect provides physical paradigm support for nonlinear optimal path selection. We propose four testable quantitative predictions using existing zeta-zero data.

Riemann Hypothesis, Information-Potential Conversion, Primitive Root Cycles, Absolute Zero, pi, Fine-Structure Constant, Mpemba Effect