{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:K7XX422YSSUGU2V2TI4HC62C7F","short_pith_number":"pith:K7XX422Y","schema_version":"1.0","canonical_sha256":"57ef7e6b5894a86a6aba9a38717b42f96e5f35ca68a4ff83e76cb02dcccc7421","source":{"kind":"arxiv","id":"2511.02324","version":1},"attestation_state":"computed","paper":{"title":"Revisiting the $\\beta_1$-action on the $3$-primary stable homotopy groups of spheres","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Jack Morgan Davies","submitted_at":"2025-11-04T07:21:10Z","abstract_excerpt":"Let $\\beta_1$ be the first $3$-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that $\\beta_1^5 \\neq 0$, whilst $\\beta_1^6 = 0$. Shimomura generalised this to the $144$-periodic family generated by $\\beta_1$, written as $\\{\\beta_{1+9s}\\}_{s\\geq 0}$, and showed that any $5$-fold product $\\prod_5 \\beta_{1+9s} \\neq 0$, whilst all $6$-fold products $\\prod_6 \\beta_{1+9s} = 0$. In this article, we give a simple proof of these results as well as some generalisations to other $144$-periodic families. Our tools include BP-synthetic spectra, and the well-known Adams--No"},"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":"2511.02324","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2025-11-04T07:21:10Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"7039902a5c4284563848627f7724fe25cb05cf1005b5f063d931a7e10b800392","abstract_canon_sha256":"753a9f1936f32da9fca4812f32b47b3c0f589d536d4c24aa0dd6e2e06a61c92f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T02:13:18.133638Z","signature_b64":"vco0v4kCNvQ+UcKQV9oRh3IvsZvhf23lJaBQlRX1eW+KSAoIgUpys0C6X6MO4bMGQlHVipJLY4ifYmbiPQCzAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"57ef7e6b5894a86a6aba9a38717b42f96e5f35ca68a4ff83e76cb02dcccc7421","last_reissued_at":"2026-06-23T02:13:18.133089Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T02:13:18.133089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Revisiting the $\\beta_1$-action on the $3$-primary stable homotopy groups of spheres","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Jack Morgan Davies","submitted_at":"2025-11-04T07:21:10Z","abstract_excerpt":"Let $\\beta_1$ be the first $3$-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that $\\beta_1^5 \\neq 0$, whilst $\\beta_1^6 = 0$. Shimomura generalised this to the $144$-periodic family generated by $\\beta_1$, written as $\\{\\beta_{1+9s}\\}_{s\\geq 0}$, and showed that any $5$-fold product $\\prod_5 \\beta_{1+9s} \\neq 0$, whilst all $6$-fold products $\\prod_6 \\beta_{1+9s} = 0$. In this article, we give a simple proof of these results as well as some generalisations to other $144$-periodic families. Our tools include BP-synthetic spectra, and the well-known Adams--No"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.02324","kind":"arxiv","version":1},"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/2511.02324/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":"2511.02324","created_at":"2026-06-23T02:13:18.133159+00:00"},{"alias_kind":"arxiv_version","alias_value":"2511.02324v1","created_at":"2026-06-23T02:13:18.133159+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2511.02324","created_at":"2026-06-23T02:13:18.133159+00:00"},{"alias_kind":"pith_short_12","alias_value":"K7XX422YSSUG","created_at":"2026-06-23T02:13:18.133159+00:00"},{"alias_kind":"pith_short_16","alias_value":"K7XX422YSSUGU2V2","created_at":"2026-06-23T02:13:18.133159+00:00"},{"alias_kind":"pith_short_8","alias_value":"K7XX422Y","created_at":"2026-06-23T02:13:18.133159+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/K7XX422YSSUGU2V2TI4HC62C7F","json":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F.json","graph_json":"https://pith.science/api/pith-number/K7XX422YSSUGU2V2TI4HC62C7F/graph.json","events_json":"https://pith.science/api/pith-number/K7XX422YSSUGU2V2TI4HC62C7F/events.json","paper":"https://pith.science/paper/K7XX422Y"},"agent_actions":{"view_html":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F","download_json":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F.json","view_paper":"https://pith.science/paper/K7XX422Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2511.02324&json=true","fetch_graph":"https://pith.science/api/pith-number/K7XX422YSSUGU2V2TI4HC62C7F/graph.json","fetch_events":"https://pith.science/api/pith-number/K7XX422YSSUGU2V2TI4HC62C7F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F/action/storage_attestation","attest_author":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F/action/author_attestation","sign_citation":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F/action/citation_signature","submit_replication":"https://pith.science/pith/K7XX422YSSUGU2V2TI4HC62C7F/action/replication_record"}},"created_at":"2026-06-23T02:13:18.133159+00:00","updated_at":"2026-06-23T02:13:18.133159+00:00"}