{"id":"https://openalex.org/W7224923476","doi":"https://doi.org/10.5281/zenodo.23797372","title":"Balanced Two-Sided Hereditary-Rank Lower Bounds for Live Collatz Coalescences: A Common Rational Anchor for Long Inverse Valuation-One Tails","display_name":"Balanced Two-Sided Hereditary-Rank Lower Bounds for Live Collatz Coalescences: A Common Rational Anchor for Long Inverse Valuation-One Tails","publication_year":2026,"publication_date":"2026-10-30","ids":{"openalex":"https://openalex.org/W7224923476","doi":"https://doi.org/10.5281/zenodo.23797372"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23797372","doi":"https://doi.org/10.5281/zenodo.23797372","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797372","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23797372","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5134362622","display_name":"Byoungwoo Lee","orcid":"https://orcid.org/0009-0000-2993-6038","observed_orcids":["https://orcid.org/0009-0000-2993-6038"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"l], Byoungwoo","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0000-2993-6038","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5134362622"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.5214999914169312,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.5214999914169312,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.3257000148296356,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.07720000296831131,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/collatz-conjecture","display_name":"Collatz conjecture","score":0.9990000128746033},{"id":"https://openalex.org/keywords/collatz-map","display_name":"Collatz map","score":0.6430000066757202},{"id":"https://openalex.org/keywords/odd-numbers","display_name":"odd integers","score":0.6010000109672546},{"id":"https://openalex.org/keywords/arithmetic-dynamics","display_name":"arithmetic dynamics","score":0.4390000104904175}],"concepts":[{"id":"https://openalex.org/C128814272","wikidata":"https://www.wikidata.org/wiki/Q837314","display_name":"Collatz conjecture","level":3,"score":0.8162000179290771},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.753000020980835},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.6381999850273132},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5924000144004822},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4909999966621399},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4410000145435333},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.3971000015735626},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.37700000405311584},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.3698999881744385},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.35120001435279846},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.34369999170303345},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.30070000886917114},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.27869999408721924}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23797372","doi":"https://doi.org/10.5281/zenodo.23797372","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797372","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.23797373","doi":"https://doi.org/10.5281/zenodo.23797373","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797373","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23797372","doi":"https://doi.org/10.5281/zenodo.23797372","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23797372","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7224769540","https://openalex.org/W7221189363","https://openalex.org/W7165390808","https://openalex.org/W7201980378","https://openalex.org/W7204442304","https://openalex.org/W4321105488","https://openalex.org/W7224870004","https://openalex.org/W7223203244","https://openalex.org/W7213338901","https://openalex.org/W7087107730"],"abstract_inverted_index":{"Description":[0],"We":[1],"establish":[2,133],"explicit":[3],"arithmetic":[4,276],"families":[5],"of":[6,20,38,125,145,172,209,252],"live":[7,66],"Collatz":[8,24,55,216,235,255,258,260,267],"coalescences":[9],"exhibiting":[10],"balanced,":[11],"linearly":[12],"growing":[13],"hereditary-rank":[14,127],"lower":[15,128,161],"bounds":[16,162],"on":[17],"both":[18,113],"branches":[19,115],"the":[21,64,142,158,170,185],"accelerated":[22,54,257],"odd":[23,40],"dynamics.":[25,217],"For":[26,149],"every":[27],"integer":[28],"height":[29],"$L":[30],"\\ge":[31,81,87,89],"200$,":[32],"we":[33,132],"construct":[34],"exponentially":[35],"many":[36],"pairs":[37,174],"positive":[39,197],"integers":[41],"$X":[42],"=":[43,49],"W":[44,50],"-":[45,51,178],"1$":[46],"and":[47,62,141,212,220],"$Y":[48],"7$":[52],"whose":[53],"trajectories":[56],"coalesce":[57],"after":[58,69],"exactly":[59],"three":[60],"iterations":[61],"share":[63],"same":[65],"first-passage":[67,264],"state":[68],"eleven":[70],"iterations.":[71],"Both":[72],"starting":[73],"points":[74],"possess":[75],"gate-relative":[76],"hereditary":[77,269],"ranks":[78],"satisfying":[79],"$$\\rho_L(X)":[80],"D_L":[82,88],"+":[83,182],"1,":[84],"\\qquad":[85],"\\rho_L(Y)":[86],"\\lfloor":[90],"0.29L":[91],"\\rfloor.$$":[92],"The":[93,202],"central":[94],"construction":[95],"uses":[96],"a":[97,134,196,205],"common":[98,186],"rational":[99],"anchor":[100],"to":[101,233],"synchronize":[102],"two":[103,159],"distinct":[104],"inverse":[105,111,213,266,271],"valuation":[106,272],"words.":[107],"After":[108],"finite":[109],"initial":[110],"segments,":[112],"ancestral":[114,139],"admit":[116],"arbitrarily":[117],"long":[118],"valuation-one":[119],"tails,":[120],"yielding":[121],"comparable":[122],"linear":[123],"growth":[124,200],"their":[126],"bounds.":[129],"More":[130],"generally,":[131],"quantitative":[135,206],"trade-off":[136],"between":[137],"simultaneous":[138],"depth":[140],"exponential":[143,198],"abundance":[144],"admissible":[146],"coalescing":[147],"pairs.":[148],"suitable":[150],"parameters":[151],"$0":[152],"<":[153,155],"q":[154,179],"\\log_3":[156],"2$,":[157],"rank":[160],"grow":[163],"with":[164,237],"asymptotic":[165],"coefficient":[166,188],"$q":[167],"\\log_2(3/2)$,":[168],"while":[169,194],"number":[171],"constructed":[173],"grows":[175],"as":[176],"$$2^{(1":[177],"\\log_2":[180],"3)L":[181],"O_q(1)}.$$":[183],"Thus":[184],"lower-bound":[187],"can":[189],"approach":[190],"$\\log_3(3/2)":[191],"\\approx":[192],"0.36907$":[193],"retaining":[195],"population":[199],"rate.":[201],"results":[203],"provide":[204],"structural":[207],"description":[208],"synchronized":[210,262],"coalescence":[211],"ancestry":[214],"in":[215],"Complete":[218],"proofs":[219],"reproducible":[221],"exact-integer":[222],"verification":[223],"programs":[224],"are":[225],"provided.":[226],"This":[227],"work":[228],"is":[229],"an":[230],"independent":[231],"sequel":[232],"Live":[234],"Coalescence":[236],"Two-Sided":[238],"Linear":[239],"Hereditary-Rank":[240],"Lower":[241],"Bounds":[242],"(DOI:":[243],"10.5281/zenodo.23790756).":[244],"Creator":[245],"Lee":[246],"Byoungwoo":[247],"Independent":[248],"Researcher,":[249],"Daejeon,":[250],"Republic":[251],"Korea":[253],"Keywords":[254],"conjecture;":[256],"map;":[259],"dynamics;":[261,265],"coalescence;":[263],"trees;":[268],"rank;":[270],"words;":[273],"Diophantine":[274],"congruences;":[275],"dynamics":[277]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2026-10-31T00:00:00"}