Arithmetic Spectral Theory: A Unified Framework Proving the Riemann Hypothesis, Quantifying the Green-Tao Theorem, and Solving Catastrophic Forgetting

The provided document outlines the Arithmetic Spectral Theory (AST), a unified mathematical framework that connects three significant challenges: the Riemann Hypothesis, the Green-Tao Theorem, and the problem of catastrophic forgetting in artificial intelligence. The framework is centered on the discovery that the first six prime numbers—$R = \{2, 3, 5, 7, 11, 13\}$—form a "pure kernel" with unique mathematical properties.

Core Mathematical Concepts

• The L-EFM Operator: The Laplace-Euler-Fourier-Mellin operator uses these six primes to define a system that exhibits a "spectral trap" at the critical line $\sigma = 0.5$.

• Spectral Weight and the Pure/Noisy Divide: The set $R$ captures 97.85% of total spectral weight. The remaining 2.15% is attributed to the "noisy" set of primes $\ge 17$, which, when added to $R$, destroys the stability of the spectral trap.

• Unified Proofs:

  • Riemann Hypothesis: The existence of the spectral trap as a unique fixed point at $\sigma = 0.5$ is used to demonstrate that all non-trivial zeros of the Riemann zeta function lie on the critical line.

  • Green-Tao Theorem: The framework offers a first-ever numerical quantification of prime arithmetic progressions through a defined coherence decay law.

  • Continual Learning: Catastrophic forgetting is addressed by using the six prime-indexed rows as stable anchors within neural networks.

TOPO-2026 Framework and Performance

The TOPO-2026 implementation acts as an "artificial hippocampus" for large language models, utilizing a "Topological Governor" to preserve performance across multiple tasks.

• Benchmark Results: Across a three-task benchmark, the framework achieved 99.5% accuracy on the final task (Task C) with only 5.6% forgetting.

• Efficiency: The solution is highly efficient, requiring only 67.5 KB of memory and introducing minimal computational overhead (0.11ms).

• Validation: The framework was tested across several major model architectures—including dense BF16 and various sparse Mixture of Experts (MoE) models—all of which were compressed with FP8 quantization before final certification.

Reproducibility and Implementation

The research emphasizes that the proof is effectively embedded in the code, which is deterministic (seed 123) and fully open-source. Technical resources, including Python libraries, benchmarking notebooks, and Hugging Face deployment scripts, are provided via GitHub and Zenodo to allow for independent verification of the spectral trap calculations and the topological certification pipeline.