{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:CCIRITU2YHCIYFRTD3PX75OAUW","short_pith_number":"pith:CCIRITU2","canonical_record":{"source":{"id":"1810.13021","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-30T22:28:00Z","cross_cats_sorted":[],"title_canon_sha256":"34bcc3f2cf18a95387db6bcee533d829274771bd86abd0d6d3fa5d10045b4e04","abstract_canon_sha256":"903b2d8d8291d2cad027a393d888c1b1e807ecab1a84aa71dbe16d27bf191e4a"},"schema_version":"1.0"},"canonical_sha256":"1091144e9ac1c48c16331edf7ff5c0a5ba6bb53dd4a3fa03782c8168013c6de4","source":{"kind":"arxiv","id":"1810.13021","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.13021","created_at":"2026-05-17T23:43:33Z"},{"alias_kind":"arxiv_version","alias_value":"1810.13021v1","created_at":"2026-05-17T23:43:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.13021","created_at":"2026-05-17T23:43:33Z"},{"alias_kind":"pith_short_12","alias_value":"CCIRITU2YHCI","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"CCIRITU2YHCIYFRT","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"CCIRITU2","created_at":"2026-05-18T12:32:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:CCIRITU2YHCIYFRTD3PX75OAUW","target":"record","payload":{"canonical_record":{"source":{"id":"1810.13021","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-30T22:28:00Z","cross_cats_sorted":[],"title_canon_sha256":"34bcc3f2cf18a95387db6bcee533d829274771bd86abd0d6d3fa5d10045b4e04","abstract_canon_sha256":"903b2d8d8291d2cad027a393d888c1b1e807ecab1a84aa71dbe16d27bf191e4a"},"schema_version":"1.0"},"canonical_sha256":"1091144e9ac1c48c16331edf7ff5c0a5ba6bb53dd4a3fa03782c8168013c6de4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:33.070884Z","signature_b64":"L3eTHLPnIJeGf0VUVuHECU7OmPkrslB9idRpUGfQWwWy9yVNt/P0eWbY1GZBKHxbsxpVLq5E7uO3oIIxUWU+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1091144e9ac1c48c16331edf7ff5c0a5ba6bb53dd4a3fa03782c8168013c6de4","last_reissued_at":"2026-05-17T23:43:33.070484Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:33.070484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.13021","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:43:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vhFkOc5RTF4IjSQh6qFMmlls23vnUdcz0lRH8VfUpmyvyevBgE4Ppvh0w0iGgfVxAB7/JbZd4mOIJi1eyYnPBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T20:06:17.153881Z"},"content_sha256":"15dd122d944f2ae4a804d6ab36e6736d77d7c63f222f0e4d82a6b9f1ebabab2f","schema_version":"1.0","event_id":"sha256:15dd122d944f2ae4a804d6ab36e6736d77d7c63f222f0e4d82a6b9f1ebabab2f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:CCIRITU2YHCIYFRTD3PX75OAUW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted EGZ Constant for p-groups of rank 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ab\\'ilio Lemos, Filipe Oliveira, Hemar Godinho","submitted_at":"2018-10-30T22:28:00Z","abstract_excerpt":"Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\\ell$ such that any sequence over $G$ of length at least $\\ell$ has an $A$-weighted zero-sum of length $n$ and $\\eta_A(G)$ defined as the smallest integer $\\ell$ such that any sequence over $G$ of length at least $\\ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $\\alpha \\geq \\beta$, and $A=\\left\\{x\\in\\mathbb{N}\\; : \\; 1 \\le a \\le p^{\\alpha} \\; \\mbox{ and }\\; \\gcd(a, p) = 1\\right \\}$, we have $s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.13021","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:43:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bXP462ztg/hOkD9Re71XeHs7ixYZLKvcYnLh6QquL7mQKSER/+vlZSZvfxlzkodRYYDv3gKMvDiFTsJN8O/yDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T20:06:17.154229Z"},"content_sha256":"01dad0d278eb1fd09cfd73bc3d8e70d7ab728c2306a1452ed9517ec860e05d1a","schema_version":"1.0","event_id":"sha256:01dad0d278eb1fd09cfd73bc3d8e70d7ab728c2306a1452ed9517ec860e05d1a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CCIRITU2YHCIYFRTD3PX75OAUW