{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:WIB4MZSZMPQ7TIUXIN6VU5XS4C","short_pith_number":"pith:WIB4MZSZ","canonical_record":{"source":{"id":"1012.2473","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-12-11T17:12:59Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"a6ba949288880715c588c59ac246fc2ef59b658ceea8f3907c4bc7f4522a4c9b","abstract_canon_sha256":"dae247268a9634e39e7877aacb403ccbae7092461870ce7445c81b174c374c15"},"schema_version":"1.0"},"canonical_sha256":"b203c6665963e1f9a297437d5a76f2e0a18a7d8b359d0ac40c6f34487e91a833","source":{"kind":"arxiv","id":"1012.2473","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2473","created_at":"2026-05-18T04:15:50Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2473v2","created_at":"2026-05-18T04:15:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2473","created_at":"2026-05-18T04:15:50Z"},{"alias_kind":"pith_short_12","alias_value":"WIB4MZSZMPQ7","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WIB4MZSZMPQ7TIUX","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WIB4MZSZ","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:WIB4MZSZMPQ7TIUXIN6VU5XS4C","target":"record","payload":{"canonical_record":{"source":{"id":"1012.2473","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-12-11T17:12:59Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"a6ba949288880715c588c59ac246fc2ef59b658ceea8f3907c4bc7f4522a4c9b","abstract_canon_sha256":"dae247268a9634e39e7877aacb403ccbae7092461870ce7445c81b174c374c15"},"schema_version":"1.0"},"canonical_sha256":"b203c6665963e1f9a297437d5a76f2e0a18a7d8b359d0ac40c6f34487e91a833","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:50.162455Z","signature_b64":"L7XIVMvhFgFWKX8HMXYAFG7kRFVDo4+a15Daqu78cLM8zbwcv91GQEupomIYSQVQ9wwWUPalCpk3uv5f1WmICA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b203c6665963e1f9a297437d5a76f2e0a18a7d8b359d0ac40c6f34487e91a833","last_reissued_at":"2026-05-18T04:15:50.161945Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:50.161945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.2473","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:15:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gOQ0sluXPcrDYVDbcN7gQH92PI+QjQtpQ2fwrZj+eLGT6zV7KeLJWYvMxqeVZPYo46NowXkPOhNhZ+9hhg2oDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T23:47:40.094213Z"},"content_sha256":"ee756aa026308f7f7f01bcc2487d2371fb32603fd5b6110f2ad72dfbdc8f3061","schema_version":"1.0","event_id":"sha256:ee756aa026308f7f7f01bcc2487d2371fb32603fd5b6110f2ad72dfbdc8f3061"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:WIB4MZSZMPQ7TIUXIN6VU5XS4C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some results concerning the $p$-Royden and $p$-harmonic boundaries of a graph of bounded degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Michael Puls","submitted_at":"2010-12-11T17:12:59Z","abstract_excerpt":"Let $p$ be a real number greater than one and let $\\Gamma$ be a connected graph of bounded degree. We show that the $p$-Royden boundary of $\\Gamma$ with the $p$-harmonic boundary removed is a $F_{\\sigma}$-set. We also characterize the $p$-harmonic boundary of $\\Gamma$ in terms of the intersection of the extreme points of a certain subset of one-sided infinite paths in $\\Gamma$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2473","kind":"arxiv","version":2},"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":"2026-05-18T04:15:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tBNw0A7XMHJjz/CrKQg027ECbf5q0MQqTiDYk7PclXS0HdrmnULl5vLvenDjEXkpjgF++IBXsD57kkcJszm8Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T23:47:40.094575Z"},"content_sha256":"c6cdd4ea21d411c052903dc2c10c65a45d67a70cd90f60f63aeca3a4ce218a0f","schema_version":"1.0","event_id":"sha256:c6cdd4ea21d411c052903dc2c10c65a45d67a70cd90f60f63aeca3a4ce218a0f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WIB4MZSZMPQ7TIUXIN6VU5XS4C/bundle.json","state_url":"https://pith.science/pith/WIB4MZSZMPQ7TIUXIN6VU5XS4C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WIB4MZSZMPQ7TIUXIN6VU5XS4C/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":"2026-06-23T23:47:40Z","links":{"resolver":"https://pith.science/pith/WIB4MZSZMPQ7TIUXIN6VU5XS4C","bundle":"https://pith.science/pith/WIB4MZSZMPQ7TIUXIN6VU5XS4C/bundle.json","state":"https://pith.science/pith/WIB4MZSZMPQ7TIUXIN6VU5XS4C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WIB4MZSZMPQ7TIUXIN6VU5XS4C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:WIB4MZSZMPQ7TIUXIN6VU5XS4C","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":"dae247268a9634e39e7877aacb403ccbae7092461870ce7445c81b174c374c15","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-12-11T17:12:59Z","title_canon_sha256":"a6ba949288880715c588c59ac246fc2ef59b658ceea8f3907c4bc7f4522a4c9b"},"schema_version":"1.0","source":{"id":"1012.2473","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.2473","created_at":"2026-05-18T04:15:50Z"},{"alias_kind":"arxiv_version","alias_value":"1012.2473v2","created_at":"2026-05-18T04:15:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2473","created_at":"2026-05-18T04:15:50Z"},{"alias_kind":"pith_short_12","alias_value":"WIB4MZSZMPQ7","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WIB4MZSZMPQ7TIUX","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WIB4MZSZ","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:c6cdd4ea21d411c052903dc2c10c65a45d67a70cd90f60f63aeca3a4ce218a0f","target":"graph","created_at":"2026-05-18T04:15:50Z","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"},"paper":{"abstract_excerpt":"Let $p$ be a real number greater than one and let $\\Gamma$ be a connected graph of bounded degree. We show that the $p$-Royden boundary of $\\Gamma$ with the $p$-harmonic boundary removed is a $F_{\\sigma}$-set. We also characterize the $p$-harmonic boundary of $\\Gamma$ in terms of the intersection of the extreme points of a certain subset of one-sided infinite paths in $\\Gamma$.","authors_text":"Michael Puls","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-12-11T17:12:59Z","title":"Some results concerning the $p$-Royden and $p$-harmonic boundaries of a graph of bounded degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2473","kind":"arxiv","version":2},"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:ee756aa026308f7f7f01bcc2487d2371fb32603fd5b6110f2ad72dfbdc8f3061","target":"record","created_at":"2026-05-18T04:15:50Z","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":"dae247268a9634e39e7877aacb403ccbae7092461870ce7445c81b174c374c15","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-12-11T17:12:59Z","title_canon_sha256":"a6ba949288880715c588c59ac246fc2ef59b658ceea8f3907c4bc7f4522a4c9b"},"schema_version":"1.0","source":{"id":"1012.2473","kind":"arxiv","version":2}},"canonical_sha256":"b203c6665963e1f9a297437d5a76f2e0a18a7d8b359d0ac40c6f34487e91a833","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b203c6665963e1f9a297437d5a76f2e0a18a7d8b359d0ac40c6f34487e91a833","first_computed_at":"2026-05-18T04:15:50.161945Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:50.161945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L7XIVMvhFgFWKX8HMXYAFG7kRFVDo4+a15Daqu78cLM8zbwcv91GQEupomIYSQVQ9wwWUPalCpk3uv5f1WmICA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:50.162455Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.2473","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee756aa026308f7f7f01bcc2487d2371fb32603fd5b6110f2ad72dfbdc8f3061","sha256:c6cdd4ea21d411c052903dc2c10c65a45d67a70cd90f60f63aeca3a4ce218a0f"],"state_sha256":"71c0e6ba4eda555d54a3c10a63e7c5ce7b951192a4d8e79857b4ad406899a59c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JdfUGkf14wqy8U2b7zlXz9YTkgicFpjgr6e7JVWsV/MZgGv3EJGDBwp01MKON95wQP8F1i42rEpd1JJghaa0Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T23:47:40.096542Z","bundle_sha256":"10a91b19cbfd55872e67ece16c6b9abfdc4cc8c7e00b936fd3486f0f209eba5f"}}