{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:FMMDOCNIWTW64GGDE62PZOKYFR","short_pith_number":"pith:FMMDOCNI","canonical_record":{"source":{"id":"1701.02819","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-11T01:18:02Z","cross_cats_sorted":[],"title_canon_sha256":"997911866f14841c5a7feef1f5eb9e749354b98deee0a52ea5244f2f9abcc649","abstract_canon_sha256":"0c3c99cb09f40fbd2903669b6446692b1ba2f18a4fe4558dd767d2eacc07a700"},"schema_version":"1.0"},"canonical_sha256":"2b183709a8b4edee18c327b4fcb9582c5aab1d663380870b63e20c879db8bfc2","source":{"kind":"arxiv","id":"1701.02819","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.02819","created_at":"2026-05-18T00:53:00Z"},{"alias_kind":"arxiv_version","alias_value":"1701.02819v1","created_at":"2026-05-18T00:53:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.02819","created_at":"2026-05-18T00:53:00Z"},{"alias_kind":"pith_short_12","alias_value":"FMMDOCNIWTW6","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FMMDOCNIWTW64GGD","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FMMDOCNI","created_at":"2026-05-18T12:31:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:FMMDOCNIWTW64GGDE62PZOKYFR","target":"record","payload":{"canonical_record":{"source":{"id":"1701.02819","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-11T01:18:02Z","cross_cats_sorted":[],"title_canon_sha256":"997911866f14841c5a7feef1f5eb9e749354b98deee0a52ea5244f2f9abcc649","abstract_canon_sha256":"0c3c99cb09f40fbd2903669b6446692b1ba2f18a4fe4558dd767d2eacc07a700"},"schema_version":"1.0"},"canonical_sha256":"2b183709a8b4edee18c327b4fcb9582c5aab1d663380870b63e20c879db8bfc2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:00.093104Z","signature_b64":"QU98Yy/mcSbInFwdJEIwHpiMnzpaDm2raF/CRKuv8+iSXbes60MQGLwfOSiRtzl6thazRf54zdOaTKziLndbDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b183709a8b4edee18c327b4fcb9582c5aab1d663380870b63e20c879db8bfc2","last_reissued_at":"2026-05-18T00:53:00.092659Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:00.092659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.02819","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":"2026-05-18T00:53:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jyGuK00GEH7VZf7fn1LT5ExM1ZdaDFkul18IdmBqkuk2mF+9WU9VQLlDCagWX5G1aBY1W8IU7ufEVE3aysDYAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:10:42.685112Z"},"content_sha256":"8388e659abbb0c675f6ecca3aa49e8d365b6c809cc26e7430c8b032c75acd8b0","schema_version":"1.0","event_id":"sha256:8388e659abbb0c675f6ecca3aa49e8d365b6c809cc26e7430c8b032c75acd8b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:FMMDOCNIWTW64GGDE62PZOKYFR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tetris Hypergraphs and Combinations of Impartial Games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Endre Boros, Kazuhisa Makino, Nhan Bao Ho, Peter Mursic, Vladimir Gurvich","submitted_at":"2017-01-11T01:18:02Z","abstract_excerpt":"The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\\cH$-combinations of impartial games for any hypergraph $\\cH$. In particular, we introduce the game $NIM_\\cH$ which is the $\\cH$-combination of single pile $NIM$ games. An impartial game is called SG decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce the $\\cH$-combination of SG decreasing games to $NIM_\\cH$. We call $\\cH$ a Tetris hypergraph if $NIM_\\cH$ is SG decreasing. We provide some necessary and some suffi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02819","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":"2026-05-18T00:53:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v/1/dtMbDBvQq+XnRWnh7nAJiA76jPMbKOwsWJqV0vQgjUULGWhxULR3YalQKUYLCL/f65H3uR0wt006T6jHBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:10:42.685484Z"},"content_sha256":"7b20a360cbc309f17e62b844d8d066073bb37cb150da49888ac2b9387779f348","schema_version":"1.0","event_id":"sha256:7b20a360cbc309f17e62b844d8d066073bb37cb150da49888ac2b9387779f348"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FMMDOCNIWTW64GGDE62PZOKYFR/bundle.json","state_url":"https