{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:J3CGKQGQXL326SWMZ6EYZPTDAN","short_pith_number":"pith:J3CGKQGQ","canonical_record":{"source":{"id":"1809.02198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-06T19:59:19Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"fdfbcdd932d6c903c675c25899a42c000a69660d147ca5e3812c78b598753822","abstract_canon_sha256":"bdf78f00a5ad52069cdba7ef0054e507f0e251591fbf9033539aa13731fa1c14"},"schema_version":"1.0"},"canonical_sha256":"4ec46540d0baf7af4acccf898cbe630363b2ddd62ed6e96cb833795609e11a6c","source":{"kind":"arxiv","id":"1809.02198","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.02198","created_at":"2026-05-18T00:06:17Z"},{"alias_kind":"arxiv_version","alias_value":"1809.02198v1","created_at":"2026-05-18T00:06:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.02198","created_at":"2026-05-18T00:06:17Z"},{"alias_kind":"pith_short_12","alias_value":"J3CGKQGQXL32","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"J3CGKQGQXL326SWM","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"J3CGKQGQ","created_at":"2026-05-18T12:32:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:J3CGKQGQXL326SWMZ6EYZPTDAN","target":"record","payload":{"canonical_record":{"source":{"id":"1809.02198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-06T19:59:19Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"fdfbcdd932d6c903c675c25899a42c000a69660d147ca5e3812c78b598753822","abstract_canon_sha256":"bdf78f00a5ad52069cdba7ef0054e507f0e251591fbf9033539aa13731fa1c14"},"schema_version":"1.0"},"canonical_sha256":"4ec46540d0baf7af4acccf898cbe630363b2ddd62ed6e96cb833795609e11a6c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:17.684284Z","signature_b64":"0FKp/Qt1YoulntIjuO1uP362Xzr0TRB3HAwsDh07xhqo3FTJr+eJ7VKkUHTKLbUORjPKD9Eyik1LOO9DXhUfCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ec46540d0baf7af4acccf898cbe630363b2ddd62ed6e96cb833795609e11a6c","last_reissued_at":"2026-05-18T00:06:17.683740Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:17.683740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.02198","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":"2026-05-18T00:06:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fN3mlLAmDWVopXjpcuBNHlNw8pzwu5qUXb6o60hhRsPL0gfQtpL0nn6PIq2nTle4OtvaLSqGhvjo600HV9/1Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T09:31:39.873921Z"},"content_sha256":"43740e7cfe62b81b8aac5274f1a06c6791febb423ec74ad28448e182bfd7adde","schema_version":"1.0","event_id":"sha256:43740e7cfe62b81b8aac5274f1a06c6791febb423ec74ad28448e182bfd7adde"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:J3CGKQGQXL326SWMZ6EYZPTDAN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"ABP inequalities for singular submanifolds of bounded mean curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Mario Santilli","submitted_at":"2018-09-06T19:59:19Z","abstract_excerpt":"Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous results by O. Savin for viscosity solutions of the minimal surface equation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02198","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":"2026-05-18T00:06:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BTtFvMGh+bKfm1+j5oNRfQ4skLZYYOOL4FXYo/5i50uJTlwhimuCeaStC+lzyJ6ytwc9LIUgRva1sSVroMdHDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T09:31:39.874288Z"},"content_sha256":"c2a2e8dc934b7a287c76fec3685a0c56fb870e92bdf039e6069c88aa4118ef78","schema_version":"1.0","event_id":"sha256:c2a2e8dc934b7a287c76fec3685a0c56fb870e92bdf039e6069c88aa4118ef78"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J3CGKQGQXL326SWMZ6EYZPTDAN/bundle.json","state_url":"https://pith.science/pith/J3CGKQGQXL326SWMZ6EYZPTDAN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J3CGKQGQXL326SWMZ6EYZPTDAN/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":"2026-06-23T09:31:39Z","links":{"resolver":"https://pith.science/pith/J3CGKQGQXL326SWMZ6EYZPTDAN","bundle":"https://pith.science/pith/J3CGKQGQXL326SWMZ6EYZPTDAN/bundle.json","state":"https://pith.science/pith/J3CGKQGQXL326SWMZ6EYZPTDAN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J3CGKQGQXL326SWMZ6EYZPTDAN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:J3CGKQGQXL326SWMZ6EYZPTDAN","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":"bdf78f00a5ad52069cdba7ef0054e507f0e251591fbf9033539aa13731fa1c14","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-06T19:59:19Z","title_canon_sha256":"fdfbcdd932d6c903c675c25899a42c000a69660d147ca5e3812c78b598753822"},"schema_version":"1.0","source":{"id":"1809.02198","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.02198","created_at":"2026-05-18T00:06:17Z"},{"alias_kind":"arxiv_version","alias_value":"1809.02198v1","created_at":"2026-05-18T00:06:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.02198","created_at":"2026-05-18T00:06:17Z"},{"alias_kind":"pith_short_12","alias_value":"J3CGKQGQXL32","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"J3CGKQGQXL326SWM","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"J3CGKQGQ","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:c2a2e8dc934b7a287c76fec3685a0c56fb870e92bdf039e6069c88aa4118ef78","target":"graph","created_at":"2026-05-18T00:06:17Z","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":"Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous results by O. Savin for viscosity solutions of the minimal surface equation.","authors_text":"Mario Santilli","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-06T19:59:19Z","title":"ABP inequalities for singular submanifolds of bounded mean curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02198","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":"sha256:43740e7cfe62b81b8aac5274f1a06c6791febb423ec74ad28448e182bfd7adde","target":"record","created_at":"2026-05-18T00:06:17Z","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":"bdf78f00a5ad52069cdba7ef0054e507f0e251591fbf9033539aa13731fa1c14","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-06T19:59:19Z","title_canon_sha256":"fdfbcdd932d6c903c675c25899a42c000a69660d147ca5e3812c78b598753822"},"schema_version":"1.0","source":{"id":"1809.02198","kind":"arxiv","version":1}},"canonical_sha256":"4ec46540d0baf7af4acccf898cbe630363b2ddd62ed6e96cb833795609e11a6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ec46540d0baf7af4acccf898cbe630363b2ddd62ed6e96cb833795609e11a6c","first_computed_at":"2026-05-18T00:06:17.683740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:17.683740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0FKp/Qt1YoulntIjuO1uP362Xzr0TRB3HAwsDh07xhqo3FTJr+eJ7VKkUHTKLbUORjPKD9Eyik1LOO9DXhUfCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:17.684284Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.02198","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43740e7cfe62b81b8aac5274f1a06c6791febb423ec74ad28448e182bfd7adde","sha256:c2a2e8dc934b7a287c76fec3685a0c56fb870e92bdf039e6069c88aa4118ef78"],"state_sha256":"b2ecf47119c7e722aef07945a355fd75ceddfb35775b4712a7b6adb760ba4777"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u9DkHvDVris0gFnNIuwQlDEkGLjWdV5B/DCUuC+C2Qm+tdJxS2VCozXkeHxcWSUJ/cm/X0MGXQ15iplgwairDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T09:31:39.876373Z","bundle_sha256":"245bd61468a7aa8f2858005e522696c65815cbde23c64616448bbc5a9b43ccc5"}}