{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:2CV5GGVRAJQHK62NFFYDNJP4AI","short_pith_number":"pith:2CV5GGVR","canonical_record":{"source":{"id":"2404.07276","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-04-10T18:09:15Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a07afab723cd4a8a99f8b909984cb6afd6b3675e6c3c4f884fc438ada09ca511","abstract_canon_sha256":"d5141a6f70f73e7d7b0a306528df004cbd31cb4a3fa1cd92e87e897dce9f4391"},"schema_version":"1.0"},"canonical_sha256":"d0abd31ab10260757b4d297036a5fc0209a016a194dfbfbcce8c3df829a81b0d","source":{"kind":"arxiv","id":"2404.07276","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.07276","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"arxiv_version","alias_value":"2404.07276v1","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.07276","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"pith_short_12","alias_value":"2CV5GGVRAJQH","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"pith_short_16","alias_value":"2CV5GGVRAJQHK62N","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"pith_short_8","alias_value":"2CV5GGVR","created_at":"2026-07-05T08:06:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:2CV5GGVRAJQHK62NFFYDNJP4AI","target":"record","payload":{"canonical_record":{"source":{"id":"2404.07276","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-04-10T18:09:15Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a07afab723cd4a8a99f8b909984cb6afd6b3675e6c3c4f884fc438ada09ca511","abstract_canon_sha256":"d5141a6f70f73e7d7b0a306528df004cbd31cb4a3fa1cd92e87e897dce9f4391"},"schema_version":"1.0"},"canonical_sha256":"d0abd31ab10260757b4d297036a5fc0209a016a194dfbfbcce8c3df829a81b0d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:06:50.160465Z","signature_b64":"pew9UXExRbXDzaMsNEz/I5Sk+QOGoUzGgczeouYPxbrdktKjOY/50mm6wJ6p+QCfHCuNLFtyz77c5xtDinvMBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d0abd31ab10260757b4d297036a5fc0209a016a194dfbfbcce8c3df829a81b0d","last_reissued_at":"2026-07-05T08:06:50.159989Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:06:50.159989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.07276","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":"2026-07-05T08:06:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IAVr617m42fCGax80gEFVYed6g6eoQZnNSsaEoica+BktGdIizyVWVnUzZEyIVFyg3Xl3Ypc3YoKlbLzMkmYCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T16:44:02.251766Z"},"content_sha256":"798f9f422a6d89c21798ac71ab2c295ade0f360a4732fa81582001fe631ab04e","schema_version":"1.0","event_id":"sha256:798f9f422a6d89c21798ac71ab2c295ade0f360a4732fa81582001fe631ab04e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:2CV5GGVRAJQHK62NFFYDNJP4AI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pointwise two-point function estimates and a non-pertubative proof of mean-field critical behaviour for long-range percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Tom Hutchcroft","submitted_at":"2024-04-10T18:09:15Z","abstract_excerpt":"In long-range percolation on $\\mathbb{Z}^d$, we connect each pair of distinct points $x$ and $y$ by an edge independently at random with probability $1-\\exp(-\\beta\\|x-y\\|^{-d-\\alpha})$, where $\\alpha>0$ is fixed and $\\beta\\geq 0$ is a parameter. In a previous paper, we proved that if $0<\\alpha<d$ then the critical two-point function satisfies the spatially averaged upper bound \\[ \\frac{1}{r^d}\\sum_{x\\in [-r,r]^d} \\mathbb{P}_{\\beta_c}(0\\leftrightarrow x) \\preceq r^{-d+\\alpha} \\] for every $r\\geq 1$. This upper bound is believed to be sharp for values of $\\alpha$ strictly below the crossover val"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.07276","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.07276/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":"2026-07-05T08:06:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fmJu+Tfpec3g3XyoMnw4mMjoTxvTQpzdXY9ygZQMNgVtVBLsNmePkfbBJkPcrWsT3VpFGLqdFYxP50aNJtbjBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T16:44:02.252758Z"},"content_sha256":"62057a312ab52f7097df50b253645f7c34c9b7b8cea8b8711245134ccc197621","schema_version":"1.0","event_id":"sha256:62057a312ab52f7097df50b253645f7c34c9b7b8cea8b8711245134ccc197621"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2CV5GGVRAJQHK62NFFYDNJP4AI/bundle.json","state_url":"https://pith