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

TL;DR

This paper introduces PrimeFamilyNet, a neural network model that probabilistically filters prime families across large scales, demonstrating improved recall and generalization consistent with Hardy-Littlewood asymptotics.

Contribution

The authors develop a multi-head residual network conditioned on prime properties, achieving scalable, density-driven prime family identification and empirical validation of Hardy-Littlewood predictions.

Findings

Recall for isolated primes increased from 80.9% to 98.4% from scale 10^8 to 10^16.

Model trained up to 10^9 generalized correctly to 10^16 without density supervision.

Causal model retained over 95% recall at 10^10, reducing search space by up to 88%.

Abstract

Special prime families (twin, Sophie Germain, safe, cousin, sexy, Chen, and isolated primes) are central objects of analytic number theory, yet no efficiently computable probabilistic filter exists for identifying likely members among known primes at large scale. Classical sieves assign no probability weights to surviving candidates, and prior machine learning approaches are limited by the algorithmic randomness of the prime indicator sequence, yielding near-zero true positive rates. We present PrimeFamilyNet, a multi-head residual network conditioned on the backward prime gap and modular primorial residues of a known prime $p$, learning probabilistic filters for all seven families simultaneously and generalising across nine orders of magnitude from training ($10^7$--$10^9$) to evaluation at $10^{16}$. Isolated prime recall increased monotonically from $0.809$ at $5\times10^8$ to $0.984$ at $10^{16}$, a gain of $17.5$ percentage points and the only family among seven to improve with scale. Because recall is invariant to class prevalence, this reflects genuine decision boundary sharpening, not the rising isolated-prime fraction at extreme scales. A model trained only to $10^9$ reproduced the correct asymptotic direction without density supervision, corroborating Hardy--Littlewood $k$-tuple predictions. The causal model retained over $95\%$ recall for five families near $10^{10}$ while reducing the search space by $62$--$88\%$. For Chen primes, causal recall exceeded non-causal recall at every scale (margin $+0.245$ at $10^{16}$) because $g^+=2$ encodes only the prime case of the Chen condition. Focal Loss collapsed sparse algebraic family recall to $0.000$. Asymmetric Loss outperformed weighted BCE in-distribution but degraded more steeply out-of-distribution, showing that in-distribution recall alone is a misleading criterion for scale-generalisation tasks.