Prime Anchor: Solving Catastrophic Forgetting in GPT-2, TinyLlama, and Mistral via Spectral Topology and the Riemann Hypothesis

Executive Summary

The paper introduces the Prime-Anchored Large Language Model (LLM), a novel, model-agnostic architecture designed to mathematically eliminate catastrophic forgetting during continual learning. By anchoring specific hidden representation vectors to prime number coordinates and enforcing strict topological constraints derived from the Riemann Hypothesis, the architecture ensures that previously acquired knowledge remains perfectly intact during subsequent training phases. The framework was successfully validated across four distinct model families ranging from 124 million to 7 billion parameters.

Core Mathematical Foundations

The architecture transitions continual learning from probabilistic mitigation to deterministic mathematical guarantees by utilizing three core principles:

1. The L-EFM Spectral Operator

The Laplace-Euler-Fourier-Mellin (L-EFM) operator synthesizes four classical transforms into a single spectral instrument to evaluate the Euler product for a given prime set at a complex argument $s = \sigma + i\gamma$:

Logarithmic frequencies map the additive frequency domain to the multiplicative group structure of the primes. A central invariant dictates that at the Riemann Hypothesis critical line ($\sigma = 0.5$ and $\gamma = 0$), the spectral coherence collapses to exactly $0.5$, completely independent of the selected primes.

2. The Spectral Trap

When evaluating the operator at $\gamma = 0$, the normalized magnitude diverges exponentially for any state off the critical line:

This exponential divergence creates a geometric "trap" that establishes $\sigma = 0.5$ as the unique stable, admissible spectral state for hidden representations.

3. The H2E Closure Constant ($\Lambda_{12}$)

Derived from the first six primes $\mathcal{P} = \{2, 3, 5, 7, 11, 13\}$, an absolute safety threshold is established as an invariant boundary:

This constant represents the maximum allowable spectral dispersion before a latent representation is categorized as incoherent.

Architectural Implementation

The Prime-Anchored system operates on top of existing causal language models via a dual-loop layer governance mechanism:

• Model-Agnostic Layer Detection: The architecture features an automated routing mechanism that scans Hugging Face models to locate the token embedding matrix, regardless of the specific underlying setup (e.g., locating transformer.wte for GPT-2 or model.embed_tokens for TinyLlama and Mistral).

• The Prime Anchor Vector: Rows in the embedding matrix corresponding to the first six prime indices $\mathcal{P} = \{2, 3, 5, 7, 11, 13\}$ are designated as an immutable coordinate origin. These rows are explicitly locked; after every valid gradient step, they are restored to their cached values.

• Dual-Loop Governance: * Inner Loop: Standard cross-entropy loss ($L_{CE}$) handles next-token prediction.

  • Outer Loop: A topological regularization penalty ($L_{topological} = |Var(h)-0.5|\cdot\lambda$) forces the final-layer hidden states ($h$) toward the $\sigma = 0.5$ line.

• The H2E Safety Gate (Sheriff Gate): At each gradient step, the system calculates the Spectral Radius of Incoherence (SROI):

  $$SROI=1-0.5\cdot \tanh(\text{std}(h))$$

  If $SROI \ge \Lambda_{12}$, the gradient update is allowed via optimizer.step(). If it falls below the threshold, the update is rejected and neutralized via optimizer.zero_grad(), blocking destructive updates before they can corrupt the anchored manifold.

• Cryptographic Manifold Lock: The prime-anchored embedding rows are fingerprinted using a SHA-256 hash. If the hash remains identical down to the final bit after training on conflicting data, structural invariance is mathematically proven.

Experimental Validation & Results

Testing followed a strict two-stage deterministic protocol (SEED = 123). Models were first trained on Dataset A (establishing core mathematical framework memory) and subsequently exposed to Dataset B (random names and noise designed to induce catastrophic forgetting).

GPT-2 Comparative Performance

A baseline GPT-2 model was compared against the Prime-Anchored version. While the baseline model's embedding rows drifted significantly after exposure to Dataset B (altering its cryptographic hash and causing forgetting), the Prime-Anchored model maintained a perfectly invariant hash and successfully recalled core concepts from Dataset A.

Cross-Architecture Verification

To demonstrate the model-agnostic nature of the architecture, validation was conducted across four distinct families, spanning two orders of magnitude in parameter scale:

The framework succeeds across different architectures because low-integer prime indices represent valid vocabulary rows within the first 50 to 100 tokens of virtually any causal LLM, bypassing variations in attention mechanisms or activation functions.

Limitations and Future Directions

Current Limitations

• Validation for larger scale models (Mistral-7B) has currently been executed at a single training-step level.

• The anchor set is fixed to the first six primes.

• Training requires continuous, direct access to hidden states to calculate the SROI.

Future Work

1. Scaling the framework to production fine-tuning on modern models like Llama 3 and Qwen.

2. Implementing dynamic prime anchoring where the anchor set scales proportionally with model parameter size.

3. Expanding anchoring principles to multimodal architectures, including vision (ViT) and audio transformers.

4. Developing automated theorem provers for absolute spectral safety verification.

5. Adapting the H2E gating mechanism for secure, distributed governance in federated learning environments.