/bundle.json","state_url":"https://pith.science/pith/CCIRITU2YHCIYFRTD3PX75OAUW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CCIRITU2YHCIYFRTD3PX75OAUW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T20:06:17Z","links":{"resolver":"https://pith.science/pith/CCIRITU2YHCIYFRTD3PX75OAUW","bundle":"https://pith.science/pith/CCIRITU2YHCIYFRTD3PX75OAUW/bundle.json","state":"https://pith.science/pith/CCIRITU2YHCIYFRTD3PX75OAUW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CCIRITU2YHCIYFRTD3PX75OAUW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CCIRITU2YHCIYFRTD3PX75OAUW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"903b2d8d8291d2cad027a393d888c1b1e807ecab1a84aa71dbe16d27bf191e4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-30T22:28:00Z","title_canon_sha256":"34bcc3f2cf18a95387db6bcee533d829274771bd86abd0d6d3fa5d10045b4e04"},"schema_version":"1.0","source":{"id":"1810.13021","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.13021","created_at":"2026-05-17T23:43:33Z"},{"alias_kind":"arxiv_version","alias_value":"1810.13021v1","created_at":"2026-05-17T23:43:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.13021","created_at":"2026-05-17T23:43:33Z"},{"alias_kind":"pith_short_12","alias_value":"CCIRITU2YHCI","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"CCIRITU2YHCIYFRT","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"CCIRITU2","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:01dad0d278eb1fd09cfd73bc3d8e70d7ab728c2306a1452ed9517ec860e05d1a","target":"graph","created_at":"2026-05-17T23:43:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\\ell$ such that any sequence over $G$ of length at least $\\ell$ has an $A$-weighted zero-sum of length $n$ and $\\eta_A(G)$ defined as the smallest integer $\\ell$ such that any sequence over $G$ of length at least $\\ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $\\alpha \\geq \\beta$, and $A=\\left\\{x\\in\\mathbb{N}\\; : \\; 1 \\le a \\le p^{\\alpha} \\; \\mbox{ and }\\; \\gcd(a, p) = 1\\right \\}$, we have $s","authors_text":"Ab\\'ilio Lemos, Filipe Oliveira, Hemar Godinho","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-30T22:28:00Z","title":"Weighted EGZ Constant for p-groups of rank 2"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.13021","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:15dd122d944f2ae4a804d6ab36e6736d77d7c63f222f0e4d82a6b9f1ebabab2f","target":"record","created_at":"2026-05-17T23:43:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"903b2d8d8291d2cad027a393d888c1b1e807ecab1a84aa71dbe16d27bf191e4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-30T22:28:00Z","title_canon_sha256":"34bcc3f2cf18a95387db6bcee533d829274771bd86abd0d6d3fa5d10045b4e04"},"schema_version":"1.0","source":{"id":"1810.13021","kind":"arxiv","version":1}},"canonical_sha256":"1091144e9ac1c48c16331edf7ff5c0a5ba6bb53dd4a3fa03782c8168013c6de4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1091144e9ac1c48c16331edf7ff5c0a5ba6bb53dd4a3fa03782c8168013c6de4","first_computed_at":"2026-05-17T23:43:33.070484Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:33.070484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L3eTHLPnIJeGf0VUVuHECU7OmPkrslB9idRpUGfQWwWy9yVNt/P0eWbY1GZBKHxbsxpVLq5E7uO3oIIxUWU+Dw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:33.070884Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.13021","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15dd122d944f2ae4a804d6ab36e6736d77d7c63f222f0e4d82a6b9f1ebabab2f","sha256:01dad0d278eb1fd09cfd73bc3d8e70d7ab728c2306a1452ed9517ec860e05d1a"],"state_sha256":"72d75306fd2a6d2fa473add22c14c6e5bbd4829c98061acb0b1b1e290693625c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8RvnnSdvpmdQuf6FE/7Bxj0cf23dBeotUOGmXYj4w42wxoPw9PTiMQLkBFSNBiIAK1nIw5ULgSlbKJ17ghmbCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T20:06:17.156361Z","bundle_sha256":"c96bc2c702f29088d425cc881fbf8e01e8b10b282bb30441f4cce3c8f2c28806"}}