[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82053-en":3,"doc-seo-82053-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82053,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Intrinsic Redundancy and Local Robustness in Finite β-Expansion Systems","Redundancy in non-standard numeration systems often links to robustness, yet its practical value in finite digital arithmetic depends on how representation, storage, and repair are defined. The paper studies intrinsic redundancy in finite β-expansion systems using an abounded-window model that separates ambient semantic non-uniqueness from canonical codebook admissibility. It evaluates structural detectability, value-preserving re-admissibilization, and semantic survival of the original value, then quantifies trade-offs across binary, NAF, and multinacci bases.","arXiv :2607 .08795v 1 [ cs .IT] 8 Jul 2026  \nIntrinsic Redundancy and Local Robustness in Finite β-Expansion Systems  \nAdilbek Taizhanov 1 and Miras Seilkhan2  \n[1](1 adilbek300108@gmail.com)[ adilbek300108@gmail.com](1 adilbek300108@gmail.com)  \n[2](2 seilkhan.miras6117@gmail.com)[ seilkhan.miras6117@gmail.com](2 seilkhan.miras6117@gmail.com)  \nAbstract. Redundancy in non-standard numeration systems is often associated with robustness, but its operational value in finite digital arithmetic depends on how representation, storage, and repair are defined. This paper studies intrinsic redundancy in finite β-expansion systems through abounded-window model that separates ambient semantic non-uniqueness from canonical codebook admissibility. The model distinguishes arithmetic canonicalization from corruption repair and evaluates three outcomes:  \nstructural detectability, value-preserving re-admissibilization of the observed state, and semantic survival of the original value.  \nFor the golden-ratio system and related multinacci bases, we formalize a single-digit impossibility result: within a canonically injective finite codebook, genuine single-digit corruptions cannot be semantically recovered by exact repair without external information. Under exact structural repair, semantic survival is possible only for localized multi-digit perturbations whose error vector lies in the algebraic kernel of the evaluation map, as in local rewrite identities such as 100 ↔ 011. Comparative experiments across standard binary, signed-digit non-adjacent form (NAF), and multinacci systems (φ, T3 , T4) quantify the trade-offs among codebook sparsity, structural fault visibility, bounded-window canonicalization cost, residual error, and boundary loss. The results identify intrinsic β-redundancy asa constrained-language resource for structural digital integrity, distinct from classical error-control redundancy.  \nKeywords: Numeration systems · β-expansions · Canonicalization · Fault detection · Golden-ratio base · Structural robustness  \n1 Introduction  \nModern digital computation is built almost entirely on binary representation. Atthe same time, positional numeration systems with a real base β > 1 provide abroader design space in which the base, the digit alphabet, and the admissible language can be chosen independently. This makes it possible to study not only compactness of representation, but also the structural constraints that a representation imposes on stored words and arithmetic intermediates. Representationsin non-integer bases, commonly referred to as β-expansions, were introduced  \n2 A. Taizhanov and M. Seilkhan  \nand systematically studied in the literature on numeration systems and symbolic dynamics [34 ,31 ,37 ,23] . A central feature of many β-systems is semantic nonuniqueness: for bases in the interval 1 \u003C β \u003C 2, one numerical value may admit several distinct expansions [39,26] .  \nThis non-uniqueness has motivated several notions of robustness. In analog β-encoders and golden-ratio encoders, redundant expansions can provide stability against imperfect circuit components during analog-to-digital conversion [39, 12] . In non-standard numeration systems, algebraic bases and extended digit alphabetscan also support structured arithmetic, including parallel addition [19] . These results motivate a finite digital question that is narrower but operationally sharper: when numbers are stored as finite canonical codewords, what does intrinsic β-redundancy actually provide?  \nThe key distinction is between ambient semantic non-uniqueness and redundancy inside a stored canonical codebook. In the finite model studied here, the admissible codebook selects canonical representatives from a larger ambient set of digit strings. A perturbed string can often be re-admissibilized while preserving its observed value. This operation is re-admissibilization of the observed state. It is distinct from recovering the pre-fault value. The paper therefore studies ","cbCaivxDF7FSkJdQ","https://ap.wps.com/l/cbCaivxDF7FSkJdQ","pdf",1540897,1,73,"English","en",105,"# Abstract\n# Introduction\n## Ambient semantic non-uniqueness vs canonical codebook redundancy\n## Operational model: bounded-window framework\n## Addition canonicalization as a benchmark\n## Corruption repair and semantic survival","[{\"question\":\"What operational outcomes does the paper evaluate for finite β-expansion systems?\",\"answer\":\"It evaluates three outcomes: structural detectability, value-preserving re-admissibilization of the observed state, and semantic survival of the original value under corruption and exact repair settings.\"},{\"question\":\"How does the paper separate semantic non-uniqueness from redundancy inside a stored codebook?\",\"answer\":\"It distinguishes an ambient set of digit strings that may represent the same value from a finite admissible canonical codebook, where re-admissibilization preserves the observed value but does not necessarily recover the pre-fault value.\"},{\"question\":\"What is the main impossibility and recovery limitation discussed for the golden-ratio and related bases?\",\"answer\":\"For the golden-ratio system and related multinacci bases, the paper formalizes that within a canonically injective finite codebook, genuine single-digit corruptions cannot be semantically recovered by exact repair without external 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operational outcomes does the paper evaluate for finite β-expansion systems?","Question",{"text":74,"@type":75},"It evaluates three outcomes: structural detectability, value-preserving re-admissibilization of the observed state, and semantic survival of the original value under corruption and exact repair settings.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the paper separate semantic non-uniqueness from redundancy inside a stored codebook?",{"text":79,"@type":75},"It distinguishes an ambient set of digit strings that may represent the same value from a finite admissible canonical codebook, where re-admissibilization preserves the observed value but does not necessarily recover the pre-fault value.",{"name":81,"@type":72,"acceptedAnswer":82},"What is the main impossibility and recovery limitation discussed for the golden-ratio and related bases?",{"text":83,"@type":75},"For the golden-ratio system and related multinacci bases, the paper formalizes that within a canonically injective finite codebook, genuine single-digit corruptions cannot be semantically recovered by exact repair without external information.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & 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