{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:W3D5K7GAA7QXDF5SW4QKBR3VAY","short_pith_number":"pith:W3D5K7GA","canonical_record":{"source":{"id":"2412.00635","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-01T01:22:47Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"823e6b0a72216c11098c1ca05d2786b12373b964f0cd7cafb60149e9b34947a0","abstract_canon_sha256":"658a9ce26c785b61e24fc1d9256ac2d823886095a80d18a1254e28372bcc3c11"},"schema_version":"1.0"},"canonical_sha256":"b6c7d57cc007e17197b2b720a0c775060336f9edea1060da03a47a1ebdc89182","source":{"kind":"arxiv","id":"2412.00635","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.00635","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"2412.00635v2","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00635","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"W3D5K7GAA7QX","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"pith_short_16","alias_value":"W3D5K7GAA7QXDF5S","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"pith_short_8","alias_value":"W3D5K7GA","created_at":"2026-07-05T09:59:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:W3D5K7GAA7QXDF5SW4QKBR3VAY","target":"record","payload":{"canonical_record":{"source":{"id":"2412.00635","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-01T01:22:47Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"823e6b0a72216c11098c1ca05d2786b12373b964f0cd7cafb60149e9b34947a0","abstract_canon_sha256":"658a9ce26c785b61e24fc1d9256ac2d823886095a80d18a1254e28372bcc3c11"},"schema_version":"1.0"},"canonical_sha256":"b6c7d57cc007e17197b2b720a0c775060336f9edea1060da03a47a1ebdc89182","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:59:01.207214Z","signature_b64":"VYUO5Hl+1WYi0j0HxNHz6dnEZs5c8kvWjjZcgfArvUexkJBITlCgw+A0wzAHICC1K6gkqCapC18OwrXrT1LTAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6c7d57cc007e17197b2b720a0c775060336f9edea1060da03a47a1ebdc89182","last_reissued_at":"2026-07-05T09:59:01.206770Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:59:01.206770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.00635","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":"2026-07-05T09:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U9Clh9Lwk2BK+Q3jfMloPQ07w8KANnMqVb310stUPW+5V+u/ouXgbYvgnweBKmPM1hYYguYKyV8jtSP9yz3PAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T19:08:02.747545Z"},"content_sha256":"b7687a4cf1feb17015f3e05c25b9208082a74a7ff91dcfe9ed980f9da5cf6113","schema_version":"1.0","event_id":"sha256:b7687a4cf1feb17015f3e05c25b9208082a74a7ff91dcfe9ed980f9da5cf6113"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:W3D5K7GAA7QXDF5SW4QKBR3VAY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical threshold for regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ishaan Bhadoo","submitted_at":"2024-12-01T01:22:47Z","abstract_excerpt":"In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \\geq \\frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all quasi-transitive $d$-regular graphs, the equality $p_c(G) = \\frac{1}{d-1}$ holds if and only if $G$ is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00635","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.00635/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":"2026-07-05T09:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T8T4h6kVmhpeD+AxVJu3f6hGhc426J7loPMEMUpeMmZmQkknpKEBCznC9nBmuySPrFsb+jhxx+bZ3pQUvNj1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T19:08:02.748065Z"},"content_sha256":"51b3126b54b550ea8aca201188a225eccc418c10b07c0817acffbee2e7566e6a","schema_version":"1.0","event_id":"sha256:51b3126b54b550ea8aca201188a225eccc418c10b07c0817acffbee2e7566e6a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/bundle.json","state_url":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/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":"2026-08-12T19:08:02Z","links":{"resolver":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY","bundle":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/bundle.json","state":"https://pith.science/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W3D5K7GAA7QXDF5SW4QKBR3VAY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:W3D5K7GAA7QXDF5SW4QKBR3VAY","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":"658a9ce26c785b61e24fc1d9256ac2d823886095a80d18a1254e28372bcc3c11","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-01T01:22:47Z","title_canon_sha256":"823e6b0a72216c11098c1ca05d2786b12373b964f0cd7cafb60149e9b34947a0"},"schema_version":"1.0","source":{"id":"2412.00635","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.00635","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"2412.00635v2","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00635","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"W3D5K7GAA7QX","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"pith_short_16","alias_value":"W3D5K7GAA7QXDF5S","created_at":"2026-07-05T09:59:01Z"},{"alias_kind":"pith_short_8","alias_value":"W3D5K7GA","created_at":"2026-07-05T09:59:01Z"}],"graph_snapshots":[{"event_id":"sha256:51b3126b54b550ea8aca201188a225eccc418c10b07c0817acffbee2e7566e6a","target":"graph","created_at":"2026-07-05T09:59:01Z","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/2412.00635/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \\geq \\frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all quasi-transitive $d$-regular graphs, the equality $p_c(G) = \\frac{1}{d-1}$ holds if and only if $G$ is a tree. Furthermore, we provide counterexamples that illustrate the necessity of the quasi-transitive assumption.","authors_text":"Ishaan Bhadoo","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-01T01:22:47Z","title":"Critical threshold for regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00635","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:b7687a4cf1feb17015f3e05c25b9208082a74a7ff91dcfe9ed980f9da5cf6113","target":"record","created_at":"2026-07-05T09:59:01Z","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":"658a9ce26c785b61e24fc1d9256ac2d823886095a80d18a1254e28372bcc3c11","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-01T01:22:47Z","title_canon_sha256":"823e6b0a72216c11098c1ca05d2786b12373b964f0cd7cafb60149e9b34947a0"},"schema_version":"1.0","source":{"id":"2412.00635","kind":"arxiv","version":2}},"canonical_sha256":"b6c7d57cc007e17197b2b720a0c775060336f9edea1060da03a47a1ebdc89182","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b6c7d57cc007e17197b2b720a0c775060336f9edea1060da03a47a1ebdc89182","first_computed_at":"2026-07-05T09:59:01.206770Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:59:01.206770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VYUO5Hl+1WYi0j0HxNHz6dnEZs5c8kvWjjZcgfArvUexkJBITlCgw+A0wzAHICC1K6gkqCapC18OwrXrT1LTAw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:59:01.207214Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.00635","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b7687a4cf1feb17015f3e05c25b9208082a74a7ff91dcfe9ed980f9da5cf6113","sha256:51b3126b54b550ea8aca201188a225eccc418c10b07c0817acffbee2e7566e6a"],"state_sha256":"0c3d6be3324ebcd015ba36d363254b128a660379adb43055bae6356eb218c4c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CaP4jHqRs+CcBDin3tNqTV+RnZwnlIPNppFYZwNoyY2NvJLDmYvpW3JDK/xdcefpBv7tPvrJzTC9HyBWOBx6AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T19:08:02.752805Z","bundle_sha256":"b714fb2ca4bc6cc9c715fed00706435cee44572a7bccae0b338b459b00955bab"}}