[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85170-en":3,"doc-seo-85170-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"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},85170,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Adaptive Search in Collatz Exponent-Code Space via 2-adic and 3-adic Constraints","Adaptive Search in Collatz Exponent-Code Space investigates symbolic search objects for the Collatz conjecture using finite exponent codes of the accelerated map. Each code tracks the number of divisions by two after every 3n+1 step and yields three diagnostics: real drift, a 2-adic start representative, and a 3-adic endpoint representative combined into the 2–3–∞ diagnostic. A counterexample-like code must align with near-critical growth while forcing compatible 2-adic and 3-adic representatives. Experiments for k=100,200,400 show adaptive evolutionary search improves finite trade-offs, yet all methods retain positive residue rates; the work provides a diagnostic framework, not a verification method.","arXiv :2607 . 1004 1v 1 [ cs .NE] 10 Jul 2026  \nAdaptive Search in Collatz Exponent-Code Space via 2-adic and 3-adic Constraints  \nOliver Kramer  \nComputational Intelligence  \nDepartment of Computer Science  \nUniversity of Oldenburg  \n[oliver.kramer@uol.de](oliver.kramer@uol.de)  \nAbstract. We study a symbolic search space for the Collatz conjecture: finite exponent codes of the accelerated map. Such a code records the number of divisions by two after each 3n+1 step. Every code induces three diagnostics: real drift, a 2-adic start representative, and a 3-adic endpoint representative. We call their combination the 2–3–∞ diagnostic. A counterexample-like code should have near-critical drift, small 2-adic start representatives, and endpoint representatives compatible with the growth bound (3/2)k . We prove that every infinite code generated by a fixed positive integer has asymptotically vanishing 2-adic and 3-adic residue rates. Experiments compare random critical codes, mechanical critical codes, and adaptive evolutionary search for k = 100 , 200 , 400. Adaptive search improves finite-length trade-offs, but all methods retain clearly positive residue rates. The contribution is not a Collatz verification method, but a symbolic diagnostic framework for probing obstruction structures in exponent-code space.  \n1 Introduction  \nThe Collatz conjecture states that repeated application of T(n) = n/2 for even n, and T (n) = 3n + 1 for odd n, eventually reaches 1 for every positive integer n. Most computational approaches test large ranges of starting values. Here, we instead search over symbolic trajectory structures.  \nWe use the accelerated Collatz map on odd integers,  \n 3n + 1   \nC (n) =  \n2v2 (3n+1) ,  \nwhere v2 (m) is the largest exponent such that 2v2 (m) | m. For a trajectory xk+1 = C (xk ), define ak+1 = v2 (3xk + 1) . The finite sequence (a1 ,..., ak ) is an exponent code.  \nExample. For x0 = 5, one obtains 3·5+1 = 16 = 24 , hence a 1 = 4 and x 1 = 1 . Since 3 · 1 + 1 = 4 = 22 , the accelerated orbit then stays at 1, giving the code  \n(4 , 2 , 2 , 2 ,...) .  \nThus exponent codes encode odd-to-odd Collatz dynamics symbolically.  \nA finite code determines a start residue modulo 2Ak , where Ak = a 1 +···+ak , and an endpoint residue modulo 3k . A counterexample-like code should therefore satisfy three compatibility conditions: near-critical average exponent, small forced 2-adic start representative, and 3-adic endpoint representative compatible with real growth. We call this the 2–3–∞ diagnostic.  \n2 Related Work  \nThe (3x+1) problem has been studied from probabilistic, dynamical-systems, and symbolic perspectives. Standard references include the survey by Lagarias [1], the reference volume edited by Lagarias [2], and the dynamical-systems treatment of Wirsching [3] . The accelerated map on odd integers and symbolic trajectory encodings are well established in this literature.  \nOur work differs by treating finite exponent codes themselves as search objects. Rather than exploring starting integers, we explore symbolic trajectory structures and evaluate them through a combination of real-valued drift, 2-adic start representatives, and 3-adic endpoint representatives.  \nThe adaptive component is inspired by evolutionary computation, particularly evolution strategies [4] and evolutionary optimization methods [5] . The goal is not to prove a number-theoretic statement by optimization, but to use adaptive search as a tool for exploring structural properties of exponent-code space.  \n3 Exponent-Code Diagnostics  \nWe consider accelerated trajectories  \n3xk + 1  \nxk+1 = C (xk ) = , ak+1 = v2 (3xk + 1) .  \n2ak+1  \nFor a finite code (a1 , ... , ak ), define Ak = Pki=1 ai and A0 = 0 . Repeated substitution gives  \nxk =  \n3k x0 + Bk 2Ak ,  \nk−1 Bk = X 3k−1−j2Aj .  \nj=0  \nThus the code controls both the factor 3k /2Ak and the additive offset Bk .  \n3.1 Real Drift  \nThe critical average exponent is α = log2 3, since Ak /k = α makes 3k /2Ak = 1 ","cbCaivPIBhGqlC5p","https://ap.wps.com/l/cbCaivPIBhGqlC5p","pdf",291554,1,6,"English","en",105,"# Introduction\n## Accelerated Collatz map and exponent codes\n# Related Work\n## Symbolic perspective and evolutionary inspiration\n# Exponent-Code Diagnostics\n## Real drift\n## 2-adic start representative\n## 3-adic endpoint representative\n## Necessary vanishing of residue rates","[{\"question\":\"What is an exponent code in this work?\",\"answer\":\"An exponent code is the finite sequence of exponents ak+1=v2(3xk+1) produced by repeatedly applying the accelerated Collatz map to odd integers, recording how many times the result is divisible by 2 after each 3n+1 step.\"},{\"question\":\"How is the 2–3–∞ diagnostic defined?\",\"answer\":\"The 2–3–∞ diagnostic combines three compatibility measures for a finite code: real drift based on how Ak/k compares to log2 3, a 2-adic start representative capturing feasible starting residues modulo 2Ak, and a 3-adic endpoint representative capturing endpoint residues modulo 3k relative to the expected (3/2)k growth bound.\"},{\"question\":\"Does the adaptive search method verify the Collatz conjecture?\",\"answer\":\"No. The paper states the contribution is a symbolic diagnostic framework for probing obstruction structures in exponent-code space; adaptive search is used to explore finite-length trade-offs rather than to produce a Collatz verification.\"}]",1784201521,15,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"adaptive-search-in-collatz-exponent-code-space-via-2-adic-and-3-adic-constraints","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/adaptive-search-in-collatz-exponent-code-space-via-2-adic-and-3-adic-constraints/85170/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is an exponent code in this work?","Question",{"text":75,"@type":76},"An exponent code is the finite sequence of exponents ak+1=v2(3xk+1) produced by repeatedly applying the accelerated Collatz map to odd integers, recording how many times the result is divisible by 2 after each 3n+1 step.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the 2–3–∞ diagnostic defined?",{"text":80,"@type":76},"The 2–3–∞ diagnostic combines three compatibility measures for a finite code: real drift based on how Ak/k compares to log2 3, a 2-adic start representative capturing feasible starting residues modulo 2Ak, and a 3-adic endpoint representative capturing endpoint residues modulo 3k relative to the expected (3/2)k growth bound.",{"name":82,"@type":73,"acceptedAnswer":83},"Does the adaptive search method verify the Collatz conjecture?",{"text":84,"@type":76},"No. The paper states the contribution is a symbolic diagnostic framework for probing obstruction structures in exponent-code space; adaptive search is used to explore finite-length trade-offs rather than to produce a Collatz verification.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]