{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:RY7HANTQTNFC72A6BDXGRLRQIM","short_pith_number":"pith:RY7HANTQ","canonical_record":{"source":{"id":"1506.07969","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-26T07:19:27Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1fa030b882e852de765daff2fceb3150922bb0bb2ab92ea0a1142874d7db821f","abstract_canon_sha256":"7c092a2719d8b06a2a7d2bd6d044383fc367dbb54172e9db5cf357002ada5927"},"schema_version":"1.0"},"canonical_sha256":"8e3e7036709b4a2fe81e08ee68ae30432fcc88232d8dbb69be44df7ccd0af66d","source":{"kind":"arxiv","id":"1506.07969","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.07969","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"arxiv_version","alias_value":"1506.07969v2","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.07969","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"pith_short_12","alias_value":"RY7HANTQTNFC","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RY7HANTQTNFC72A6","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RY7HANTQ","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:RY7HANTQTNFC72A6BDXGRLRQIM","target":"record","payload":{"canonical_record":{"source":{"id":"1506.07969","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-26T07:19:27Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1fa030b882e852de765daff2fceb3150922bb0bb2ab92ea0a1142874d7db821f","abstract_canon_sha256":"7c092a2719d8b06a2a7d2bd6d044383fc367dbb54172e9db5cf357002ada5927"},"schema_version":"1.0"},"canonical_sha256":"8e3e7036709b4a2fe81e08ee68ae30432fcc88232d8dbb69be44df7ccd0af66d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:28.770607Z","signature_b64":"K8SU1x6883TGlPk75GBZbRyNPNqx+uhkcJl7oJkkLulZSm8TKYvRn1Q96KbBqJuZnNVi/2Jwq42k246bHybBDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e3e7036709b4a2fe81e08ee68ae30432fcc88232d8dbb69be44df7ccd0af66d","last_reissued_at":"2026-05-18T00:55:28.769920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:28.769920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.07969","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":"2026-05-18T00:55:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bh8TGLelChlGaWVzDnVmnrwVkqvtBolk+3XMnaaKwKZz0XVB+mYDRRA2UZqAijaJ8znjkOFLQZmLFSl99avrDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:14:05.125175Z"},"content_sha256":"a800ae17386ee28134e01f60cc862503fb6b2576ed258c7561a9447f2c04e599","schema_version":"1.0","event_id":"sha256:a800ae17386ee28134e01f60cc862503fb6b2576ed258c7561a9447f2c04e599"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:RY7HANTQTNFC72A6BDXGRLRQIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized approach to the non-backtracking lace expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Remco van der Hofstad, Robert Fitzner","submitted_at":"2015-06-26T07:19:27Z","abstract_excerpt":"The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the $\\mathbb Z^d$-lattice for percolation, lattice trees and lattice animals.\n  The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.07969","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":"2026-05-18T00:55:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b4BXAjEEqFIvRrCLc2vxXJaLglX+OlhrEfOkVh8yE7PRvbAaPFyBo5oHC+pG02VY3VmPmm+8pNH2ZhePfVWHDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:14:05.125546Z"},"content_sha256":"c58b02d94a5fa5a53b70479fe0dd3d92ad054dfc0888e01075a4217680828ee0","schema_version":"1.0","event_id":"sha256:c58b02d94a5fa5a53b70479fe0dd3d92ad054dfc0888e01075a4217680828ee0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RY7HANTQTNFC72A6BDXGRLRQIM/bundle.json","state_url":"https://pith.science/pith/RY7HANTQTNFC72A6BDXGRLRQIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RY7HANTQTNFC72A6BDXGRLRQIM/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":"2026-06-20T12:14:05Z","links":{"resolver":"https://pith.science/pith/RY7HANTQTNFC72A6BDXGRLRQIM","bundle":"https://pith.science/pith/RY7HANTQTNFC72A6BDXGRLRQIM/bundle.json","state":"https://pith.science/pith/RY7HANTQTNFC72A6BDXGRLRQIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RY7HANTQTNFC72A6BDXGRLRQIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RY7HANTQTNFC72A6BDXGRLRQIM","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":"7c092a2719d8b06a2a7d2bd6d044383fc367dbb54172e9db5cf357002ada5927","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-26T07:19:27Z","title_canon_sha256":"1fa030b882e852de765daff2fceb3150922bb0bb2ab92ea0a1142874d7db821f"},"schema_version":"1.0","source":{"id":"1506.07969","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.07969","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"arxiv_version","alias_value":"1506.07969v2","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.07969","created_at":"2026-05-18T00:55:28Z"},{"alias_kind":"pith_short_12","alias_value":"RY7HANTQTNFC","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RY7HANTQTNFC72A6","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RY7HANTQ","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:c58b02d94a5fa5a53b70479fe0dd3d92ad054dfc0888e01075a4217680828ee0","target":"graph","created_at":"2026-05-18T00:55:28Z","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 lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the $\\mathbb Z^d$-lattice for percolation, lattice trees and lattice animals.\n  The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the i","authors_text":"Remco van der Hofstad, Robert Fitzner","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-26T07:19:27Z","title":"Generalized approach to the non-backtracking lace expansion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.07969","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:a800ae17386ee28134e01f60cc862503fb6b2576ed258c7561a9447f2c04e599","target":"record","created_at":"2026-05-18T00:55:28Z","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":"7c092a2719d8b06a2a7d2bd6d044383fc367dbb54172e9db5cf357002ada5927","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-26T07:19:27Z","title_canon_sha256":"1fa030b882e852de765daff2fceb3150922bb0bb2ab92ea0a1142874d7db821f"},"schema_version":"1.0","source":{"id":"1506.07969","kind":"arxiv","version":2}},"canonical_sha256":"8e3e7036709b4a2fe81e08ee68ae30432fcc88232d8dbb69be44df7ccd0af66d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e3e7036709b4a2fe81e08ee68ae30432fcc88232d8dbb69be44df7ccd0af66d","first_computed_at":"2026-05-18T00:55:28.769920Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:28.769920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K8SU1x6883TGlPk75GBZbRyNPNqx+uhkcJl7oJkkLulZSm8TKYvRn1Q96KbBqJuZnNVi/2Jwq42k246bHybBDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:28.770607Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.07969","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a800ae17386ee28134e01f60cc862503fb6b2576ed258c7561a9447f2c04e599","sha256:c58b02d94a5fa5a53b70479fe0dd3d92ad054dfc0888e01075a4217680828ee0"],"state_sha256":"1b700ffa64f72a77d881ab8b66cbe828c205b6b5dd435eb2b782d97a6fa70d19"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z4m58hQl7o5Gv0EWzB1yMbscyGkw0IkAzGhe6rEKRLya7NzUiQTD9wrv/poO8L1/cbYq7zaigEekfEwxt5kkDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T12:14:05.127560Z","bundle_sha256":"5382e37975edd1e57ac05347c7efc860c4722a94c4a119c14fbcbc15617e6633"}}