{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:GM4GO2UJ4QKLR67YQBNR3KM66C","short_pith_number":"pith:GM4GO2UJ","canonical_record":{"source":{"id":"2109.10309","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-09-21T16:42:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e936598d589712b0139f65e65182803b3e74bb100ae589c54754bd7c838bdbec","abstract_canon_sha256":"1d846cc5735b21eee93c06f55aeb8de4f6268639ac5e4bcf64b8de948c19ced9"},"schema_version":"1.0"},"canonical_sha256":"3338676a89e414b8fbf8805b1da99ef08b20c1fcfdce8225042dea26f84e9fb9","source":{"kind":"arxiv","id":"2109.10309","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.10309","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"arxiv_version","alias_value":"2109.10309v1","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.10309","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"pith_short_12","alias_value":"GM4GO2UJ4QKL","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"pith_short_16","alias_value":"GM4GO2UJ4QKLR67Y","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"pith_short_8","alias_value":"GM4GO2UJ","created_at":"2026-07-05T03:16:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:GM4GO2UJ4QKLR67YQBNR3KM66C","target":"record","payload":{"canonical_record":{"source":{"id":"2109.10309","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-09-21T16:42:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e936598d589712b0139f65e65182803b3e74bb100ae589c54754bd7c838bdbec","abstract_canon_sha256":"1d846cc5735b21eee93c06f55aeb8de4f6268639ac5e4bcf64b8de948c19ced9"},"schema_version":"1.0"},"canonical_sha256":"3338676a89e414b8fbf8805b1da99ef08b20c1fcfdce8225042dea26f84e9fb9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:16:27.831067Z","signature_b64":"rHRuG3LgbVjDlu2UpG32v1td2nyRGvsPXPLk2vX5wDsE6hR+Gfd334EHqnuAdsboUeNyTX+XevWGgTLjepHzCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3338676a89e414b8fbf8805b1da99ef08b20c1fcfdce8225042dea26f84e9fb9","last_reissued_at":"2026-07-05T03:16:27.830638Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:16:27.830638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.10309","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":"2026-07-05T03:16:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c3KezF+5vq9ehee3e1olDh3HYNbhcU+/2F9vyj7w98HDMDg4I69t4ETfsfYXZYfXzt1yE2W/Fr/noiVZdQyiCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T07:54:05.655249Z"},"content_sha256":"31d7e4a0947b4d423fba12d7b185de8983f1ce59639eed951399c3685ff2d349","schema_version":"1.0","event_id":"sha256:31d7e4a0947b4d423fba12d7b185de8983f1ce59639eed951399c3685ff2d349"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:GM4GO2UJ4QKLR67YQBNR3KM66C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Multiplicative Property for Zero-Sums II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Chao Liu, David J. Grynkiewicz","submitted_at":"2021-09-21T16:42:26Z","abstract_excerpt":"Let $G=C_n\\oplus C_{mn}$ with $n\\geq 2$ and $m\\geq 1$, and let $k\\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to characterize those sequences with maximal length $mn+n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $mn+n-1-k$. For $k\\leq 1$, this is the inverse question for the Davenport Constant. For $k=n-1$, this is the inverse question for the $\\eta(G)$ invariant concerning short zero-sum subsequences. The structure in both these case"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.10309","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/2109.10309/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:16:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8fnzoL9LyRgbatT9XzosSO0Oc8kS6umo2NszeseHCQklEN1iDtBp5Yyv+dCqrapnjfjsUfmCrjs4TiDgMqJXCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T07:54:05.655762Z"},"content_sha256":"187914a24cd57f846bbc318ea44092c9891fc3da237db3f26462457fb04fc749","schema_version":"1.0","event_id":"sha256:187914a24cd57f846bbc318ea44092c9891fc3da237db3f26462457fb04fc749"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GM4GO2UJ4QKLR67YQBNR3KM66C/bundle.json","state_url":"https://pith.