{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:V47TJJ22S3B4FN4CIKR7H3N6RA","short_pith_number":"pith:V47TJJ22","canonical_record":{"source":{"id":"1201.2344","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-11T16:33:21Z","cross_cats_sorted":[],"title_canon_sha256":"6bddda283236c0c3e094199c6f3e784a585cd378ef494f6b6f054db8758b4606","abstract_canon_sha256":"043cfa956fc1efaef0f27eea9a527cda4dc7b9d923b7ffa184a8143c65904920"},"schema_version":"1.0"},"canonical_sha256":"af3f34a75a96c3c2b78242a3f3edbe88193c1f4664db6320ab7fc7358431c890","source":{"kind":"arxiv","id":"1201.2344","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.2344","created_at":"2026-05-18T04:04:52Z"},{"alias_kind":"arxiv_version","alias_value":"1201.2344v1","created_at":"2026-05-18T04:04:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.2344","created_at":"2026-05-18T04:04:52Z"},{"alias_kind":"pith_short_12","alias_value":"V47TJJ22S3B4","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"V47TJJ22S3B4FN4C","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"V47TJJ22","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:V47TJJ22S3B4FN4CIKR7H3N6RA","target":"record","payload":{"canonical_record":{"source":{"id":"1201.2344","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-11T16:33:21Z","cross_cats_sorted":[],"title_canon_sha256":"6bddda283236c0c3e094199c6f3e784a585cd378ef494f6b6f054db8758b4606","abstract_canon_sha256":"043cfa956fc1efaef0f27eea9a527cda4dc7b9d923b7ffa184a8143c65904920"},"schema_version":"1.0"},"canonical_sha256":"af3f34a75a96c3c2b78242a3f3edbe88193c1f4664db6320ab7fc7358431c890","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:52.553839Z","signature_b64":"oyrkpAdHslRhjKT05ehqB8MOt7m6TSVkI229k6PVwrFF0qDuIFPLp0pJLqp2tB39EVfQeoGQjr+1JlEKzzYdBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af3f34a75a96c3c2b78242a3f3edbe88193c1f4664db6320ab7fc7358431c890","last_reissued_at":"2026-05-18T04:04:52.553113Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:52.553113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.2344","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":"2026-05-18T04:04:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fc2I9UpoSczaIo7fP183MPVIOMwFm3EYwx6JTD643YIBF6tT/3FF/oceV5mbuckkacOQeZxmZ/nQy2FwhzI2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T21:31:42.191975Z"},"content_sha256":"8284d861498b77529ffcdefc0f32c171edb91fbd9480aa91c3ad8cc73f460b4b","schema_version":"1.0","event_id":"sha256:8284d861498b77529ffcdefc0f32c171edb91fbd9480aa91c3ad8cc73f460b4b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:V47TJJ22S3B4FN4CIKR7H3N6RA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Continuum Percolation for Quermass Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Coupier, David Dereudre","submitted_at":"2012-01-11T16:33:21Z","abstract_excerpt":"The continuum percolation for Markov (or Gibbs) germ-grain models is investigated. The grains are assumed circular with random radii on a compact support. The morphological interaction is the so-called Quermass interaction defined by a linear combination of the classical Minkowski functionals (area, perimeter and Euler-Poincar\\'e characteristic). We show that the percolation occurs for any coefficient of this linear combination and for a large enough activity parameter. An application to the phase transition of the multi-type Quermass model is given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2344","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":""},"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":"2026-05-18T04:04:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OujQd4tlrloVbg+Xk4PlnNucIjq+Z92bQvhC/Su47jxqByXyQsG/lM3xKQ0xMp53iTR2h8j/+dOCGqy75+t5AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T21:31:42.192303Z"},"content_sha256":"11d775fe35fe6d82ea7923ac265f09b9ccdd55b0e123cfe640d0837aab8a02ca","schema_version":"1.0","event_id":"sha256:11d775fe35fe6d82ea7923ac265f09b9ccdd55b0e123cfe640d0837aab8a02ca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V47TJJ22S3B4FN4CIKR7H3N6RA/bundle.json","state_url":