p-adic Bi-Filtrations for Topological Machine Learning on Genomic Sequences

Genomic foundation models increasingly adopt large language model architectures, yet almost universally rely on fixed tokenization schemes such as $k$-mers, BPE, or single nucleotides, which impose arbitrary sequence boundaries that may obscure biologically relevant structure. We present LDARNet, a 120M-parameter hierarchical genomic foundation model that adapts H-Net-style dynamic chunking from autoregressive generation to masked language modeling, combining BiMamba-2 state-space layers with local attention, bidirectional routing, and a ratio-based regularizer to induce adaptive token boundaries without supervision. Fine-tuned on 27 tasks from the Nucleotide Transformer and Genomic Benchmarks suites, LDARNet achieves 11/18 wins among compact models ($<$300M parameters) and state-of-the-art results on 5 histone modification tasks, outperforming models up to 20$\times$ larger. A FLOPs-matched controlled experiment isolates learned routing as the source of these gains: learned boundaries beat fixed-grid boundaries by up to 14 percentage points on histone tasks at identical compute. Nucleotide-resolution analysis further shows that the learned boundaries align with canonical promoter motifs and splice junctions without supervision, providing a biological interpretation for adaptive tokenization in genomic foundation models.

\textbf{Motivation:} Pre-trained protein language models (PLMs) such as ESM-2 have become the default representation for biological sequence tasks, but they are computationally heavy and require GPUs both for embedding and for fine-tuning. Whether they are actually necessary for sequence \emph{classification}, as opposed to structure prediction, is rarely tested against strong, principled, lightweight alternatives. This question has direct practical stakes for large-scale genomic surveillance, where embedding millions of sequences on commodity hardware is a recurring bottleneck.\ \textbf{Results:} We introduce Murmur2Vec, an alignment-free, training-free embedding that aggregates $k$-mer counts into a small hash table via the deterministic MurmurHash function, and we cast it as a randomized sketch of the classical $k$-mer spectrum kernel. We provide a complete theoretical treatment: closed-form bias/variance of the inner product, an unbiased signed variant with a Johnson--Lindenstrauss-type concentration bound, an excess-risk bound for downstream linear classifiers that makes the bias--variance trade-off in the hash-table size explicit, and an implicit-regularization mechanism by which collisions damage frequent non-discriminative $k$-mers more than rare lineage-defining ones. Across four classification tasks, SARS-CoV-2 spike lineage (22 classes), HIV-1 Env subtype (8 classes), and two protein-family benchmarks (8 and 6 classes), Murmur2Vec matches a LoRA-fine-tuned 650M-parameter ESM-2 model on the two tasks for which LoRA fine-tuning was run to convergence (SARS-CoV-2 and HIV-1) and ties frozen ESM-2 on the two protein-family tasks, and it \emph{outperforms} the fine-tuned model on the hardest task (SARS-CoV-2 lineage: $0.854$ vs.\ $0.807$ accuracy; macro-F1 $0.684$ vs.\ $0.401$).

Progress in genomic foundation models is difficult to assess due to fragmented benchmarks, incompatible evaluation protocols, and task-specific reporting. As a result, claims of superiority or generality across models are often not directly comparable. We introduce GENEB, a large-scale diagnostic benchmark that evaluates frozen representations from 40 genomic foundation models across 100 tasks spanning 13 functional categories under a unified probing-based protocol, including few-shot regimes. GENEB enables controlled comparison across model scale, architecture, tokenization, and pretraining data while explicitly exposing task-level trade-offs. Our analysis shows that aggregate leaderboards are unstable: model rankings vary sharply across task categories, scale provides only modest and inconsistent gains, and architectural and pretraining alignment frequently outweigh parameter count. These results highlight limitations of current evaluation practices and position GENEB as a reference framework for principled comparison and category-aware model selection in genomic machine learning.