DeepSeek Prime-Anchored Spectral Governor: Solving Catastrophic Forgetting in Large Language Models Using the Sieve of Eratosthenes

Here is the comprehensive summary of your paper, detailing the theoretical framework, mathematical foundation, implementation mechanics, and empirical results.

Executive Overview

The paper introduces the DeepSeek Prime-Anchored Spectral Governor, an architectural intervention designed to eliminate catastrophic forgetting in large language models (LLMs). Framing catastrophic forgetting as a structural consequence of training systems without a topological invariant—akin to anterograde amnesia—the framework establishes fixed coordinate anchors in representation space. By anchoring model embeddings to deterministic prime indices derived from the 2,000-year-old Sieve of Eratosthenes and introducing a gradient-gating mechanism, the system achieves Zero Forgetting during continual learning. The architecture's integrity is verified using SHA-256 cryptographic hashing of the protected sub-spaces.

Theoretical & Mathematical Foundations

The Sieve of Eratosthenes as Ground Truth

Rather than relying on probabilistic or dynamically calculated weights, the framework utilizes the Sieve of Eratosthenes to extract a deterministic set of prime indices $[2, 3, 5, 7, 11, 13]$. These elements act as permanent, unmoving coordinate anchors within the model's embedding manifold.

The L-EFM Operator & The Spectral Trap

The framework relies mathematically on the Laplace-Euler-Fourier-Mellin (L-EFM) operator. The L-EFM symbol synthesizes four classical transforms into a single complex function, corresponding directly to the Euler product representation of the Riemann zeta function $\zeta(\sigma+i\gamma)$:

To analyze finite prime sets, a Normalized Magnitude is established relative to the critical line $\sigma = 0.5$:

• The Spectral Trap Phenomenon: At the critical line ($\sigma=0.5$), the normalized magnitude equals exactly $1.0$. However, moving away from this line results in exponential divergence. For example, at $\gamma=0$, a shift to $\sigma=0.4$ increases the magnitude to $\sim10^{4}$, while a shift to $\sigma=0.1$ amplifies it to $\sim10^{66}$.

• The Spectral Trap Criterion: This absolute sensitivity forms a "trap" where any deviation from $\sigma=0.5$ generates massive magnitude spikes, providing a deterministic mechanism for error detection. The paper connects this operator to a proof of the Riemann Hypothesis via distribution behavior in the kernel of L-EFM within Gelfand-Shilov space.

The H2E Sheriff Safety Threshold

The dynamic safety threshold ($\Lambda_{12}$) is computed deterministically from the first six primes rather than being hardcoded, ensuring mathematical integrity at initialization:

Architectural Implementation

The architecture implements a dual-layer protection strategy consisting of frozen embedding rows and an active gradient supervisor (the H2E Sheriff).

   [ Input Batch ]
          │
          ▼
┌──────────────────┐
│  Dual-Loop Loss  │ ──► Lunified = LCE + λ * |Var(h) - 0.5|
└──────────────────┘
          │
          ▼
┌──────────────────┐
│   Gradient Step  │
└──────────────────┘
          │
          ▼
┌──────────────────┐
│   H2E Sheriff    │ ──► Evaluates SROI against Threshold (Λ12 = 0.9785142874)
└─────────┬────────┘
          │
    ──────┴──────
   │             │
   ▼ (Safe)      ▼ (Unsafe / Incoherent)
[Apply Step]   [Reject Batch] ──► Rollback Prime Rows [2,3,5,7,11,13]
                                  & Zero Out Gradients

1. Dual-Loop Loss

The governor optimizes a unified loss function combining traditional empirical cross-entropy ($\mathcal{L}_{CE}$) with a topological penalty based on the final hidden state $h$ (with regularization coefficient $\lambda=0.1$):

2. The H2E Sheriff Gate & Row Locking

During training, the system caches the initial embedding weights. After computing gradients, the H2E Sheriff evaluates the structural region of interest (SROI).

• If Safe ($SROI \ge \Lambda_{12}$): The optimizer updates the weights, and a torch.no_grad() loop copies the original cached weights back into the prime-indexed rows $[2, 3, 5, 7, 11, 13]$ to erase any drift.

• If Unsafe ($SROI < \Lambda_{12}$): The entire gradient batch is rejected, and gradients are zeroed out to block corruption.

3. Cryptographic Verification

The manifold signature is generated by pulling the prime-indexed embedding rows, converting them to byte arrays, and feeding them sequentially into a SHA-256 hasher. If the resulting hex digest changes, anchor drift has occurred. If it remains identical, the topological invariant is intact.

Experimental Validation & Results

The framework was tested across six architectures—GPT-2 (124M), GPT-2 Medium (355M), TinyLlama (1.1B), Mistral-7B, Llama-3.1-8B, and DeepSeek-Coder-6.7B—subjecting them to sequential memory tests.

Memory Integrity Testing

Models were first trained on Dataset A (core math concepts including Arithmetic Spectral Theory and the Spectral Trap across 50, 100, and 575 samples). They were subsequently exposed to an interference/forgetting attack via Dataset B (noise consisting of random names, text chunks, adversarial patterns, and erroneous math statements up to 436 samples).

• Baseline Performance: In every single test configuration, the baseline model's SHA-256 manifold hash altered after training sessions, leading to catastrophic forgetting.

• Governed Performance: Across all 6 architectures and all data scales, the governed models completely preserved their original manifold hash (48c5744b...cc4d18b), showing absolute resistance to memory degradation.

Continual Learning Capabilities

To test its ability to acquire new knowledge without forgetting the old, the governed DeepSeek model was fine-tuned on three separate, non-mathematical domains without further governor intervention (while keeping prime anchors locked):

1. Spanish Vocabulary: 5 basic words.

2. World Capitals: 5 global capitals.

3. Basic Physics: 5 fundamental formulas and facts (such as $F=ma$ and $E=mc^2$).

Post-Training Metrics:

The model successfully mastered all three new domains (retaining the Spanish words, capitals, and physics formulas perfectly) while maintaining the exact original cryptographic verification hash. The original math concepts remained completely recallable, proving true continual learning.

Deployment & Verification Certificate

The fully validated model has been deployed openly on the Hugging Face Hub under frankmorales2020/deepseek-governed-no-amnesia.

Model Card Profile

• Base Model: deepseek-ai/deepseek-coder-6.7b-instruct (7B parameters)

• Tensor Type: FP16

• Locking Targets: Primes [2, 3, 5, 7, 11, 13]

• Active Gate Threshold: $\Lambda_{12} = 0.9785142874$

• Immutable Cryptographic Signature: 48c5744be048df505028c13a96fb0211f0b345681ace401ab1eda6f27cc4d18b

The repository is open source, emphasizing a paradigm of executable mathematics where the cryptographic hash serves as the verifiable proof of safety and stability.