{"id":"https://openalex.org/W7225100326","doi":"https://doi.org/10.5281/zenodo.23898175","title":"The Parity Dual-Sphere Invariance Theorem: Constructive Multi-Sphere Packings Beyond Behrend's Bound for Arithmetic Progression-Free Sets","display_name":"The Parity Dual-Sphere Invariance Theorem: Constructive Multi-Sphere Packings Beyond Behrend's Bound for Arithmetic Progression-Free Sets","publication_year":2026,"publication_date":"2026-11-02","ids":{"openalex":"https://openalex.org/W7225100326","doi":"https://doi.org/10.5281/zenodo.23898175"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23898175","doi":"https://doi.org/10.5281/zenodo.23898175","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23898175","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.23898175","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159470751","display_name":"Hussam Gad-Allah","orcid":"https://orcid.org/0009-0002-6823-2951","observed_orcids":["https://orcid.org/0009-0002-6823-2951"]},"institutions":[{"id":"https://openalex.org/I145487455","display_name":"Cairo University","ror":"https://ror.org/03q21mh05","country_code":"EG","type":"education","lineage":["https://openalex.org/I145487455"]}],"countries":["EG"],"is_corresponding":true,"raw_author_name":"Gad-Allah, Hussam","raw_affiliation_strings":["Faculty of Graduate Studies for Statistical Research (FGSSR), Cairo University, Giza, Egypt"],"raw_orcid":"https://orcid.org/0009-0002-6823-2951","affiliations":[{"raw_affiliation_string":"Faculty of Graduate Studies for Statistical Research (FGSSR), Cairo University, Giza, Egypt","institution_ids":["https://openalex.org/I145487455"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5159470751"],"corresponding_institution_ids":["https://openalex.org/I145487455"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.7943999767303467,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.7943999767303467,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.19850000739097595,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0031999999191612005,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/three-term-arithmetic-progressions","display_name":"three-term arithmetic progressions","score":0.8220000267028809},{"id":"https://openalex.org/keywords/integer-lattice","display_name":"integer lattice","score":0.6110000014305115},{"id":"https://openalex.org/keywords/additive-combinatorics","display_name":"additive combinatorics","score":0.6589999794960022}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7901999950408936},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.6782000064849854},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.6499000191688538},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5249000191688538},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4438999891281128},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4415000081062317},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.40790000557899475},{"id":"https://openalex.org/C131521367","wikidata":"https://www.wikidata.org/wiki/Q625502","display_name":"Parity bit","level":2,"score":0.3910999894142151},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3862000107765198},{"id":"https://openalex.org/C202854965","wikidata":"https://www.wikidata.org/wiki/Q3044470","display_name":"Constructive proof","level":2,"score":0.3828999996185303},{"id":"https://openalex.org/C203858083","wikidata":"https://www.wikidata.org/wiki/Q188347","display_name":"Harmonic mean","level":2,"score":0.3806999921798706},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.3783000111579895},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.3736000061035156},{"id":"https://openalex.org/C130498168","wikidata":"https://www.wikidata.org/wiki/Q170008","display_name":"Arithmetic progression","level":2,"score":0.3709000051021576},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.35109999775886536},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.3483999967575073},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.33869999647140503},{"id":"https://openalex.org/C142885752","wikidata":"https://www.wikidata.org/wiki/Q1252145","display_name":"Integer lattice","level":3,"score":0.3133000135421753},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.29249998927116394},{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.28940001130104065},{"id":"https://openalex.org/C56863137","wikidata":"https://www.wikidata.org/wiki/Q743139","display_name":"Half-integer","level":2,"score":0.2777000069618225},{"id":"https://openalex.org/C134949993","wikidata":"https://www.wikidata.org/wiki/Q2068617","display_name":"Pencil (optics)","level":2,"score":0.2721000015735626},{"id":"https://openalex.org/C188139237","wikidata":"https://www.wikidata.org/wiki/Q50706","display_name":"Square-free integer","level":2,"score":0.2711000144481659}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23898175","doi":"https://doi.org/10.5281/zenodo.23898175","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23898175","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.23898176","doi":"https://doi.org/10.5281/zenodo.23898176","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23898176","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.23898175","doi":"https://doi.org/10.5281/zenodo.23898175","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23898175","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/W2118490018","https://openalex.org/W4415031546","https://openalex.org/W4415021901","https://openalex.org/W3023445990","https://openalex.org/W7151778120","https://openalex.org/W2568381359","https://openalex.org/W7081429023","https://openalex.org/W7097015080","https://openalex.org/W3104302389","https://openalex.org/W2041646143"],"abstract_inverted_index":{"We":[0],"introduce":[1],"the":[2,9,21,46,63],"Parity":[3],"Dual-Sphere":[4],"Invariance":[5],"Theorem,":[6],"establishing":[7],"that":[8],"union":[10],"of":[11,104],"two":[12],"concentric":[13],"hyperspheres":[14],"S(R^2)":[15],"and":[16],"S(R^2":[17],"+":[18],"1)":[19],"in":[20,62],"integer":[22,64],"lattice":[23],"[0,":[24],"b-1]^m":[25],"contains":[26],"no":[27,60],"non-trivial":[28],"three-term":[29],"arithmetic":[30],"progressions.":[31],"The":[32],"proof":[33],"rests":[34],"on":[35,81],"a":[36],"fundamental":[37],"parity":[38],"contradiction:":[39],"any":[40],"progression":[41],"spanning":[42],"both":[43],"shells":[44],"forces":[45],"midpoint":[47],"difference":[48],"norm":[49],"squared":[50],"to":[51,70,92],"satisfy":[52],"2||x":[53],"-":[54],"y||^2":[55],"=":[56],"1,":[57],"which":[58],"has":[59],"solution":[61],"lattice.":[65],"By":[66],"extending":[67],"this":[68],"invariance":[69],"an":[71,82],"independent":[72],"radius":[73],"hypergraph,":[74],"we":[75],"formulate":[76],"maximal":[77],"multi-sphere":[78],"families":[79],"executed":[80],"NVIDIA":[83],"GeForce":[84],"RTX":[85],"3090":[86],"GPU,":[87],"achieving":[88],"constructive":[89],"densities":[90],"+250%":[91],"+402.9%":[93],"beyond":[94],"Behrend's":[95],"classic":[96],"single-sphere":[97],"lower":[98],"bound":[99],"with":[100],"100%":[101],"certified":[102],"absence":[103],"3-APs.":[105]},"counts_by_year":[],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2026-11-03T00:00:00"}