://pith.science/pith/FMMDOCNIWTW64GGDE62PZOKYFR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FMMDOCNIWTW64GGDE62PZOKYFR/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":"2026-06-01T04:10:42Z","links":{"resolver":"https://pith.science/pith/FMMDOCNIWTW64GGDE62PZOKYFR","bundle":"https://pith.science/pith/FMMDOCNIWTW64GGDE62PZOKYFR/bundle.json","state":"https://pith.science/pith/FMMDOCNIWTW64GGDE62PZOKYFR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FMMDOCNIWTW64GGDE62PZOKYFR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:FMMDOCNIWTW64GGDE62PZOKYFR","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":"0c3c99cb09f40fbd2903669b6446692b1ba2f18a4fe4558dd767d2eacc07a700","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-11T01:18:02Z","title_canon_sha256":"997911866f14841c5a7feef1f5eb9e749354b98deee0a52ea5244f2f9abcc649"},"schema_version":"1.0","source":{"id":"1701.02819","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.02819","created_at":"2026-05-18T00:53:00Z"},{"alias_kind":"arxiv_version","alias_value":"1701.02819v1","created_at":"2026-05-18T00:53:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.02819","created_at":"2026-05-18T00:53:00Z"},{"alias_kind":"pith_short_12","alias_value":"FMMDOCNIWTW6","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FMMDOCNIWTW64GGD","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FMMDOCNI","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:7b20a360cbc309f17e62b844d8d066073bb37cb150da49888ac2b9387779f348","target":"graph","created_at":"2026-05-18T00:53:00Z","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":"The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\\cH$-combinations of impartial games for any hypergraph $\\cH$. In particular, we introduce the game $NIM_\\cH$ which is the $\\cH$-combination of single pile $NIM$ games. An impartial game is called SG decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce the $\\cH$-combination of SG decreasing games to $NIM_\\cH$. We call $\\cH$ a Tetris hypergraph if $NIM_\\cH$ is SG decreasing. We provide some necessary and some suffi","authors_text":"Endre Boros, Kazuhisa Makino, Nhan Bao Ho, Peter Mursic, Vladimir Gurvich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-11T01:18:02Z","title":"Tetris Hypergraphs and Combinations of Impartial Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02819","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":"sha256:8388e659abbb0c675f6ecca3aa49e8d365b6c809cc26e7430c8b032c75acd8b0","target":"record","created_at":"2026-05-18T00:53:00Z","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":"0c3c99cb09f40fbd2903669b6446692b1ba2f18a4fe4558dd767d2eacc07a700","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-11T01:18:02Z","title_canon_sha256":"997911866f14841c5a7feef1f5eb9e749354b98deee0a52ea5244f2f9abcc649"},"schema_version":"1.0","source":{"id":"1701.02819","kind":"arxiv","version":1}},"canonical_sha256":"2b183709a8b4edee18c327b4fcb9582c5aab1d663380870b63e20c879db8bfc2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b183709a8b4edee18c327b4fcb9582c5aab1d663380870b63e20c879db8bfc2","first_computed_at":"2026-05-18T00:53:00.092659Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:00.092659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QU98Yy/mcSbInFwdJEIwHpiMnzpaDm2raF/CRKuv8+iSXbes60MQGLwfOSiRtzl6thazRf54zdOaTKziLndbDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:00.093104Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.02819","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8388e659abbb0c675f6ecca3aa49e8d365b6c809cc26e7430c8b032c75acd8b0","sha256:7b20a360cbc309f17e62b844d8d066073bb37cb150da49888ac2b9387779f348"],"state_sha256":"f2454f009679c5c402519d4f8543116fd146f619ee3cc2c37f9af8647b290b34"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T+9bSr41KqTQfNfBlGzo5fUTKYKzKXhyI/1xuSmgBZ/XER7RTMceJP5xLJpL/N4WNunXZtxJVlBrOL1pCjzNBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T04:10:42.687409Z","bundle_sha256":"a263f74923a309917031134c9c9b7a944aa7652e0898d05c187b7e542ce66286"}}