TOPO-2026: Universal Continual Learning Across Modalities Empirical Validation on Vision-Language Models and Formal Proof via Arithmetic Spectral Theory

The research in "topo-2026-fixed.pdf" validates the TOPO-2026 continual learning framework across six architecturally distinct systems, totalling approximately 124B parameters. By anchoring six embedding rows at prime indices {2, 3, 5, 7, 11, 13}, the framework successfully mitigates catastrophic forgetting.

Key highlights include:

• Cross-Modal Validation: The study provides the first demonstration of TOPO-2026 on a vision-language model, the Gemma-4 E4B Vision, which achieved 100.0% Task C accuracy with 0.0% forgetting across five runs.

• Mathematical Foundation: The framework is grounded in Arithmetic Spectral Theory, identifying a spectral trap via the L-EFM operator at $\sigma=0.5$ and quantifying the Green-Tao Theorem.

• Artificial Hippocampus: The mechanism replicates biological memory consolidation, protection, and integration through a take_snapshot() function, gradient zeroing, and anchor enforcement.

• Efficiency: The solution maintains an $O(1)$ memory guarantee, requiring only 451.5 KB of total anchor memory across all six models, resulting in an overhead of 0.00000036%.

• Certified Performance: All 30 experimental runs passed the certification requirements, with several models exhibiting backward transfer rather than forgetting.

• Accessibility: The certified Gemma-4-E4B-Vision model is available at https://huggingface.co/frankmorales2020/gemma-4-e4b-topo-2026, with full implementations provided via GitHub and Zenodo.