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Also, if $a,b\\in D(M)$, then $ab\\in D(M)$; a set $A\\subseteq\\mathbb Z$ so that $ab\\in A$ for each $a,b\\in A$ is called multiplicative. On the one hand, not every infinite set of integers (containing $0$) is a mapping degree set [NWW] and, on the other hand, every finite set of integers (containing $0$) is the mapping degree set of some $3$-manifolds [CMV]. We show the following:\n  (i) Not every multiplicative set $A$ con"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2303.11922","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2023-03-21T15:14:20Z","cross_cats_sorted":["math.AT","math.NT"],"title_canon_sha256":"d859e7b10541cdbe013097f3945d02b884ce4081d0314ce9ef5dee85f4392e28","abstract_canon_sha256":"e850b162df935a4ba28e8524af12d801f8ff6870ce11edbd202e4be1bfb4f304"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:53:26.489550Z","signature_b64":"LMKfTmw29rsjfJDt+09eVQNuuR5hP5scuuNIvrJECGUQqIlFdwLwlOvJFykQ6Kosgk+T6B/aDl0mK/cqmflICA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc0bcb4255a8740dce7b33716d90e50d82500babefa8dc0751827929b1e0081a","last_reissued_at":"2026-07-05T11:53:26.489049Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:53:26.489049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the realisation problem for mapping degree sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.NT"],"primary_cat":"math.GT","authors_text":"Christoforos Neofytidis, Hongbin Sun, Shicheng Wang, Ye Tian, Zhongzi Wang","submitted_at":"2023-03-21T15:14:20Z","abstract_excerpt":"The set of degrees of maps $D(M,N)$, where $M,N$ are closed oriented $n$-manifolds, always contains $0$ and the set of degrees of self-maps $D(M)$ always contains $0$ and $1$. Also, if $a,b\\in D(M)$, then $ab\\in D(M)$; a set $A\\subseteq\\mathbb Z$ so that $ab\\in A$ for each $a,b\\in A$ is called multiplicative. On the one hand, not every infinite set of integers (containing $0$) is a mapping degree set [NWW] and, on the other hand, every finite set of integers (containing $0$) is the mapping degree set of some $3$-manifolds [CMV]. We show the following:\n  (i) Not every multiplicative set $A$ con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.11922","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.11922/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2303.11922","created_at":"2026-07-05T11:53:26.489103+00:00"},{"alias_kind":"arxiv_version","alias_value":"2303.11922v2","created_at":"2026-07-05T11:53:26.489103+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.11922","created_at":"2026-07-05T11:53:26.489103+00:00"},{"alias_kind":"pith_short_12","alias_value":"3QF4WQSVVB2A","created_at":"2026-07-05T11:53:26.489103+00:00"},{"alias_kind":"pith_short_16","alias_value":"3QF4WQSVVB2A3TT3","created_at":"2026-07-05T11:53:26.489103+00:00"},{"alias_kind":"pith_short_8","alias_value":"3QF4WQSV","created_at":"2026-07-05T11:53:26.489103+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW","json":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW.json","graph_json":"https://pith.science/api/pith-number/3QF4WQSVVB2A3TT3GNYW3EHFBW/graph.json","events_json":"https://pith.science/api/pith-number/3QF4WQSVVB2A3TT3GNYW3EHFBW/events.json","paper":"https://pith.science/paper/3QF4WQSV"},"agent_actions":{"view_html":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW","download_json":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW.json","view_paper":"https://pith.science/paper/3QF4WQSV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2303.11922&json=true","fetch_graph":"https://pith.science/api/pith-number/3QF4WQSVVB2A3TT3GNYW3EHFBW/graph.json","fetch_events":"https://pith.science/api/pith-number/3QF4WQSVVB2A3TT3GNYW3EHFBW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW/action/storage_attestation","attest_author":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW/action/author_attestation","sign_citation":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW/action/citation_signature","submit_replication":"https://pith.science/pith/3QF4WQSVVB2A3TT3GNYW3EHFBW/action/replication_record"}},"created_at":"2026-07-05T11:53:26.489103+00:00","updated_at":"2026-07-05T11:53:26.489103+00:00"}