science/pith/GM4GO2UJ4QKLR67YQBNR3KM66C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GM4GO2UJ4QKLR67YQBNR3KM66C/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":"2026-08-06T07:54:05Z","links":{"resolver":"https://pith.science/pith/GM4GO2UJ4QKLR67YQBNR3KM66C","bundle":"https://pith.science/pith/GM4GO2UJ4QKLR67YQBNR3KM66C/bundle.json","state":"https://pith.science/pith/GM4GO2UJ4QKLR67YQBNR3KM66C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GM4GO2UJ4QKLR67YQBNR3KM66C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:GM4GO2UJ4QKLR67YQBNR3KM66C","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":"1d846cc5735b21eee93c06f55aeb8de4f6268639ac5e4bcf64b8de948c19ced9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-09-21T16:42:26Z","title_canon_sha256":"e936598d589712b0139f65e65182803b3e74bb100ae589c54754bd7c838bdbec"},"schema_version":"1.0","source":{"id":"2109.10309","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.10309","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"arxiv_version","alias_value":"2109.10309v1","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.10309","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"pith_short_12","alias_value":"GM4GO2UJ4QKL","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"pith_short_16","alias_value":"GM4GO2UJ4QKLR67Y","created_at":"2026-07-05T03:16:27Z"},{"alias_kind":"pith_short_8","alias_value":"GM4GO2UJ","created_at":"2026-07-05T03:16:27Z"}],"graph_snapshots":[{"event_id":"sha256:187914a24cd57f846bbc318ea44092c9891fc3da237db3f26462457fb04fc749","target":"graph","created_at":"2026-07-05T03:16:27Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2109.10309/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $G=C_n\\oplus C_{mn}$ with $n\\geq 2$ and $m\\geq 1$, and let $k\\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to characterize those sequences with maximal length $mn+n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $mn+n-1-k$. For $k\\leq 1$, this is the inverse question for the Davenport Constant. For $k=n-1$, this is the inverse question for the $\\eta(G)$ invariant concerning short zero-sum subsequences. The structure in both these case","authors_text":"Chao Liu, David J. Grynkiewicz","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-09-21T16:42:26Z","title":"A Multiplicative Property for Zero-Sums II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.10309","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":"sha256:31d7e4a0947b4d423fba12d7b185de8983f1ce59639eed951399c3685ff2d349","target":"record","created_at":"2026-07-05T03:16:27Z","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":"1d846cc5735b21eee93c06f55aeb8de4f6268639ac5e4bcf64b8de948c19ced9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-09-21T16:42:26Z","title_canon_sha256":"e936598d589712b0139f65e65182803b3e74bb100ae589c54754bd7c838bdbec"},"schema_version":"1.0","source":{"id":"2109.10309","kind":"arxiv","version":1}},"canonical_sha256":"3338676a89e414b8fbf8805b1da99ef08b20c1fcfdce8225042dea26f84e9fb9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3338676a89e414b8fbf8805b1da99ef08b20c1fcfdce8225042dea26f84e9fb9","first_computed_at":"2026-07-05T03:16:27.830638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:16:27.830638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rHRuG3LgbVjDlu2UpG32v1td2nyRGvsPXPLk2vX5wDsE6hR+Gfd334EHqnuAdsboUeNyTX+XevWGgTLjepHzCA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:16:27.831067Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.10309","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31d7e4a0947b4d423fba12d7b185de8983f1ce59639eed951399c3685ff2d349","sha256:187914a24cd57f846bbc318ea44092c9891fc3da237db3f26462457fb04fc749"],"state_sha256":"c76fcaf2173206f7fd8bc6f278f35cb5be91067f729325700cb303a7b2e8ebf9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d2bXGEMGhWcrdUbFsnu/V3C/txZ+3rHcxOvSdr7YF7Mlbg6sLwcn+o6UpND8nxkLN64VOpwuqQ0S48PbgcVGAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T07:54:05.661184Z","bundle_sha256":"5a9a6aefa83d84f5a52ac020d8bbad5f5320ca4ae2f9c0e997d94416283dc824"}}