{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2003:6K5474QF2Q6KGIVZJJLIAXUI2P","short_pith_number":"pith:6K5474QF","canonical_record":{"source":{"id":"math/0309318","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2003-09-19T12:05:15Z","cross_cats_sorted":[],"title_canon_sha256":"7b3805081a86c61fc89c869415d78cbf72d4eb11607b4aafec73072676ba7f8d","abstract_canon_sha256":"2b806ca710e03b146c68c486912afe2f3f5e8b7e96218015296f228e75e67776"},"schema_version":"1.0"},"canonical_sha256":"f2bbcff205d43ca322b94a56805e88d3e6bc43e091c04bda02263102acdc861f","source":{"kind":"arxiv","id":"math/0309318","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0309318","created_at":"2026-05-18T01:05:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0309318v3","created_at":"2026-05-18T01:05:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0309318","created_at":"2026-05-18T01:05:28Z"},{"alias_kind":"pith_short_12","alias_value":"6K5474QF2Q6K","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"6K5474QF2Q6KGIVZ","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"6K5474QF","created_at":"2026-05-18T12:25:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2003:6K5474QF2Q6KGIVZJJLIAXUI2P","target":"record","payload":{"canonical_record":{"source":{"id":"math/0309318","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2003-09-19T12:05:15Z","cross_cats_sorted":[],"title_canon_sha256":"7b3805081a86c61fc89c869415d78cbf72d4eb11607b4aafec73072676ba7f8d","abstract_canon_sha256":"2b806ca710e03b146c68c486912afe2f3f5e8b7e96218015296f228e75e67776"},"schema_version":"1.0"},"canonical_sha256":"f2bbcff205d43ca322b94a56805e88d3e6bc43e091c04bda02263102acdc861f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:28.627726Z","signature_b64":"YxvIhZ2lERDehx6dBufHsPFTKjf2oBVP0SZAx8PZIk1gpMIOT04bPDN3DxW8ZUy8ySHkTvJK0Sm2D6Ea+FJRAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2bbcff205d43ca322b94a56805e88d3e6bc43e091c04bda02263102acdc861f","last_reissued_at":"2026-05-18T01:05:28.627208Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:28.627208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0309318","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bmo3YfTdtu6Kdgn8c3/cpSKN7je9iV8nlfc3HOxcKfygnI8ni3c8zXFkvggtFPqaJFlbOhUj+EW/1njmoCbUBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:05:05.884832Z"},"content_sha256":"99289c12b242f09c7e27c76367bb13e00a616dffca7176b850c6e4325c009fc0","schema_version":"1.0","event_id":"sha256:99289c12b242f09c7e27c76367bb13e00a616dffca7176b850c6e4325c009fc0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2003:6K5474QF2Q6KGIVZJJLIAXUI2P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a set-theoretic invariant","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dion Gijswijt, Pieter Moree","submitted_at":"2003-09-19T12:05:15Z","abstract_excerpt":"Let a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 solutions of the linear inequality |a_i-a_j| < e_ka_k < a_i+a_j, with e_k=-1 of 1 and k running over the integers 1,...,m with i and j skipped. They introduced some invariants and near invariants related to this situation (invariant meaning here: not depending on the choice of i and j).\n The main result of their paper is extended here to a much more general setting, namely that of certain maps from finite sets to {-1,1}. Some applications are given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0309318","kind":"arxiv","version":3},"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":"2026-05-18T01:05:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l/gRkUo+TFsMHYdFdBVrSILBrz4Mb2dSyiWsHG/eZmG8J7xrMgxnvhOdXXi4gNus/gTmy0y5mymXU8wX1Fz2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:05:05.885538Z"},"content_sha256":"e1f5b7214b17d6277cee41814c632f98c1701b1d9e5a84dca1219c83fe2ed424","schema_version":"1.0","event_id":"sha256:e1f5b7214b17d6277cee41814c632f98c1701b1d9e5a84dca1219c83fe2ed424"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6K5474QF2Q6KGIVZJJLIAXUI2P/bundle.json","state_url":"https://pith.science/pith/6K5474QF2Q6KGIVZJJLIAXUI2P