{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:EQC6LWMMV6H4G5G4QUPASHX2IB","short_pith_number":"pith:EQC6LWMM","canonical_record":{"source":{"id":"1312.5473","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-19T10:42:46Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"cf8c5eced4bc9fc798ebe6dddbf7d9d6e05e767d11ba96a3028aa5cd64d154ff","abstract_canon_sha256":"9cfe5cf180660df539cb13456c294a4887b99f7028dbd6f210ea3b196cae2e74"},"schema_version":"1.0"},"canonical_sha256":"2405e5d98caf8fc374dc851e091efa407160c6b788151fe38ff40c79a3ecbd4e","source":{"kind":"arxiv","id":"1312.5473","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.5473","created_at":"2026-05-17T23:56:15Z"},{"alias_kind":"arxiv_version","alias_value":"1312.5473v6","created_at":"2026-05-17T23:56:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5473","created_at":"2026-05-17T23:56:15Z"},{"alias_kind":"pith_short_12","alias_value":"EQC6LWMMV6H4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EQC6LWMMV6H4G5G4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EQC6LWMM","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:EQC6LWMMV6H4G5G4QUPASHX2IB","target":"record","payload":{"canonical_record":{"source":{"id":"1312.5473","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-19T10:42:46Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"cf8c5eced4bc9fc798ebe6dddbf7d9d6e05e767d11ba96a3028aa5cd64d154ff","abstract_canon_sha256":"9cfe5cf180660df539cb13456c294a4887b99f7028dbd6f210ea3b196cae2e74"},"schema_version":"1.0"},"canonical_sha256":"2405e5d98caf8fc374dc851e091efa407160c6b788151fe38ff40c79a3ecbd4e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:15.843637Z","signature_b64":"qBPwS03OVcutMVMN2t3Q9UGS98jnTlgB5bLEUCJxlINnXFX+/zGKtM8qVBgxrmMBch/sohJ1ILMAaV61i/3UCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2405e5d98caf8fc374dc851e091efa407160c6b788151fe38ff40c79a3ecbd4e","last_reissued_at":"2026-05-17T23:56:15.843013Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:15.843013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.5473","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:56:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GsXtVf+ziJpaLPpW73zGaysJKlIlcrFatqkrNXqtoE8ngbPMTB6oj2zCcDkx69eWm4VOC8u5kUOoEfbcLAz7AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:18:58.805421Z"},"content_sha256":"39dbb0e8e09ce81c719711858c9b9834e23577a93bfa2a0831e073e5824324ec","schema_version":"1.0","event_id":"sha256:39dbb0e8e09ce81c719711858c9b9834e23577a93bfa2a0831e073e5824324ec"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:EQC6LWMMV6H4G5G4QUPASHX2IB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harnack inequalities on weighted graphs and some applications to the random conductance model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Jean-Dominique Deuschel, Martin Slowik, Sebastian Andres","submitted_at":"2013-12-19T10:42:46Z","abstract_excerpt":"We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk $X$ in an environment of ergodic random conductances taking values in $[0, \\infty)$ satisfying some moment conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5473","kind":"arxiv","version":6},"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":"2026-05-17T23:56:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sYWu0cuHlURtoOOu9LI2ETkkRVI2o/fITcoK2CzWkPTE1YbYzwknpGzCKmdaw+le7ul3a5Q8JneB7LTlWg0IDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:18:58.805774Z"},"content_sha256":"8195ffcd0204cd2302bd59f04912c8bfe06cbf1c84eb9c15686428211411564d","schema_version":"1.0","event_id":"sha256:8195ffcd0204cd2302bd59f04912c8bfe06cbf1c84eb9c15686428211411564d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EQC6LWMMV6H4G5G4QUPASHX2IB/bundle.json","state_url":"https://pith.science/pith/EQC6LWMMV6H4G5G4QUPASHX2IB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EQC6LWMMV6H4G5G4QUPASHX2IB/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":"2026-06-20T07:18:58Z","links":{"resolver":"https://pith.science/pith/EQC6LWMMV6H4G5G4QUPASHX2IB","bundle":"https://pith.science/pith/EQC6LWMMV6H4G5G4QUPASHX2IB/bundle.json","state":"https://pith.science/pith/EQC6LWMMV6H4G5G4QUPASHX2IB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EQC6LWMMV6H4G5G4QUPASHX2IB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EQC6LWMMV6H4G5G4QUPASHX2IB","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":"9cfe5cf180660df539cb13456c294a4887b99f7028dbd6f210ea3b196cae2e74","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-19T10:42:46Z","title_canon_sha256":"cf8c5eced4bc9fc798ebe6dddbf7d9d6e05e767d11ba96a3028aa5cd64d154ff"},"schema_version":"1.0","source":{"id":"1312.5473","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.5473","created_at":"2026-05-17T23:56:15Z"},{"alias_kind":"arxiv_version","alias_value":"1312.5473v6","created_at":"2026-05-17T23:56:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5473","created_at":"2026-05-17T23:56:15Z"},{"alias_kind":"pith_short_12","alias_value":"EQC6LWMMV6H4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EQC6LWMMV6H4G5G4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EQC6LWMM","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:8195ffcd0204cd2302bd59f04912c8bfe06cbf1c84eb9c15686428211411564d","target":"graph","created_at":"2026-05-17T23:56:15Z","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":"We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk $X$ in an environment of ergodic random conductances taking values in $[0, \\infty)$ satisfying some moment conditions.","authors_text":"Jean-Dominique Deuschel, Martin Slowik, Sebastian Andres","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-19T10:42:46Z","title":"Harnack inequalities on weighted graphs and some applications to the random conductance model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5473","kind":"arxiv","version":6},"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":"sha256:39dbb0e8e09ce81c719711858c9b9834e23577a93bfa2a0831e073e5824324ec","target":"record","created_at":"2026-05-17T23:56:15Z","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":"9cfe5cf180660df539cb13456c294a4887b99f7028dbd6f210ea3b196cae2e74","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-19T10:42:46Z","title_canon_sha256":"cf8c5eced4bc9fc798ebe6dddbf7d9d6e05e767d11ba96a3028aa5cd64d154ff"},"schema_version":"1.0","source":{"id":"1312.5473","kind":"arxiv","version":6}},"canonical_sha256":"2405e5d98caf8fc374dc851e091efa407160c6b788151fe38ff40c79a3ecbd4e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2405e5d98caf8fc374dc851e091efa407160c6b788151fe38ff40c79a3ecbd4e","first_computed_at":"2026-05-17T23:56:15.843013Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:15.843013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qBPwS03OVcutMVMN2t3Q9UGS98jnTlgB5bLEUCJxlINnXFX+/zGKtM8qVBgxrmMBch/sohJ1ILMAaV61i/3UCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:15.843637Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.5473","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:39dbb0e8e09ce81c719711858c9b9834e23577a93bfa2a0831e073e5824324ec","sha256:8195ffcd0204cd2302bd59f04912c8bfe06cbf1c84eb9c15686428211411564d"],"state_sha256":"32b95dbbda6e0de568798634d7eceac6645ba0c6dd862515923557fca206995b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RHOKyd8kd9fceF4E/gzw1RY4N7eDfpyIntqVUv1WAWM27EVzRMg49NPXTLc+QCkleBdyZLY7VcpNctcyEIMgAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T07:18:58.807757Z","bundle_sha256":"52dac96eee8fafee7825891c5ad0e48aeb9388e296e1eeb2b2832f47a2e900a4"}}