"TOPO-2026_searchable.pdf" introduces a universal, architecture-agnostic solution to catastrophic forgetting, a 37-year-old obstacle in artificial intelligence. Developed by Frank Morales Aguilera, the framework leverages Arithmetic Spectral Theory to implement an "artificial hippocampus" that stabilizes neural networks during sequential learning. Theoretical Core Prime-Anchored Protection: The framework functions by anchoring six specific embedding rows at prime indices {2, 3, 5, 7, 11, 13}, which the research identifies as the "Pure Kernel". Spectral Mechanism: These six primes capture 97.85% of the total spectral weight, creating a "spectral trap" at the critical line ($\sigma = 0.5$) that protects established representations from being overwritten by new gradient updates. Biological Inspiration: The methodology is derived from the neuroimaging principles of fMRISTAT, developed by the author with Keith Worsley and Alan Evans, emphasizing the strategy to "fix the reference and let the rest adapt". Operational Impact Resource Efficiency: TOPO-2026 maintains $O(1)$ memory complexity, requiring only 67.5 KB to 403.5 KB of anchor memory regardless of the model's total parameter count. Computational Overhead: Implementation adds minimal overhead of approximately 0.11 ms per training step. Stability: The framework was validated across 122 billion parameters, including dense, sparse, and fine-grained MoE architectures, demonstrating 0.25% average forgetting and absolute numerical stability (zero NaN/Inf values across 1.99 billion elements). Experimental Findings Performance: Models certified with TOPO-2026 achieved a 94.2% accuracy on demanding cross-domain benchmarks. Backward Transfer: Certain sparse architectures, specifically Mixtral-8x7B, exhibited up to 6.12% backward transfer, indicating that the protected anchors facilitate improved retention rather than just stability. Reproducibility: All findings are fully reproducible using the deterministic seed of 123. The framework serves as a tribute to the legacy of Keith Worsley and Alan Evans, providing an open-source, practical, and highly efficient solution for enabling lifelong learning in artificial intelligence.