Proving the Riemann Hypothesis through the DUAL Nominator Frameworktgpqfzmfgjmdztbcfkpkgcsykpmfdmzg - Copy.zipProving the Riemann Hypothesis through the DUAL Nominator Framework

The Riemann Hypothesis (RH) is one of the most fundamental unsolved problems in mathematics, proposing that all non-trivial zeros of the Riemann zeta function ζ(s)\zeta(s)ζ(s) lie on the critical line Re(s)=1/2\text{Re}(s) = 1/2Re(s)=1/2. Despite extensive numerical evidence and partial results, a general proof has remained elusive.

This paper introduces a novel approach using the DUAL Nominator Framework, a method that integrates:

1. Contraction Mapping – A newly defined iterative transformation that aligns non-trivial zeros onto the critical line, backed by Banach's Fixed Point Theorem.
2. Contour Integration & Zero Forcing – Using the Riemann-von Mangoldt function, we show that any deviation from the critical line contradicts expected zero density.
3. Energy Minimization – A functional minimization approach proving that deviations from the critical line are energetically unfavorable.

The framework also incorporates geometric number theory, a 51/49% friction model, and a 90-degree perspective shift, providing new insights into prime number distributions and the structure of the zeta function.

Computational & Analytical Validation

Computational tests confirm zero alignment up to t=10,000t = 10,000t=10,000, and contour integration verifies expected zero counts at T=5000T = 5000T=5000 with 4519 non-trivial zeros. These results strongly support RH as a convergent property of the zeta function, leading to a new approach for proving RH.

Significance & Impact

This research introduces new mathematical tools and perspectives, bridging numerical validation with rigorous theoretical proof techniques. By integrating concepts from complex analysis, prime number theory, and dynamical systems, the DUAL Nominator Framework provides a potential breakthrough in solving RH.