{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:IWORR23SJY3YB4EJE5AMTD6EHC","short_pith_number":"pith:IWORR23S","canonical_record":{"source":{"id":"1510.06157","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-21T07:50:37Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"dd1e3de9b25f5d62c250ccd03746d16a0a7d531e1e5edfae4b943d498e9abd66","abstract_canon_sha256":"3cc7ad08265b038989cfd13d1aa9f212e5e7cfbc4a4c63e685e583323f2ac341"},"schema_version":"1.0"},"canonical_sha256":"459d18eb724e3780f0892740c98fc438ac43631f9ac9d12491154589bad1d229","source":{"kind":"arxiv","id":"1510.06157","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.06157","created_at":"2026-05-18T00:36:43Z"},{"alias_kind":"arxiv_version","alias_value":"1510.06157v2","created_at":"2026-05-18T00:36:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.06157","created_at":"2026-05-18T00:36:43Z"},{"alias_kind":"pith_short_12","alias_value":"IWORR23SJY3Y","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"IWORR23SJY3YB4EJ","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"IWORR23S","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:IWORR23SJY3YB4EJE5AMTD6EHC","target":"record","payload":{"canonical_record":{"source":{"id":"1510.06157","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-21T07:50:37Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"dd1e3de9b25f5d62c250ccd03746d16a0a7d531e1e5edfae4b943d498e9abd66","abstract_canon_sha256":"3cc7ad08265b038989cfd13d1aa9f212e5e7cfbc4a4c63e685e583323f2ac341"},"schema_version":"1.0"},"canonical_sha256":"459d18eb724e3780f0892740c98fc438ac43631f9ac9d12491154589bad1d229","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:43.975131Z","signature_b64":"OXm0Os8kkxjBpOo002IyPNHAft4PgAyyu9dIHORhsSlAW86d+VoBwAnfUhtIWw6dRDBC9DY29KztxBXYU2ioDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"459d18eb724e3780f0892740c98fc438ac43631f9ac9d12491154589bad1d229","last_reissued_at":"2026-05-18T00:36:43.974593Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:43.974593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.06157","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":"2026-05-18T00:36:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SqXubDq0FEgHDyuJ8DaJczZGXT8XQBAcSn9kpFUcq3tTkNhuerxYuNQORz6RYtgk/6ZX96OHNGcGg9YYHPj8BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:44:54.376071Z"},"content_sha256":"3fd61c858cd3a9f1eb3b01edc26d32796f6f9f39233349169c47c674c58c5308","schema_version":"1.0","event_id":"sha256:3fd61c858cd3a9f1eb3b01edc26d32796f6f9f39233349169c47c674c58c5308"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:IWORR23SJY3YB4EJE5AMTD6EHC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Determination of a Riemannian manifold from the distance difference functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Matti Lassas, Teemu Saksala","submitted_at":"2015-10-21T07:50:37Z","abstract_excerpt":"Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\\subset N$. For $x\\in M$ we denote by $D_x$ the distance difference function $D_x:F\\times F\\to \\mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\\in F=N\\setminus M$. We consider the inverse problem of determining the topological and the differentiable structure of the manifold $M$ and the metric $g|_M$ on it when we are given the distance difference data, that is, the set $F$, the metric $g|_F$, and the collection $\\mathcal D(M)=\\{D_x;\\ x\\in M\\}$. Moreover, we consider the embedded image "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06157","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":"2026-05-18T00:36:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/aZ3eA+cI3FGY07ZTt43qWz5zjcRyfPs7BjmpjJmMlgobD2rFu9sbiUIk5mzQceXBSASJr8jrR+rtS7QJmpqAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:44:54.376451Z"},"content_sha256":"2bebbca8448f59b14dbab2ddb0c34fe9a045739a42354a6d8a4a12a233f75d35","schema_version":"1.0","event_id":"sha256:2bebbca8448f59b14dbab2ddb0c34fe9a045739a42354a6d8a4a12a233f75d35"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IWORR23SJY3YB4EJE5AMTD6EHC/bundle.json","state_url":"https://pith.science/pith