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Weight binarization has emerged as a promising strategy to reduce the complexity of large language models (LLMs). Existing approaches fall into post-training binarization, which is simple but causes severe performance loss, and training-aware methods, which depend on full-precision latent weights, adding complexity and limiting efficiency. We propose a novel framework that represents LLMs with multi-kernel Boolean parameters and, for the first time, enables direct finetuning LMMs in the Boolean domain, eliminating the need for latent weights. This enhances representational capacity and dramatically reduces complexity during both finetuning and inference. Extensive experiments across diverse LLMs show our method outperforms recent ultra low-bit quantization and binarization techniques.

We study token dynamics in deep Mamba models, with a particular focus on the attracting role of the first token. Unlike attention mechanisms, where token interactions are governed by positive scalar coefficients, Mamba dynamics involve full matrix-valued interaction coefficients with no positivity or definiteness guarantees. This structural difference makes Mamba dynamics substantially harder to analyze and prevents the direct application of existing techniques for attention-based models. To overcome this challenge, we introduce a new analytical framework based on a high-dimensional invariant box and a sharp exit-time argument, which together recover an effective form of sign-definiteness. Using this framework, we characterize the possible limit points of token trajectories and prove explicit exponential convergence rates. Our analysis reveals that the first token has privileged dynamics: its limiting direction attracts subsequent tokens, inducing token clustering and attention-sink phenomena in deep Mamba models. We further show that this attraction mechanism extends to broader sequence dynamics, including state-space models, causal attention, and classical time-series models. Experiments on Falcon-Mamba-7B validate our predictions, showing that concentration on the first token increases with depth and dominates in later layers.