Published October 1, 2026 | Version v1
Preprint Open

Zero-Drift Continual Backpropagation via Arithmetic Spectral Theory: Eliminating Catastrophic Forgetting Through Prime-Indexed Weight Anchors

Authors/Creators

  • 1. Sovereign Machine Lab (SOMALA)

Description


Paper Overview

  • Title: Zero-Drift Continual Backpropagation via Arithmetic Spectral Theory: Eliminating Catastrophic Forgetting Through Prime-Indexed Weight Anchors
  • Author: Frank Morales Aguilera (Sovereign Machine Laboratory)

1. The Core Problem

Traditional neural networks suffer from catastrophic forgetting and latent coordinate shear during continuous training. Standard gradient descent and optimizer weight decay treat every parameter as equally malleable, which instantly disrupts foundational representations on the very first optimization step. This forces networks onto a shifting foundation, leading to progressive degradation and memory corruption over time.

2. The Topological Solution: Topo-CBP

Instead of relying on heuristic safety wrappers or brute-force parameter scaling, the paper introduces Topology-Continual Backpropagation (Topo-CBP), governed by Arithmetic Spectral Theory (AST).

  • Prime-Indexed Anchoring: Core structural geometry is locked to specific prime coordinate rows ($\mathcal{P} = \{2, 3, 5, 7, 11, 13\}$).
  • Dual-Phase Control Loop:

    1. Pre-Optimizer Gradient Zeroing: Forcing gradients to zero at prime indices ($\nabla_{\mathcal{P}} \mathcal{L} = 0$) so gradient descent cannot rewrite anchor states.
    2. Post-Optimizer Anchor Clamping: Hard-clamping tensors back to the registered anchor_bedrock immediately after the optimizer step to neutralize numerical precision drift.
  • Protective Unit Recycling: Shielding prime coordinates with infinite utility scores ($\infty$) during dormant unit reclamation, ensuring structural permanence during network renewal cycles.

3. Empirical Validation & Results

Tested on NVIDIA L4 Tensor Core GPUs across 1,000 continuous training steps and 100 metabolic recycling cycles, the framework delivered absolute structural permanence:

Step Baseline Hash Baseline Status Baseline Drift Topo-CBP Hash Topo-CBP Status Topo-CBP Drift
Initial 2ce28ffcf131d5c9 LOCKED 0.0000000000 2ce28ffcf131d5c9 LOCKED 0.0000000000
1 6cccef039e2c45c7 CORRUPTED 0.0283359457 2ce28ffcf131d5c9 LOCKED 0.0000000000
200 e33122698286b741 CORRUPTED 0.3726324439 2ce28ffcf131d5c9 LOCKED 0.0000000000
400 307a650433833888 CORRUPTED 0.4863870442 2ce28ffcf131d5c9 LOCKED 0.0000000000
600 a43a99f48bf9ef4d CORRUPTED 0.5534539223 2ce28ffcf131d5c9 LOCKED 0.0000000000
800 bf68ba7e07f536c6 CORRUPTED 0.6082839370 2ce28ffcf131d5c9 LOCKED 0.0000000000
1000 51b609f1ca1eb786 CORRUPTED 0.6635395885 2ce28ffcf131d5c9 LOCKED 0.0000000000
  • The Baseline Failure: Suffers immediate coordinate shear at Step 1 and compounds drift up to 0.6635 by Step 1000.
  • The Topo-CBP Triumph: Holds an unyielding cryptographic digest (2ce28ffcf131d5c9) and achieves an exact 0.0000000000 Frobenius drift throughout the entire run.

4. Broader Implications & Future Deployments

  • Hardware-Level Security: By pairing internal invariant verification ($\Delta\text{Drift} = 0.0$) with eBPF transport gates, unauthorized out-of-envelope mutations trigger sub-500µs wire cuts to halt gradient updates instantly.
  • Edge & Multi-Agent Intelligence: Eliminating catastrophic forgetting through prime anchors unlocks ultra-low-bit-depth LLM compression, enabling dense, zero-drift multi-agent fleets to run locally on standard smartphone hardware.

Files

CBP-TOPO.pdf

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