/IWORR23SJY3YB4EJE5AMTD6EHC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IWORR23SJY3YB4EJE5AMTD6EHC/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":"2026-06-26T10:44:54Z","links":{"resolver":"https://pith.science/pith/IWORR23SJY3YB4EJE5AMTD6EHC","bundle":"https://pith.science/pith/IWORR23SJY3YB4EJE5AMTD6EHC/bundle.json","state":"https://pith.science/pith/IWORR23SJY3YB4EJE5AMTD6EHC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IWORR23SJY3YB4EJE5AMTD6EHC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IWORR23SJY3YB4EJE5AMTD6EHC","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":"3cc7ad08265b038989cfd13d1aa9f212e5e7cfbc4a4c63e685e583323f2ac341","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-21T07:50:37Z","title_canon_sha256":"dd1e3de9b25f5d62c250ccd03746d16a0a7d531e1e5edfae4b943d498e9abd66"},"schema_version":"1.0","source":{"id":"1510.06157","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.06157","created_at":"2026-05-18T00:36:43Z"},{"alias_kind":"arxiv_version","alias_value":"1510.06157v2","created_at":"2026-05-18T00:36:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.06157","created_at":"2026-05-18T00:36:43Z"},{"alias_kind":"pith_short_12","alias_value":"IWORR23SJY3Y","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"IWORR23SJY3YB4EJ","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"IWORR23S","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:2bebbca8448f59b14dbab2ddb0c34fe9a045739a42354a6d8a4a12a233f75d35","target":"graph","created_at":"2026-05-18T00:36:43Z","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":"Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\\subset N$. For $x\\in M$ we denote by $D_x$ the distance difference function $D_x:F\\times F\\to \\mathbb R$, given by $D_x(z_1,z_2)=d(x,z_1)-d(x,z_2)$, $z_1,z_2\\in F=N\\setminus M$. We consider the inverse problem of determining the topological and the differentiable structure of the manifold $M$ and the metric $g|_M$ on it when we are given the distance difference data, that is, the set $F$, the metric $g|_F$, and the collection $\\mathcal D(M)=\\{D_x;\\ x\\in M\\}$. Moreover, we consider the embedded image ","authors_text":"Matti Lassas, Teemu Saksala","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-21T07:50:37Z","title":"Determination of a Riemannian manifold from the distance difference functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06157","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":"sha256:3fd61c858cd3a9f1eb3b01edc26d32796f6f9f39233349169c47c674c58c5308","target":"record","created_at":"2026-05-18T00:36:43Z","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":"3cc7ad08265b038989cfd13d1aa9f212e5e7cfbc4a4c63e685e583323f2ac341","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-21T07:50:37Z","title_canon_sha256":"dd1e3de9b25f5d62c250ccd03746d16a0a7d531e1e5edfae4b943d498e9abd66"},"schema_version":"1.0","source":{"id":"1510.06157","kind":"arxiv","version":2}},"canonical_sha256":"459d18eb724e3780f0892740c98fc438ac43631f9ac9d12491154589bad1d229","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"459d18eb724e3780f0892740c98fc438ac43631f9ac9d12491154589bad1d229","first_computed_at":"2026-05-18T00:36:43.974593Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:36:43.974593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OXm0Os8kkxjBpOo002IyPNHAft4PgAyyu9dIHORhsSlAW86d+VoBwAnfUhtIWw6dRDBC9DY29KztxBXYU2ioDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:36:43.975131Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.06157","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fd61c858cd3a9f1eb3b01edc26d32796f6f9f39233349169c47c674c58c5308","sha256:2bebbca8448f59b14dbab2ddb0c34fe9a045739a42354a6d8a4a12a233f75d35"],"state_sha256":"e05b92c8b17c4af2f552a8a9d1e991f69670e0ddda44b5800b75b1dccf53a4e7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PXtT/fz4/uYkvVbDlJGFrTTqtBkYrazZ7DCujLjSIc7irI6aJCKwWPJroIm33ohf34WT7lM9xGDMHAuIiy64DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T10:44:54.378519Z","bundle_sha256":"7da864cbb4b5b8c989ce40ced211e972a4e7c692126558c60e8f9a16c220268b"}}