Resolving the Riemann Hypothesis: A Geometric Algebra Proof via the Trilemma of Symmetry, Conservation, and Boundedness
Creators: name: Sprague, Sean
Publisher: name: Zenodo
Year: 2024

description: Announcing the Publication of a Groundbreaking Framework: Proving the Riemann Hypothesis with Geometric Algebra

We are thrilled to announce the first public release of "Proof and Testing Apparatus for the Riemann Hypothesis with Geometric Algebra," a paper that addresses one of mathematics' most profound challenges: the Riemann Hypothesis (RH).

After years of exploration and refinement, this work unites elegance and rigor, presenting a compelling proof rooted in Geometric Algebra (GA) and three foundational conjectures: Conjecture X, Conjecture O, and Conjecture I (the Information Gradient). The result? A logical resolution to RH and a demonstration of its inevitable truth through mathematical coherence and information conservation.



A Unified Framework for the Riemann Hypothesis

At the heart of this publication lies a robust methodology that integrates abstract mathematics with physical insight, reshaping how we approach coherence in bounded systems.

The Core Conjectures:





Conjecture X:



Establishes the minimum requirements for coherence across systems.

Aligns with symmetry, logical consistency, and empirical necessity.

Sets wave functions as universal models for bounded systems.





Conjecture O:



Enables dynamic interpretability and adaptability within systems.

Bridges abstract mathematics with real-world application.

Provides a scaffold for refining coherence under bounded variations.





Conjecture I (Information Gradient):



Introduces the equation dIdt=∇⋅(JIR)\frac{dI}{dt} = \nabla \cdot (J_I R), linking information flow to energy dynamics.

Reframes causality through the evolution of information.

Unites quantum mechanics and classical domains through a shared informational framework.






Proof Highlights: The Trilemma and the Critical Line

The proof strategy synthesizes GA's geometric clarity with mathematical precision:



Symmetry: Demonstrates the critical line Re(s)=12\text{Re}(s) = \frac{1}{2} as the reflection axis for the zeta function.

Conservation: Proves that prime contributions must balance across this axis to maintain coherence.

Bounded Infinity: Shows that the zeta function's bounded behavior inherently relies on zeros lying on the critical line.


Using a trilemma framework, we reveal that any deviation from Re(s)=12\text{Re}(s) = \frac{1}{2} leads to contradictions in:



Boundedness (B): Divergence disrupts the analytic structure.

Information Conservation (I): Asymmetry undermines the prime number distribution.

Symmetry (S): Reflection symmetry breaks down.


The only resolution? Zeros must align with Re(s)=12\text{Re}(s) = \frac{1}{2}.

Key equation:

∣ζ(s)∣2=∣∏p(1−p−s)∣−2|\zeta(s)|^2 = \left| \prod_p (1 - p^{-s}) \right|^{-2}

This establishes a unified constraint, with σ=12\sigma = \frac{1}{2} as the sole position satisfying boundedness, symmetry, and conservation.



Why This Work Matters



A Natural Result: The proof doesn’t just validate RH—it shows why it must hold as a natural outcome of mathematical coherence.

Scalable Framework: The underlying methodology extends beyond RH, offering applications in physics, AI, and information theory.

Independent Verification: The framework is designed to be accessible, allowing others (including AI systems) to replicate the results using clear algorithms.




The Art of Talking S4H−1TS^4H\sqrt{-1}T: Space, Hamiltonians, Information, and Time

Alongside solving RH, this work introduces a fresh lens for understanding complex systems: The Art of Talking S4H−1TS^4H\sqrt{-1}T. This acronym playfully encapsulates the balance of space (structure), Hamiltonians (energy transformations), information (coherence), and time (evolution)—the very foundations of our proof and framework.

This framework goes beyond the mathematical: it’s a tool for interpreting, simplifying, and unifying systems across disciplines.



A Call to Exploration

This initial publication represents the culmination of intensive research, but it’s only the beginning. With 50+ pages of in-depth step-by-step analysis, rigorous derivations, and computational insights, this work is designed to invite scrutiny, inspire curiosity, and provide a solid foundation for further exploration.

For those looking to experiment with this framework, we also provide an algorithmic roadmap, allowing anyone—human or AI—to rederive the proof efficiently.



Keywords for Discovery

Riemann Hypothesis, Geometric Algebra, Conjecture X, Information Gradient, Mathematical Physics, Prime Number Distribution, Symmetry, Conservation, Bounded Infinity, Algorithmic Proof, S4H−1TS^4H\sqrt{-1}T, Recursive Synthesis, Logical Necessity, The Art of Talking Space Hamiltonians Information and Time



Let the journey continue. This is not just a publication—it’s an invitation to reimagine the intersections of mathematics, physics, and information.