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6K5474QF2Q6KGIVZJJLIAXUI2P/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":"2026-05-25T15:05:05Z","links":{"resolver":"https://pith.science/pith/6K5474QF2Q6KGIVZJJLIAXUI2P","bundle":"https://pith.science/pith/6K5474QF2Q6KGIVZJJLIAXUI2P/bundle.json","state":"https://pith.science/pith/6K5474QF2Q6KGIVZJJLIAXUI2P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6K5474QF2Q6KGIVZJJLIAXUI2P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:6K5474QF2Q6KGIVZJJLIAXUI2P","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":"2b806ca710e03b146c68c486912afe2f3f5e8b7e96218015296f228e75e67776","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2003-09-19T12:05:15Z","title_canon_sha256":"7b3805081a86c61fc89c869415d78cbf72d4eb11607b4aafec73072676ba7f8d"},"schema_version":"1.0","source":{"id":"math/0309318","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0309318","created_at":"2026-05-18T01:05:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0309318v3","created_at":"2026-05-18T01:05:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0309318","created_at":"2026-05-18T01:05:28Z"},{"alias_kind":"pith_short_12","alias_value":"6K5474QF2Q6K","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"6K5474QF2Q6KGIVZ","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"6K5474QF","created_at":"2026-05-18T12:25:51Z"}],"graph_snapshots":[{"event_id":"sha256:e1f5b7214b17d6277cee41814c632f98c1701b1d9e5a84dca1219c83fe2ed424","target":"graph","created_at":"2026-05-18T01:05:28Z","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 a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 solutions of the linear inequality |a_i-a_j| < e_ka_k < a_i+a_j, with e_k=-1 of 1 and k running over the integers 1,...,m with i and j skipped. They introduced some invariants and near invariants related to this situation (invariant meaning here: not depending on the choice of i and j).\n The main result of their paper is extended here to a much more general setting, namely that of certain maps from finite sets to {-1,1}. Some applications are given.","authors_text":"Dion Gijswijt, Pieter Moree","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2003-09-19T12:05:15Z","title":"On a set-theoretic invariant"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0309318","kind":"arxiv","version":3},"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":"sha256:99289c12b242f09c7e27c76367bb13e00a616dffca7176b850c6e4325c009fc0","target":"record","created_at":"2026-05-18T01:05:28Z","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":"2b806ca710e03b146c68c486912afe2f3f5e8b7e96218015296f228e75e67776","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2003-09-19T12:05:15Z","title_canon_sha256":"7b3805081a86c61fc89c869415d78cbf72d4eb11607b4aafec73072676ba7f8d"},"schema_version":"1.0","source":{"id":"math/0309318","kind":"arxiv","version":3}},"canonical_sha256":"f2bbcff205d43ca322b94a56805e88d3e6bc43e091c04bda02263102acdc861f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f2bbcff205d43ca322b94a56805e88d3e6bc43e091c04bda02263102acdc861f","first_computed_at":"2026-05-18T01:05:28.627208Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:28.627208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YxvIhZ2lERDehx6dBufHsPFTKjf2oBVP0SZAx8PZIk1gpMIOT04bPDN3DxW8ZUy8ySHkTvJK0Sm2D6Ea+FJRAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:28.627726Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0309318","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99289c12b242f09c7e27c76367bb13e00a616dffca7176b850c6e4325c009fc0","sha256:e1f5b7214b17d6277cee41814c632f98c1701b1d9e5a84dca1219c83fe2ed424"],"state_sha256":"88e05519e7c00b0cb855f55ba2af09adbf6807ec6660e8d54e65778b8f672e9f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NOlifDNfq+b9Ao/NLaMtggYutbZ+P+IoNM3YCGwfGbhwKZVIX1waEHb3gW3UUf8b2MnxtaPKqc7riB9mLBsYAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T15:05:05.888945Z","bundle_sha256":"f850af95924f02bcc605e18dd5573198c04468a7e0b00d1740d43e5b6b041c60"}}