{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:3PVFHAWTEFAK2F7DLEZBEQG2YZ","short_pith_number":"pith:3PVFHAWT","canonical_record":{"source":{"id":"0708.3628","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cs.DM","submitted_at":"2007-08-27T15:24:39Z","cross_cats_sorted":[],"title_canon_sha256":"41c11a78852ab04f12506f7a7a994a3a2b4abe6421a7f4c9821cc8decd5ea0ad","abstract_canon_sha256":"22a794382d6728ccb54cccddc7d373334b67c5aa7a398ad3bf10c494dfc85a3f"},"schema_version":"1.0"},"canonical_sha256":"dbea5382d32140ad17e359321240dac64b9e90a5b7a59c55d310b8e8fe3a81e9","source":{"kind":"arxiv","id":"0708.3628","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0708.3628","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"arxiv_version","alias_value":"0708.3628v1","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0708.3628","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"pith_short_12","alias_value":"3PVFHAWTEFAK","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"pith_short_16","alias_value":"3PVFHAWTEFAK2F7D","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"pith_short_8","alias_value":"3PVFHAWT","created_at":"2026-07-04T15:03:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:3PVFHAWTEFAK2F7DLEZBEQG2YZ","target":"record","payload":{"canonical_record":{"source":{"id":"0708.3628","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cs.DM","submitted_at":"2007-08-27T15:24:39Z","cross_cats_sorted":[],"title_canon_sha256":"41c11a78852ab04f12506f7a7a994a3a2b4abe6421a7f4c9821cc8decd5ea0ad","abstract_canon_sha256":"22a794382d6728ccb54cccddc7d373334b67c5aa7a398ad3bf10c494dfc85a3f"},"schema_version":"1.0"},"canonical_sha256":"dbea5382d32140ad17e359321240dac64b9e90a5b7a59c55d310b8e8fe3a81e9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:03:50.805580Z","signature_b64":"kC3lq9wjJQFuR+FV1wmmT+khjulb4GdIR0MmAIdJKQCDoAsO9r8CrBXKl5lQaYyUKgzgDlMVEJ+EN3xvA0B2Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbea5382d32140ad17e359321240dac64b9e90a5b7a59c55d310b8e8fe3a81e9","last_reissued_at":"2026-07-04T15:03:50.805203Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:03:50.805203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0708.3628","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":"2026-07-04T15:03:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T0wryUw3j+6CRG7QHZD5AUzekpZi7SwLldgD5qngITpfaCP2YE46rL3aY7U0ydyC5z5LIVzcszkLJ6Z8jwI1BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T07:11:01.813220Z"},"content_sha256":"4cafdd2ac9430c83a63d377927be5be0a5e9d88813fd73cbf23c0a8d9efd2895","schema_version":"1.0","event_id":"sha256:4cafdd2ac9430c83a63d377927be5be0a5e9d88813fd73cbf23c0a8d9efd2895"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:3PVFHAWTEFAK2F7DLEZBEQG2YZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Recursive Structure and Bandwidth of Hales-Numbered Hypercube","license":"","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Xiaohan Wang, Xiaolin Wu","submitted_at":"2007-08-27T15:24:39Z","abstract_excerpt":"The Hales numbered $n$-dimensional hypercube and the corresponding adjacency matrix exhibit interesting recursive structures in $n$. These structures lead to a very simple proof of the well-known bandwidth formula for hypercube, whose proof was thought to be surprisingly difficult. A related problem called hypercube antibandwidth, for which Harper proposed an algorithm, is also reexamined in the light of the above recursive structures, and a close form solution is found."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.3628","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/0708.3628/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":"2026-07-04T15:03:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AatXgyLhfdG0GfzFxDaDIzrDxhrjhx0aGEHg9/efETopoGHoD/twQt67Y1S42hzGdslcSY+q8UBpJquzWAfIBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T07:11:01.813615Z"},"content_sha256":"edcd444868921e14c1ff7563585344cfa3e1b569b646b52116f85d8e48be85f7","schema_version":"1.0","event_id":"sha256:edcd444868921e14c1ff7563585344cfa3e1b569b646b52116f85d8e48be85f7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3PVFHAWTEFAK2F7DLEZBEQG2YZ/bundle.json","state_url":"https://pith.science/pith/3