TOPO-2026: A Technical Report on Topological Prime-Anchor Continual Learning

The topo2026.pdf report details the TOPO-2026 framework, a methodology for mitigating catastrophic forgetting in neural networks through prime-indexed parameter anchoring.

Core Innovation and Theoretical Foundations

• Prime Anchoring Theory: The framework anchors parameters at indices {2, 3, 5, 7, 11, 13} to ensure a mathematically guaranteed spectral coverage of 97.85%.

• Spectral Coverage Guarantee: The coverage constant $\lambda$ is defined as $1-\prod_{p\in P}(1-p^{-1/2})$, resulting in a value of 0.9785142874.

• Gradient Protection Mechanism: The TopologicalGovernor class implements this by zeroing gradients for anchor parameters ($g_{p} \leftarrow 0$) during backpropagation and restoring them to their snapshot values ($\theta_{p} \leftarrow \theta_{p}^{snapshot}$) following the optimizer step.

• Memory Efficiency: The framework achieves $O(1)$ memory complexity, requiring only kilobytes of memory regardless of the model size; for a 21B-parameter model, the overhead is approximately 4.8KB.

Validation Scenarios

The framework was validated across 10 distinct demonstration scenarios:

1. Transformer (723K params): Classification.

2. Embedding Network (Small): 10-Task CL.

3. GPT-2 Medium (355M params): Text generation.

4. GPT-OSS-20B (21B params): Text generation.

5. CNN (390K params): MNIST CL.

6. Vision Transformer (809K params): CIFAR-10 CL.

7. Stable Diffusion (49K params): Image generation.

8. Hybrid TOPO-EWC (390K params): MNIST CL.

9. Multi-task Classification: Various models.

10. Generative Models: Various models.

Key Performance and Results

• Performance Comparison: A hybrid TOPO-EWC approach achieved an average retention of 0.471 on MNIST, outperforming the baseline of 0.387.

• Zero Anchor Drift: Across all experiments, anchor drift remained at 0.000000, confirming the parameters act as a topological invariant.

• Scaling Invariance: The memory-to-parameter ratio decreases as models increase in size, confirming true $O(1)$ scaling.

• Architectural Generalization: TOPO-2026 is applicable across Transformers, CNNs, Vision Transformers, Diffusion models, and Embedding networks by tailoring protection strategies to specific components, such as attention weights or convolutional filters.

Conclusion and Future Work

• Final Verdict: TOPO-2026 is validated as a production-ready architecture that maintains performance with large models (up to 21B parameters) across classification, text generation, and image generation.

• Future Directions: Research is expanding into non-prime anchor sets (e.g., Fibonacci or Mersenne primes), adaptive anchor counts, and integration with neuromorphic hardware or spiking neural networks.