Intrinsic Redundancy and Local Robustness in Finite $β$-Expansion Systems

Abstract

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.