PVFHAWTEFAK2F7DLEZBEQG2YZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3PVFHAWTEFAK2F7DLEZBEQG2YZ/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":"2026-07-13T07:11:01Z","links":{"resolver":"https://pith.science/pith/3PVFHAWTEFAK2F7DLEZBEQG2YZ","bundle":"https://pith.science/pith/3PVFHAWTEFAK2F7DLEZBEQG2YZ/bundle.json","state":"https://pith.science/pith/3PVFHAWTEFAK2F7DLEZBEQG2YZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3PVFHAWTEFAK2F7DLEZBEQG2YZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:3PVFHAWTEFAK2F7DLEZBEQG2YZ","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":"22a794382d6728ccb54cccddc7d373334b67c5aa7a398ad3bf10c494dfc85a3f","cross_cats_sorted":[],"license":"","primary_cat":"cs.DM","submitted_at":"2007-08-27T15:24:39Z","title_canon_sha256":"41c11a78852ab04f12506f7a7a994a3a2b4abe6421a7f4c9821cc8decd5ea0ad"},"schema_version":"1.0","source":{"id":"0708.3628","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0708.3628","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"arxiv_version","alias_value":"0708.3628v1","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0708.3628","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"pith_short_12","alias_value":"3PVFHAWTEFAK","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"pith_short_16","alias_value":"3PVFHAWTEFAK2F7D","created_at":"2026-07-04T15:03:50Z"},{"alias_kind":"pith_short_8","alias_value":"3PVFHAWT","created_at":"2026-07-04T15:03:50Z"}],"graph_snapshots":[{"event_id":"sha256:edcd444868921e14c1ff7563585344cfa3e1b569b646b52116f85d8e48be85f7","target":"graph","created_at":"2026-07-04T15:03:50Z","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/0708.3628/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Hales numbered $n$-dimensional hypercube and the corresponding adjacency matrix exhibit interesting recursive structures in $n$. These structures lead to a very simple proof of the well-known bandwidth formula for hypercube, whose proof was thought to be surprisingly difficult. A related problem called hypercube antibandwidth, for which Harper proposed an algorithm, is also reexamined in the light of the above recursive structures, and a close form solution is found.","authors_text":"Xiaohan Wang, Xiaolin Wu","cross_cats":[],"headline":"","license":"","primary_cat":"cs.DM","submitted_at":"2007-08-27T15:24:39Z","title":"Recursive Structure and Bandwidth of Hales-Numbered Hypercube"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.3628","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":"sha256:4cafdd2ac9430c83a63d377927be5be0a5e9d88813fd73cbf23c0a8d9efd2895","target":"record","created_at":"2026-07-04T15:03:50Z","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":"22a794382d6728ccb54cccddc7d373334b67c5aa7a398ad3bf10c494dfc85a3f","cross_cats_sorted":[],"license":"","primary_cat":"cs.DM","submitted_at":"2007-08-27T15:24:39Z","title_canon_sha256":"41c11a78852ab04f12506f7a7a994a3a2b4abe6421a7f4c9821cc8decd5ea0ad"},"schema_version":"1.0","source":{"id":"0708.3628","kind":"arxiv","version":1}},"canonical_sha256":"dbea5382d32140ad17e359321240dac64b9e90a5b7a59c55d310b8e8fe3a81e9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dbea5382d32140ad17e359321240dac64b9e90a5b7a59c55d310b8e8fe3a81e9","first_computed_at":"2026-07-04T15:03:50.805203Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:03:50.805203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kC3lq9wjJQFuR+FV1wmmT+khjulb4GdIR0MmAIdJKQCDoAsO9r8CrBXKl5lQaYyUKgzgDlMVEJ+EN3xvA0B2Dg==","signature_status":"signed_v1","signed_at":"2026-07-04T15:03:50.805580Z","signed_message":"canonical_sha256_bytes"},"source_id":"0708.3628","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4cafdd2ac9430c83a63d377927be5be0a5e9d88813fd73cbf23c0a8d9efd2895","sha256:edcd444868921e14c1ff7563585344cfa3e1b569b646b52116f85d8e48be85f7"],"state_sha256":"c6ab02ac449120d8cfd1c5edd931ed8080adf9f0c61e6d5ddb1e90bbbce8cbfb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PykC7xSucvNtPp5dD0wlXfyM2nTzImUgIomqEpIEIDEc00KYA7Z5JmvFFRWookBsO0Ff3uyQXQoR7UCBhWZ0Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-13T07:11:01.816058Z","bundle_sha256":"82d3a144377bd0be391c8323e296a4c1666503dbfe27847acedda29216fc2273"}}