{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7C2Q2G3KR5TNZMORP234ENF34H","short_pith_number":"pith:7C2Q2G3K","canonical_record":{"source":{"id":"2607.26830","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:21:52Z","cross_cats_sorted":["cs.NA","math.AP"],"title_canon_sha256":"3222a19922262c4de848a379fc7f1a4aba8e7d1361b48ed0b642c42af8a5709a","abstract_canon_sha256":"076b8d62a083ec3e915963892d4297fd7739d2f4d65bb8b59679a19958f4507d"},"schema_version":"1.0"},"canonical_sha256":"f8b50d1b6a8f66dcb1d17eb7c234bbe1ec6c400cdf23a222cccff2b26cab4865","source":{"kind":"arxiv","id":"2607.26830","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26830","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26830v1","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26830","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"pith_short_12","alias_value":"7C2Q2G3KR5TN","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"pith_short_16","alias_value":"7C2Q2G3KR5TNZMOR","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"pith_short_8","alias_value":"7C2Q2G3K","created_at":"2026-07-30T01:22:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7C2Q2G3KR5TNZMORP234ENF34H","target":"record","payload":{"canonical_record":{"source":{"id":"2607.26830","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:21:52Z","cross_cats_sorted":["cs.NA","math.AP"],"title_canon_sha256":"3222a19922262c4de848a379fc7f1a4aba8e7d1361b48ed0b642c42af8a5709a","abstract_canon_sha256":"076b8d62a083ec3e915963892d4297fd7739d2f4d65bb8b59679a19958f4507d"},"schema_version":"1.0"},"canonical_sha256":"f8b50d1b6a8f66dcb1d17eb7c234bbe1ec6c400cdf23a222cccff2b26cab4865","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8b50d1b6a8f66dcb1d17eb7c234bbe1ec6c400cdf23a222cccff2b26cab4865","last_reissued_at":"2026-07-30T01:22:28.005712Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:22:28.005712Z"},"source_kind":"arxiv","source_id":"2607.26830","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":"2026-07-30T01:22:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zrszpEx5GFM7tfg7LDbH/i9ZtQY5q4UMp8yvOEnLUSopw+5dEJpJ12dJd9X8QpljROHlKtoypIYMod4ggugUBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T04:05:07.920317Z"},"content_sha256":"74721f5d33079a5339781b7dbdf65a483fdd2f0d04d17733829f7e7d4ab01d12","schema_version":"1.0","event_id":"sha256:74721f5d33079a5339781b7dbdf65a483fdd2f0d04d17733829f7e7d4ab01d12"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7C2Q2G3KR5TNZMORP234ENF34H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Galerkin Approximation of the Fractional Hardy Constant","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.AP"],"primary_cat":"math.NA","authors_text":"Andreea Dima, Liviu I. Ignat","submitted_at":"2026-07-29T12:21:52Z","abstract_excerpt":"We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\\geq 1$, with fractional exponent $s\\in \\left(0,\\min\\left\\{1,\\frac{N}{2}\\right\\}\\right)$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26830","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/2607.26830/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":"2026-07-30T01:22:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n1MvJipP1ln+wW/0o/CftD7qxM/XbXES6GSVilZ3XXm0k277m6bUELboPpeBq3btBktDFTrE07tiMU/po8/nDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T04:05:07.921013Z"},"content_sha256":"af1b4f3ce0b1e594d33f76dbfc8e6e35cdce0ba860c04936969bcf2375b82433","schema_version":"1.0","event_id":"sha256:af1b4f3ce0b1e594d33f76dbfc8e6e35cdce0ba860c04936969bcf2375b82433"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7C2Q2G3KR5TNZMORP234ENF34H/bundle.json","state_url":"https://pith.science/pith/7C2Q2G3KR5TNZMORP234ENF34H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7C2Q2G3KR5TNZMORP234ENF34H/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":"2026-08-09T04:05:07Z","links":{"resolver":"https://pith.science/pith/7C2Q2G3KR5TNZMORP234ENF34H","bundle":"https://pith.science/pith/7C2Q2G3KR5TNZMORP234ENF34H/bundle.json","state":"https://pith.science/pith/7C2Q2G3KR5TNZMORP234ENF34H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7C2Q2G3KR5TNZMORP234ENF34H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7C2Q2G3KR5TNZMORP234ENF34H","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":"076b8d62a083ec3e915963892d4297fd7739d2f4d65bb8b59679a19958f4507d","cross_cats_sorted":["cs.NA","math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:21:52Z","title_canon_sha256":"3222a19922262c4de848a379fc7f1a4aba8e7d1361b48ed0b642c42af8a5709a"},"schema_version":"1.0","source":{"id":"2607.26830","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26830","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26830v1","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26830","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"pith_short_12","alias_value":"7C2Q2G3KR5TN","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"pith_short_16","alias_value":"7C2Q2G3KR5TNZMOR","created_at":"2026-07-30T01:22:28Z"},{"alias_kind":"pith_short_8","alias_value":"7C2Q2G3K","created_at":"2026-07-30T01:22:28Z"}],"graph_snapshots":[{"event_id":"sha256:af1b4f3ce0b1e594d33f76dbfc8e6e35cdce0ba860c04936969bcf2375b82433","target":"graph","created_at":"2026-07-30T01:22:28Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.26830/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\\geq 1$, with fractional exponent $s\\in \\left(0,\\min\\left\\{1,\\frac{N}{2}\\right\\}\\right)$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.","authors_text":"Andreea Dima, Liviu I. Ignat","cross_cats":["cs.NA","math.AP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:21:52Z","title":"Galerkin Approximation of the Fractional Hardy Constant"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26830","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":"sha256:74721f5d33079a5339781b7dbdf65a483fdd2f0d04d17733829f7e7d4ab01d12","target":"record","created_at":"2026-07-30T01:22:28Z","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":"076b8d62a083ec3e915963892d4297fd7739d2f4d65bb8b59679a19958f4507d","cross_cats_sorted":["cs.NA","math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:21:52Z","title_canon_sha256":"3222a19922262c4de848a379fc7f1a4aba8e7d1361b48ed0b642c42af8a5709a"},"schema_version":"1.0","source":{"id":"2607.26830","kind":"arxiv","version":1}},"canonical_sha256":"f8b50d1b6a8f66dcb1d17eb7c234bbe1ec6c400cdf23a222cccff2b26cab4865","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8b50d1b6a8f66dcb1d17eb7c234bbe1ec6c400cdf23a222cccff2b26cab4865","first_computed_at":"2026-07-30T01:22:28.005712Z","kind":"pith_receipt","last_reissued_at":"2026-07-30T01:22:28.005712Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.26830","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74721f5d33079a5339781b7dbdf65a483fdd2f0d04d17733829f7e7d4ab01d12","sha256:af1b4f3ce0b1e594d33f76dbfc8e6e35cdce0ba860c04936969bcf2375b82433"],"state_sha256":"3c68f7637e09e66f9aa6ac8470324738454e2004592a42d4e3cb979004e07ee5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Km+fegphNXZlvD/Oy5j6mtrd51ce3ZlC0qt4ojeRVc8MCijzKFz3IuKUgpsllCvt8qq72oypWttOdkTHsP6RCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T04:05:07.926798Z","bundle_sha256":"c1c5098b04673a25eb9383cc9cd03a1292efa1840d2c4945ada903b16354fb72"}}