{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:ZH3DAHUJZTKDTTI2L352F77JXH","short_pith_number":"pith:ZH3DAHUJ","canonical_record":{"source":{"id":"2111.13173","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2021-11-25T17:09:28Z","cross_cats_sorted":[],"title_canon_sha256":"994d617e167a0d8e5bbc267fae79a95af9e1deb01cc0cea9d49669781d6f79c3","abstract_canon_sha256":"b2284637faad5e5d9abd382ff943f37c634846615900714702d605c84aeec7a9"},"schema_version":"1.0"},"canonical_sha256":"c9f6301e89ccd439cd1a5efba2ffe9b9d35ffcd78410fb7cfa2786ff617eff19","source":{"kind":"arxiv","id":"2111.13173","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2111.13173","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"arxiv_version","alias_value":"2111.13173v1","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2111.13173","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"pith_short_12","alias_value":"ZH3DAHUJZTKD","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"pith_short_16","alias_value":"ZH3DAHUJZTKDTTI2","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"pith_short_8","alias_value":"ZH3DAHUJ","created_at":"2026-07-05T05:52:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:ZH3DAHUJZTKDTTI2L352F77JXH","target":"record","payload":{"canonical_record":{"source":{"id":"2111.13173","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2021-11-25T17:09:28Z","cross_cats_sorted":[],"title_canon_sha256":"994d617e167a0d8e5bbc267fae79a95af9e1deb01cc0cea9d49669781d6f79c3","abstract_canon_sha256":"b2284637faad5e5d9abd382ff943f37c634846615900714702d605c84aeec7a9"},"schema_version":"1.0"},"canonical_sha256":"c9f6301e89ccd439cd1a5efba2ffe9b9d35ffcd78410fb7cfa2786ff617eff19","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:52:06.763361Z","signature_b64":"EM2l7qJFdCswh0hUS4FA2hlkfM2e4wn65piF0SIoepLUfp3NWKlQl2zrUYBP8G15x7tOc9NdZtEeoQx8HuxmDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9f6301e89ccd439cd1a5efba2ffe9b9d35ffcd78410fb7cfa2786ff617eff19","last_reissued_at":"2026-07-05T05:52:06.762893Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:52:06.762893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2111.13173","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":"2026-07-05T05:52:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q7wTuLkXuY/Zjl0HZHn8I2YuSbfJ/IjE07Rweo0UvZSaMwWU3w5nWJT1xne4z/3DYzryH4t0j8FtqB+o/aXwAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T17:55:28.522860Z"},"content_sha256":"6c4b59d23e1eae4e32a4ddaff49b25576c4156c21ae67a6d4cef6f23c44785e9","schema_version":"1.0","event_id":"sha256:6c4b59d23e1eae4e32a4ddaff49b25576c4156c21ae67a6d4cef6f23c44785e9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:ZH3DAHUJZTKDTTI2L352F77JXH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A new lower bound on the pebbling number of the grid","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jan Petr, Julien Portier, Szymon Stolarczyk","submitted_at":"2021-11-25T17:09:28Z","abstract_excerpt":"A pebbling move on a graph consists of removing $2$ pebbles from a vertex and adding $1$ pebble to one of the neighbouring vertices. A vertex is called reachable if we can put $1$ pebble on it after a sequence of moves. The optimal pebbling number of a graph is the minimum number $m$ such that there exists a distribution of $m$ pebbles so that each vertex is reachable. For the case of a square grid $n \\times m$, Gy\\H{o}ri, Katona and Papp recently showed that its optimal pebbling number is at least $\\frac{2}{13}nm \\approx 0.1538nm$ and at most $\\frac{2}{7}nm +O(n+m) \\approx 0.2857nm$. We impro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.13173","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/2111.13173/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":"2026-07-05T05:52:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U4kQffYokkuxjtHvwQJcmq9SeYw/Ohwo4MbCw8onMoupcTN7sXFuwxI5/3vTbPLEyeg8W6REHdtcLCYoBaqrBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T17:55:28.523381Z"},"content_sha256":"ffc8c79ad3527a5dc0cc7a54e613bf86019c197ccdcedcb33bd738c9bdde4f25","schema_version":"1.0","event_id":"sha256:ffc8c79ad3527a5dc0cc7a54e613bf86019c197ccdcedcb33bd738c9bdde4f25"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZH3DAHUJZTKDTTI2L352F77JXH/bundle.json","state_url":"https://pith.science