.science/pith/2CV5GGVRAJQHK62NFFYDNJP4AI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2CV5GGVRAJQHK62NFFYDNJP4AI/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":"2026-08-14T16:44:02Z","links":{"resolver":"https://pith.science/pith/2CV5GGVRAJQHK62NFFYDNJP4AI","bundle":"https://pith.science/pith/2CV5GGVRAJQHK62NFFYDNJP4AI/bundle.json","state":"https://pith.science/pith/2CV5GGVRAJQHK62NFFYDNJP4AI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2CV5GGVRAJQHK62NFFYDNJP4AI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:2CV5GGVRAJQHK62NFFYDNJP4AI","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":"d5141a6f70f73e7d7b0a306528df004cbd31cb4a3fa1cd92e87e897dce9f4391","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-04-10T18:09:15Z","title_canon_sha256":"a07afab723cd4a8a99f8b909984cb6afd6b3675e6c3c4f884fc438ada09ca511"},"schema_version":"1.0","source":{"id":"2404.07276","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.07276","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"arxiv_version","alias_value":"2404.07276v1","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.07276","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"pith_short_12","alias_value":"2CV5GGVRAJQH","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"pith_short_16","alias_value":"2CV5GGVRAJQHK62N","created_at":"2026-07-05T08:06:50Z"},{"alias_kind":"pith_short_8","alias_value":"2CV5GGVR","created_at":"2026-07-05T08:06:50Z"}],"graph_snapshots":[{"event_id":"sha256:62057a312ab52f7097df50b253645f7c34c9b7b8cea8b8711245134ccc197621","target":"graph","created_at":"2026-07-05T08:06:50Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2404.07276/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In long-range percolation on $\\mathbb{Z}^d$, we connect each pair of distinct points $x$ and $y$ by an edge independently at random with probability $1-\\exp(-\\beta\\|x-y\\|^{-d-\\alpha})$, where $\\alpha>0$ is fixed and $\\beta\\geq 0$ is a parameter. In a previous paper, we proved that if $0<\\alpha<d$ then the critical two-point function satisfies the spatially averaged upper bound \\[ \\frac{1}{r^d}\\sum_{x\\in [-r,r]^d} \\mathbb{P}_{\\beta_c}(0\\leftrightarrow x) \\preceq r^{-d+\\alpha} \\] for every $r\\geq 1$. This upper bound is believed to be sharp for values of $\\alpha$ strictly below the crossover val","authors_text":"Tom Hutchcroft","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-04-10T18:09:15Z","title":"Pointwise two-point function estimates and a non-pertubative proof of mean-field critical behaviour for long-range percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.07276","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":"sha256:798f9f422a6d89c21798ac71ab2c295ade0f360a4732fa81582001fe631ab04e","target":"record","created_at":"2026-07-05T08:06:50Z","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":"d5141a6f70f73e7d7b0a306528df004cbd31cb4a3fa1cd92e87e897dce9f4391","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-04-10T18:09:15Z","title_canon_sha256":"a07afab723cd4a8a99f8b909984cb6afd6b3675e6c3c4f884fc438ada09ca511"},"schema_version":"1.0","source":{"id":"2404.07276","kind":"arxiv","version":1}},"canonical_sha256":"d0abd31ab10260757b4d297036a5fc0209a016a194dfbfbcce8c3df829a81b0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d0abd31ab10260757b4d297036a5fc0209a016a194dfbfbcce8c3df829a81b0d","first_computed_at":"2026-07-05T08:06:50.159989Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:06:50.159989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pew9UXExRbXDzaMsNEz/I5Sk+QOGoUzGgczeouYPxbrdktKjOY/50mm6wJ6p+QCfHCuNLFtyz77c5xtDinvMBA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:06:50.160465Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.07276","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:798f9f422a6d89c21798ac71ab2c295ade0f360a4732fa81582001fe631ab04e","sha256:62057a312ab52f7097df50b253645f7c34c9b7b8cea8b8711245134ccc197621"],"state_sha256":"7445d6925ca0265b3433f403471e0b1b0f089e75979f8eea990c2c401e06b4fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QaGNilhABhspw2wyHsMom769BEFmIvvJxTHZ508RN68afO3pkb6d7r6tkV3g9epUerYu1xmm2BZ0zM+0+B2fDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T16:44:02.260438Z","bundle_sha256":"17873f1014aa127c5a7156d20efd0275a542470b6e3b05f823f79f5961fcbcf0"}}