"https://pith.science/pith/V47TJJ22S3B4FN4CIKR7H3N6RA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V47TJJ22S3B4FN4CIKR7H3N6RA/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":"2026-06-01T21:31:42Z","links":{"resolver":"https://pith.science/pith/V47TJJ22S3B4FN4CIKR7H3N6RA","bundle":"https://pith.science/pith/V47TJJ22S3B4FN4CIKR7H3N6RA/bundle.json","state":"https://pith.science/pith/V47TJJ22S3B4FN4CIKR7H3N6RA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V47TJJ22S3B4FN4CIKR7H3N6RA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:V47TJJ22S3B4FN4CIKR7H3N6RA","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":"043cfa956fc1efaef0f27eea9a527cda4dc7b9d923b7ffa184a8143c65904920","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-11T16:33:21Z","title_canon_sha256":"6bddda283236c0c3e094199c6f3e784a585cd378ef494f6b6f054db8758b4606"},"schema_version":"1.0","source":{"id":"1201.2344","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.2344","created_at":"2026-05-18T04:04:52Z"},{"alias_kind":"arxiv_version","alias_value":"1201.2344v1","created_at":"2026-05-18T04:04:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.2344","created_at":"2026-05-18T04:04:52Z"},{"alias_kind":"pith_short_12","alias_value":"V47TJJ22S3B4","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"V47TJJ22S3B4FN4C","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"V47TJJ22","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:11d775fe35fe6d82ea7923ac265f09b9ccdd55b0e123cfe640d0837aab8a02ca","target":"graph","created_at":"2026-05-18T04:04:52Z","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":"The continuum percolation for Markov (or Gibbs) germ-grain models is investigated. The grains are assumed circular with random radii on a compact support. The morphological interaction is the so-called Quermass interaction defined by a linear combination of the classical Minkowski functionals (area, perimeter and Euler-Poincar\\'e characteristic). We show that the percolation occurs for any coefficient of this linear combination and for a large enough activity parameter. An application to the phase transition of the multi-type Quermass model is given.","authors_text":"David Coupier, David Dereudre","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-11T16:33:21Z","title":"Continuum Percolation for Quermass Model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2344","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":"sha256:8284d861498b77529ffcdefc0f32c171edb91fbd9480aa91c3ad8cc73f460b4b","target":"record","created_at":"2026-05-18T04:04:52Z","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":"043cfa956fc1efaef0f27eea9a527cda4dc7b9d923b7ffa184a8143c65904920","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-11T16:33:21Z","title_canon_sha256":"6bddda283236c0c3e094199c6f3e784a585cd378ef494f6b6f054db8758b4606"},"schema_version":"1.0","source":{"id":"1201.2344","kind":"arxiv","version":1}},"canonical_sha256":"af3f34a75a96c3c2b78242a3f3edbe88193c1f4664db6320ab7fc7358431c890","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af3f34a75a96c3c2b78242a3f3edbe88193c1f4664db6320ab7fc7358431c890","first_computed_at":"2026-05-18T04:04:52.553113Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:52.553113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oyrkpAdHslRhjKT05ehqB8MOt7m6TSVkI229k6PVwrFF0qDuIFPLp0pJLqp2tB39EVfQeoGQjr+1JlEKzzYdBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:52.553839Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.2344","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8284d861498b77529ffcdefc0f32c171edb91fbd9480aa91c3ad8cc73f460b4b","sha256:11d775fe35fe6d82ea7923ac265f09b9ccdd55b0e123cfe640d0837aab8a02ca"],"state_sha256":"3ecd25032e88bac77d2b7fd20bb6dace217a746b2e5aefabf46d830d17ba8d4a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yNxydIKRvT/X8A5HHDfT7hXhfGNBIMfH6xWM09HgkrdmpQ3AYG30WEuesD8xoISezeLVOsARf14IIyw2wGU5AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T21:31:42.194051Z","bundle_sha256":"ec53d6bc4bff5d2f56f97e9d72a75ddc90d4572cb6bd1c3c995870c6db79d794"}}