HoloByte: Continuous Hyperspherical Distillation for Tokenizer-Free Modeling

Sequence modeling universally relies on discrete subword tokenization to circumvent the O(N2) computational intractability of native byte-level attention. However, this heuristic quantization imposes artificial morphological boundaries, enforces vocabulary dependence, and fractures the continuity of the optimization landscape. To resolve this dichotomy, we introduce HoloByte: a strictly tokenizer-free framework utilizing Continuous Hyperspherical Distillation. HoloByte partitions discrete byte sequences into fixed-capacity chunks and projects them into a continuous, strictly bounded hyperspherical manifold via an invertible, dimension-preserving orthogonal rotation operator. This spatial superposition allows a macroscopic transformer to operate exclusively on compressed continuous representations, formally reducing the exact attention time complexity from O(N2D) to Oleft( fracN2W2D + ND2 right). A localized causal micro-decoder subsequently unbinds these representations to compute exact byte-level distributions. To govern this continuous trajectory, we propose a dual-objective formulation incorporating a mathematically precise Holographic Latent Mean Squared Error, which strictly bounds the gradient and guarantees asymptotic stability. Theoretically, we derive the minimal embedding dimension D = Omega(W ln |V|) required to ensure error-free discrete recovery from the continuous manifold. Empirically, under strictly matched parameter constraints, HoloByte is systematically outperforming a comparable discrete Byte-Pair Encoding (BPE) baseline. These results establish Continuous Hyperspherical Distillation as a mathematically rigorous and computationally tractable foundation for vocabulary-invariant sequence modeling. The code is available at https://github.com/VladimerKhasia/HoloByte