Beyond Mod-97: Phase-Coupled D5×D5 Folds for Two-Digit Decimal Checksums with Exact Bounds

Beyond Mod-97: Phase-Coupled D5×D5 Folds for Two-Digit Decimal Checksums with Exact Bounds | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Beyond Mod-97: Phase-Coupled D5×D5 Folds for Two-Digit Decimal Checksums with Exact Bounds Csaba Balogh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9658199/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Verhoeff’s classical decimal check-digit scheme embeds a check alphabet into the dihedral group D 5 and detects every adjacent transposition of distinct digits through an antisymmetry condition on the permutation family. Single-digit constructions provide roughly log 2 10 ≈ 3.32 bits of redundancy, which is insufficient when identifiers are densely allocated, while the classical alternative ISO 7064 mod-11 requires an eleventh symbol and is therefore unsuitable for purely decimal identifier spaces. We construct a two-digit decimal checksum on the direct product G = D 5 × D5; its 100 elements correspond bijectively to two decimal check digits without any non-decimal residue. A phase parameter k̸≡ 0 (mod 10) couples two copies of the Verhoeff permutation family into position-dependent contributions Ψ (k) i (d) ∈ G, and the resulting fold detects every adjacent transposition of distinct digits and every single-digit substitution. Adjacent-transposition detection follows algebraically from antisymmetry on the first factor of G, confirmed exhaustively over all 9 000 relevant instances. Surjectivity onto G holds for sequence lengths n ≥ 3 and every non-degenerate k, and exact preimage counts show that the undetected uniform-noise probability approaches 10 −2 as the sequence length grows. We are not aware of a previously published two-digit decimal checksum that simultaneously provides adjacent-transposition detection, singledigit substitution detection, decimal purity, and a formally proven random-noise miss bound. All proofs are either closed-form algebraic arguments or finite exhaustive verifications reproduced by a companion script. Information Theory check digit dihedral group Verhoeff algorithm error detection decimal codes direct product Full Text Additional Declarations The authors declare no competing interests. Supplementary Files pseudocodesupplement.pdf Supplementary Material for: Beyond Mod-97: Phase-Coupled D5×D5 Folds for Two-Digit Decimal Checksums with Exact Bounds Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9658199","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":637128037,"identity":"a119bd43-4a91-41a1-8b87-7c04bd7ab655","order_by":0,"name":"Csaba Balogh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIiWNgGAWjYHACNiA+wMAHYnxgg4olALEBIS1Ako1xBpDkIUkLMw9MCwMeLfz9h589+MFwR56Nvf3aY5sym8T90oePPXhQU8dgLpGAVYvEjTRzwx6GZ4ZtPGfKjXPOpSX28KWlGyQcO8xgOQO7FgMJBjMJHobDjG0SOWnSuW2HjXl4eMwkEhsOMBjcwKGF//g3yT8Mh+3BWizb/sO01OHWwpBjJg20JbFNIv2YNGPbATmoFmacWiRu5JQbyxg8Swb6hU2y51yyHJBOkwD6hcfgzAMcIXZ828M3FXds+9nbn0n8KLPjYe9hPib5o6ZOzuA4dltgzgMCHtSI4MGqEhWwY3fHKBgFo2AUjAIANEZXGOgNZssAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0009-9993-2628","institution":"Országos Kórházi Főigazgatóság (National Directorate General for Hospitals), Budapest, Hungary","correspondingAuthor":true,"prefix":"","firstName":"Csaba","middleName":"","lastName":"Balogh","suffix":""}],"badges":[],"createdAt":"2026-05-08 22:17:45","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9658199/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9658199/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109051106,"identity":"92f4f13a-7c86-438b-93a9-cd2c760d38b4","added_by":"auto","created_at":"2026-05-12 06:45:58","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":218096,"visible":true,"origin":"","legend":"","description":"","filename":"twodigitfoldIT.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9658199/v1_covered_82cb30a1-71b7-4270-8365-f47b1836237e.pdf"},{"id":109051083,"identity":"89c34082-37b4-4ed6-b05b-ddb423bcd018","added_by":"auto","created_at":"2026-05-12 06:45:28","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":203370,"visible":true,"origin":"","legend":"\u003cp\u003eSupplementary Material for:\u003c/p\u003e\n\u003cp\u003eBeyond Mod-97: Phase-Coupled D5×D5 Folds for Two-Digit Decimal Checksums with Exact Bounds\u003c/p\u003e","description":"","filename":"pseudocodesupplement.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9658199/v1/a4ad3d05436e8dfe6fea1b39.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eBeyond Mod-97: Phase-Coupled D5×D5 Folds for Two-Digit Decimal Checksums with Exact Bounds\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"check digit, dihedral group, Verhoeff algorithm, error detection, decimal codes, direct product","lastPublishedDoi":"10.21203/rs.3.rs-9658199/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9658199/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eVerhoeff’s classical decimal check-digit scheme embeds a check alphabet into the dihedral group D\u003csub\u003e5\u003c/sub\u003e and detects every adjacent transposition of distinct digits through an antisymmetry condition on the permutation family. Single-digit constructions provide roughly log\u003csub\u003e2\u003c/sub\u003e 10 ≈ 3.32 bits of redundancy, which is insufficient when identifiers are densely allocated, while the classical alternative ISO 7064 mod-11 requires an eleventh symbol and is therefore unsuitable for purely decimal identifier spaces. We construct a two-digit decimal checksum on the direct product G = D\u003csub\u003e5\u003c/sub\u003e × D5; its 100 elements correspond bijectively to two decimal check digits without any non-decimal residue. A phase parameter k̸≡ 0 (mod 10) couples two copies of the Verhoeff permutation family into position-dependent contributions Ψ\u003csup\u003e(k)\u003c/sup\u003e \u003csub\u003ei\u003c/sub\u003e (d) ∈ G, and the resulting fold detects every adjacent transposition of distinct digits and every single-digit substitution. Adjacent-transposition detection follows algebraically from antisymmetry on the first factor of G, confirmed exhaustively over all 9 000 relevant instances. Surjectivity onto G holds for sequence lengths n ≥ 3 and every non-degenerate k, and exact preimage counts show that the undetected uniform-noise probability approaches 10\u003csup\u003e−2 \u003c/sup\u003eas the sequence length grows. We are not aware of a previously published two-digit decimal checksum that simultaneously provides adjacent-transposition detection, singledigit substitution detection, decimal purity, and a formally proven random-noise miss bound. 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