A Structural Proof of the Riemann Hypothesis via Prime-Indexed Inversion Entropy

In this manuscript, we introduce a structural and computational framework establishing a comprehensive equivalence characterization of the Riemann Hypothesis (RH). We define strictly invertible arithmetic transformations indexed by prime numbers and introduce inversion entropy, a computational complexity measure capturing the difficulty of reversing these transformations. Under this framework, we demonstrate that the validity of RH is equivalent to polynomially bounded inversion entropy. Specifically, we show that RH implies classical prime-gap bounds ensuring polynomial growth of inversion entropy, thereby guaranteeing polynomial-time computability of prime-indexed inversions. Conversely, we prove that polynomially bounded inversion entropy implies RH, as super-polynomial entropy growth contradicts well-established analytic results involving prime gaps and Chebyshev's function. These results culminate in a equivalence theorem situating RH at the intersection of analytic number theory, computational complexity, and arithmetic structure. Our findings provide a deeper and unified structural interpretation of RH, revealing its foundational significance as a universal mathematical and computational law governing prime arithmetic and complexity.