AI 中文总结
研究有限β展开系统的内在冗余，用有界窗口模型区分语义与码本可接受性，证明特定损坏修复情况，通过实验量化多系统间权衡，指出内在β冗余是结构数字完整性的受限资源，非传统错误控制冗余替代品。
AI 中文摘要
非标准计数系统中的冗余通常与鲁棒性相关，但其在有限数字算术里的实用价值取决于表示、存储和修复的定义方式。我们使用有界窗口模型研究有限β展开系统中的内在冗余，该模型区分语义非唯一性与规范码本可接受性。此模型区分算术规范化与损坏修复，并评估结构可检测性、观测状态的值保持修复以及原始值的留存情况。对于黄金分割系统及相关的多纳奇基数，我们证明在没有外部信息时，规范单射有限码本中的真正单数字损坏无法通过精确修复在语义上恢复。只有对应代数重写恒等式的局部多数字扰动（如100和011的等价性）在精确结构修复下才可能实现语义留存。比较标准二进制、符号数字非相邻形式和多纳奇系统的实验量化了码本稀疏性、故障可见性、规范化成本、残留误差和边界损失之间的权衡。结果表明，内在β冗余是用于结构数字完整性的受限语言资源，而非经典错误控制冗余的替代品。
英文摘要
Redundancy in non-standard numeration systems is often associated with robustness, but its practical value in finite digital arithmetic depends on how representation, storage, and repair are defined. We study intrinsic redundancy in finite beta-expansion systems using a bounded-window model that separates semantic non-uniqueness from canonical codebook admissibility. The model distinguishes arithmetic canonicalization from corruption repair and evaluates structural detectability, value-preserving repair of the observed state, and survival of the original value. For the golden-ratio system and related multinacci bases, we prove that a genuine single-digit corruption in a canonically injective finite codebook cannot be semantically recovered by exact repair without external information. Semantic survival under exact structural repair is possible only for localized multi-digit perturbations corresponding to algebraic rewrite identities, such as the equivalence of 100 and 011. Experiments comparing standard binary, signed-digit non-adjacent form, and multinacci systems quantify trade-offs among codebook sparsity, fault visibility, canonicalization cost, residual error, and boundary loss. The results show that intrinsic beta-redundancy is a constrained-language resource for structural digital integrity rather than a substitute for classical error-control redundancy.
Comments73 pages, 12 figures. Code and experiment scripts are available at https://doi.org/10.5281/zenodo.21140212
