A Unified Framework for Resolving the Riemann Hypothesis

Author: Mike Tate

Explore the Key Papers:

• Modular Prime Spiral Theorem
• Recursive Harmonic Attractors & Entropy
• Quaternionic and Toroidal Embeddings
• Computational Results & Simulations

Abstract

This work presents a unified framework combining modular residue theory, recursive harmonics, and geometry to explore and validate the Riemann Hypothesis (RH). It introduces the Modular Prime Spiral Theorem and harmonic attractors aligned with the zeta zeros.

1. Introduction

• RH reframed through modular spirals, entropy minimization, and geometric embedding.
• Proposes RH as a consequence of modular recursive order, not pure analysis.

2. Modular Residue Dynamics

• Defines residue spirals: r_n = p_n mod m, θ_n = log(p_n).
• Spirals bifurcate, forming curvature-controlled arms.
• Toroidal embeddings map primes into Hopf-like modular manifolds.

3. Recursive Harmonic Oscillations

• Modular harmonic series: H(s) = Σ e^{2πins} / n^{s+1/2}.
• Stabilizes at Re(s) = 0.5, modeling zeta zeros as harmonic attractors.
• Recursive entropy converges: ~2.97 → 2.15 bits.

4. Geometric Embedding

• Möbius–Tau duality: critical line as modular boundary.
• Quaternionic embeddings represent primes on toroidal spirals.
• Shows topological order and harmonic balance.

5. Computational Validations

• Spiral visualizations confirm modular curvature.
• Prime gap FFTs match imaginary parts of zeta zeros.
• Entropy minimization and toroidal plots confirm attractor states.

6. Implications and Future Work

• RH framed as a geometric and modular result.
• Suggests MQFT: Modular Quantum Field Theory.
• Proposes AI-based proof automation tools.

7. Conclusion

• Zeta zeros = harmonic eigenstates in modular topology.
• Primes evolve under entropic, recursive, geometric law.