/pith/ZH3DAHUJZTKDTTI2L352F77JXH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZH3DAHUJZTKDTTI2L352F77JXH/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":"2026-08-04T17:55:28Z","links":{"resolver":"https://pith.science/pith/ZH3DAHUJZTKDTTI2L352F77JXH","bundle":"https://pith.science/pith/ZH3DAHUJZTKDTTI2L352F77JXH/bundle.json","state":"https://pith.science/pith/ZH3DAHUJZTKDTTI2L352F77JXH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZH3DAHUJZTKDTTI2L352F77JXH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:ZH3DAHUJZTKDTTI2L352F77JXH","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":"b2284637faad5e5d9abd382ff943f37c634846615900714702d605c84aeec7a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2021-11-25T17:09:28Z","title_canon_sha256":"994d617e167a0d8e5bbc267fae79a95af9e1deb01cc0cea9d49669781d6f79c3"},"schema_version":"1.0","source":{"id":"2111.13173","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2111.13173","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"arxiv_version","alias_value":"2111.13173v1","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2111.13173","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"pith_short_12","alias_value":"ZH3DAHUJZTKD","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"pith_short_16","alias_value":"ZH3DAHUJZTKDTTI2","created_at":"2026-07-05T05:52:06Z"},{"alias_kind":"pith_short_8","alias_value":"ZH3DAHUJ","created_at":"2026-07-05T05:52:06Z"}],"graph_snapshots":[{"event_id":"sha256:ffc8c79ad3527a5dc0cc7a54e613bf86019c197ccdcedcb33bd738c9bdde4f25","target":"graph","created_at":"2026-07-05T05:52:06Z","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/2111.13173/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A pebbling move on a graph consists of removing $2$ pebbles from a vertex and adding $1$ pebble to one of the neighbouring vertices. A vertex is called reachable if we can put $1$ pebble on it after a sequence of moves. The optimal pebbling number of a graph is the minimum number $m$ such that there exists a distribution of $m$ pebbles so that each vertex is reachable. For the case of a square grid $n \\times m$, Gy\\H{o}ri, Katona and Papp recently showed that its optimal pebbling number is at least $\\frac{2}{13}nm \\approx 0.1538nm$ and at most $\\frac{2}{7}nm +O(n+m) \\approx 0.2857nm$. We impro","authors_text":"Jan Petr, Julien Portier, Szymon Stolarczyk","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2021-11-25T17:09:28Z","title":"A new lower bound on the pebbling number of the grid"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.13173","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":"sha256:6c4b59d23e1eae4e32a4ddaff49b25576c4156c21ae67a6d4cef6f23c44785e9","target":"record","created_at":"2026-07-05T05:52:06Z","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":"b2284637faad5e5d9abd382ff943f37c634846615900714702d605c84aeec7a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2021-11-25T17:09:28Z","title_canon_sha256":"994d617e167a0d8e5bbc267fae79a95af9e1deb01cc0cea9d49669781d6f79c3"},"schema_version":"1.0","source":{"id":"2111.13173","kind":"arxiv","version":1}},"canonical_sha256":"c9f6301e89ccd439cd1a5efba2ffe9b9d35ffcd78410fb7cfa2786ff617eff19","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9f6301e89ccd439cd1a5efba2ffe9b9d35ffcd78410fb7cfa2786ff617eff19","first_computed_at":"2026-07-05T05:52:06.762893Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:52:06.762893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EM2l7qJFdCswh0hUS4FA2hlkfM2e4wn65piF0SIoepLUfp3NWKlQl2zrUYBP8G15x7tOc9NdZtEeoQx8HuxmDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:52:06.763361Z","signed_message":"canonical_sha256_bytes"},"source_id":"2111.13173","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c4b59d23e1eae4e32a4ddaff49b25576c4156c21ae67a6d4cef6f23c44785e9","sha256:ffc8c79ad3527a5dc0cc7a54e613bf86019c197ccdcedcb33bd738c9bdde4f25"],"state_sha256":"d07c5009815a894677d37cf7bd67ca22fff25a33cb320fd5dcf530c1f5880380"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XJrOa/saX1hX3oxu9ARUuPkFmJY1qlfNN8mC/dLym44+65o/RZve48ksP4TSaaZhlo/04Yl1v23DMt46DTA/AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T17:55:28.528351Z","bundle_sha256":"3e62b4ae9f10956f9db0d4fd562d32f03ec685b33de569a250cb2506f99a261c"}}