Neural Prime Sieves: Density-Driven Generalization and Empirical Evidence for Hardy-Littlewood Asymptotics

1. Problem Statement

The paper addresses a fundamental challenge in computational number theory: identifying members of specific prime families (e.g., twin, Sophie Germain, safe, Chen, isolated primes) from a stream of known primes.

• The Limitation of Classical Methods: Classical sieves eliminate composites but do not assign probabilistic weights to coprime candidates.
• The Limitation of Prior ML: Previous machine learning attempts to predict primality or prime membership have failed because the prime indicator sequence is algorithmically random (Kolmogorov complexity). Models trained on raw integers achieve near-zero true positive rates.
• The Core Question: Can a deep neural network, trained strictly on causal features (information available before the next prime) within a limited range ($10^7$ to $10^9$), learn a probabilistic filter that generalizes to extreme scales ($10^{16}$) and implicitly captures the Hardy–Littlewood asymptotic density trends without explicit density supervision?

2. Methodology

2.1 Data and Feature Engineering

The model, PRIMEFAMILYNET, was trained on 200,000 labeled primes from the range $[10^7, 10^9]$.

• Strictly Causal Features: The input vector $x(p)$ for a prime $p$ relies only on the history of the prime sequence (predecessors and modular residues), never on the forward gap ($g_+ = p_+ - p$). This prevents data leakage.
• Feature Vector (25 dimensions):
  • Primorial Residues: Normalized residues modulo 6, 30, 210, and 2310. These encode the deterministic sieve constraints (e.g., $p \pmod{30}$ dictates possible values for $p+2$).
  • Small Prime Residues: Residues modulo the first 12 primes.
  • Backward Gap: The gap to the previous prime ($g_-$), normalized.
  • Scale & Digit Features: Logarithmic scale, bit-length, and decimal digit properties.
• Target Families: Seven families were predicted simultaneously: Twin, Cousin, Sexy, Sophie Germain, Safe, Chen, and Isolated primes.

2.2 Model Architecture

• Architecture: A multi-head residual MLP with 1.25 million parameters.
• Structure:
  1. Input projection to 512 dimensions (LayerNorm + GELU).
  2. Two stacked residual blocks.
  3. Narrowing projections to a shared 128-dimensional embedding.
  4. Seven independent sigmoid heads, one for each prime family, outputting a probability $[0, 1]$.
• Training Protocol: Optimized with AdamW using Cosine Annealing. Three random seeds were used for robustness.
• Loss Functions: A systematic comparison was conducted between:
  • Frequency-weighted Binary Cross-Entropy (wBCE).
  • Focal Loss.
  • Asymmetric Loss (ASL).

2.3 Evaluation Strategy

• Out-of-Distribution (OOD) Testing: The model was evaluated at scales $10^{10}, 10^{12}, 10^{14},$ and $10^{16}$, spanning nine orders of magnitude beyond the training range.
• Control Models:
  • Non-Causal Upper Bound: A model given the forward gap ($g_+$), which trivially solves gap-defined families (e.g., $g_+=2 \iff$ Twin). This establishes the theoretical maximum performance.
  • XGBoost Baseline: To compare deep learning against gradient-boosted trees.

3. Key Contributions & Findings

3.1 Density-Driven Generalisation (The "Isolated Prime" Phenomenon)

The most significant finding is that Isolated Prime recall monotonically increased with scale, rising from 0.809 at $5 \times 10^8$ to 0.984 at $10^{16}$.

• Significance: This is the only family among the seven to improve as the scale increased.
• Mechanism: As $N \to \infty$, the density of twin primes decays as $1/(\log N)^2$, while the density of isolated primes increases relative to the total. The model, trained only on causal modular residues, learned that the "signature" of an isolated prime becomes sharper and easier to distinguish as twin primes become rarer.
• Proof of Learning: Since recall is a within-class ratio (invariant to class prevalence), this improvement cannot be attributed to the growing fraction of isolated primes in the dataset. It proves the model genuinely sharpened its decision boundary for isolated primes, implicitly learning the Hardy–Littlewood asymptotic trends without being told the formula.

3.2 The Cost of Causality

• For Chen primes (where $p+2$ is prime or semiprime), the causal model outperformed the non-causal model at every scale.
• Reason: The forward gap $g_+$ only indicates if $p+2$ is prime. It provides no information about the semiprime case. The causal model, using primorial residues, successfully captured the modular constraints for both prime and semiprime cases, demonstrating that modular arithmetic features are more informative than the raw gap for complex definitions.
• The causal model achieved near-perfect recall for isolated primes ($0.984$ at $10^{16}$) despite never seeing the forward gap, proving the sufficiency of backward-looking modular features.

3.3 Loss Function Sensitivity

• Focal Loss: Catastrophically failed for sparse algebraic families (Sophie Germain and Safe primes), driving recall to 0.000 at all scales. It suppressed gradients for hard examples too aggressively.
• Asymmetric Loss (ASL): Achieved superior in-distribution recall but degraded steeply out-of-distribution. For Sophie Germain and Safe primes, ASL recall collapsed at $10^{16}$, whereas wBCE maintained robust performance.
• Conclusion: In-distribution recall is a misleading criterion for scale-generalization tasks. Frequency-weighted BCE is the recommended loss for prime family prediction under distribution shift.

3.4 Feature Ablation

• Primorial Residues (Group A) were the dominant feature. Removing them caused recall for Sophie Germain and Safe primes to collapse to near zero, confirming these families are defined almost entirely by modular constraints rather than gap statistics.
• Backward Gap (Group C) was critical for identifying isolated primes.

4. Significance and Implications

1. Empirical Validation of Number Theory: This paper provides the first machine-learning evidence that a data-driven model can implicitly learn and extrapolate Hardy–Littlewood k-tuple asymptotics. The model reproduced the correct density trajectory for isolated primes purely from arithmetic features.
2. Redefining Prime Prediction: It demonstrates that predicting prime families (structured modular conditions) is fundamentally different from predicting primality (algorithmic randomness). The former is learnable and generalizable; the latter is not.
3. Probabilistic Sieving: The model acts as a "neural sieve," reducing the search space by 62–88% at validation scales while maintaining high recall, offering a new tool for filtering candidates in large-scale prime searches.
4. Guidance for Rare-Class Learning: The study highlights that for tasks with shifting class densities (like prime families at different scales), standard loss functions like Focal Loss or ASL can lead to fragile models. Frequency-weighted BCE offers better robustness.

Summary

The paper presents PRIMEFAMILYNET, a deep residual network that successfully learns to sieve prime families using only causal, backward-looking arithmetic features. Its ability to generalize from $10^9$ to $10^{16}$ and improve performance on isolated primes as twin primes become rarer serves as a novel, independent confirmation of analytic number theory predictions, bridging the gap between deep learning and classical sieve theory.