{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:VYO264BSFC2IK7X7FNMZ6O6VM7","short_pith_number":"pith:VYO264BS","canonical_record":{"source":{"id":"1812.04651","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-12-11T19:07:49Z","cross_cats_sorted":[],"title_canon_sha256":"1cb954a12316d97875c001fac0c0862bf92a09e16b8fec678b5bdcc297484c2f","abstract_canon_sha256":"51969b00597449d4173c888caded8544e6764b8a1519ae121721b0b119c694ab"},"schema_version":"1.0"},"canonical_sha256":"ae1daf703228b4857eff2b599f3bd567db0dd1340ae1ee42203a13ff1e6c0315","source":{"kind":"arxiv","id":"1812.04651","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04651","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04651v1","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04651","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"pith_short_12","alias_value":"VYO264BSFC2I","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VYO264BSFC2IK7X7","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VYO264BS","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:VYO264BSFC2IK7X7FNMZ6O6VM7","target":"record","payload":{"canonical_record":{"source":{"id":"1812.04651","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-12-11T19:07:49Z","cross_cats_sorted":[],"title_canon_sha256":"1cb954a12316d97875c001fac0c0862bf92a09e16b8fec678b5bdcc297484c2f","abstract_canon_sha256":"51969b00597449d4173c888caded8544e6764b8a1519ae121721b0b119c694ab"},"schema_version":"1.0"},"canonical_sha256":"ae1daf703228b4857eff2b599f3bd567db0dd1340ae1ee42203a13ff1e6c0315","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:29.915546Z","signature_b64":"cse01t3+SqvoR0wVkYM1YJ1LgvxuJs3d/wGPrLTAd47U2qXOgfdSIFk6cnwv+8xPhdnF68z92wLGTfZ+0n3BDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae1daf703228b4857eff2b599f3bd567db0dd1340ae1ee42203a13ff1e6c0315","last_reissued_at":"2026-05-17T23:58:29.914943Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:29.914943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.04651","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":"2026-05-17T23:58:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rSv5e7EGuNk8NLYkQUABcMCbAEc0wMzYwuGzoKLvrofvjMEisceVALbIUrrlLoXEKXeTC5SX42n5woJ9oTPjCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T23:20:28.190058Z"},"content_sha256":"f366c524797e327d9f02dc9af31b4d366454a69ad89696e067fb015157e63af2","schema_version":"1.0","event_id":"sha256:f366c524797e327d9f02dc9af31b4d366454a69ad89696e067fb015157e63af2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:VYO264BSFC2IK7X7FNMZ6O6VM7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Infinitesimally small spheres and conformally invariant metrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Alexander Yu. Solynin, Stamatis Pouliasis","submitted_at":"2018-12-11T19:07:49Z","abstract_excerpt":"The modulus metric (also called the capacity metric) on a domain $D\\subset \\mathbb{R}^n$ can be defined as $\\mu_D(x,y)=\\inf\\{{\\mbox{cap}}\\,(D,\\gamma)\\}$, where ${\\mbox{cap}}\\,(D,\\gamma)$ stands for the capacity of the condenser $(D,\\gamma)$ and the infimum is taken over all continua $\\gamma\\subset D$ containing the points $x$ and $y$. It was conjectured by J. Ferrand, G. Martin and M. Vuorinen in 1991 that every isometry in the modulus metric is a conformal mapping. In this note, we confirm this conjecture and prove new geometric properties of surfaces that are spheres in the metric space $(D,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04651","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":"2026-05-17T23:58:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8v5KATXxZEcIllvdE2LJAcnl8zhMP5x7y2UFHTntuMLY0NUnswJtZIuYapNL+sKH9DlYVIvEAjNYWRbsraZrDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T23:20:28.190412Z"},"content_sha256":"fde0edf32c698492479f207e8ec8847be4f67f2c50199a0592ded19cad2fe97f","schema_version":"1.0","event_id":"sha256:fde0edf32c698492479f207e8ec8847be4f67f2c50199a0592ded19cad2fe97f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VYO264BSFC2IK7X7FNMZ6O6VM7/bundle.json","state_url":"https://pith.science/pith/VYO264BSFC2IK7X7FNMZ6O6VM7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VYO264BSFC2IK7X7FNMZ6O6VM7/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":"2026-06-23T23:20:28Z","links":{"resolver":"https://pith.science/pith/VYO264BSFC2IK7X7FNMZ6O6VM7","bundle":"https://pith.science/pith/VYO264BSFC2IK7X7FNMZ6O6VM7/bundle.json","state":"https://pith.science/pith/VYO264BSFC2IK7X7FNMZ6O6VM7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VYO264BSFC2IK7X7FNMZ6O6VM7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VYO264BSFC2IK7X7FNMZ6O6VM7","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":"51969b00597449d4173c888caded8544e6764b8a1519ae121721b0b119c694ab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-12-11T19:07:49Z","title_canon_sha256":"1cb954a12316d97875c001fac0c0862bf92a09e16b8fec678b5bdcc297484c2f"},"schema_version":"1.0","source":{"id":"1812.04651","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04651","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04651v1","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04651","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"pith_short_12","alias_value":"VYO264BSFC2I","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VYO264BSFC2IK7X7","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VYO264BS","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:fde0edf32c698492479f207e8ec8847be4f67f2c50199a0592ded19cad2fe97f","target":"graph","created_at":"2026-05-17T23:58:29Z","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 modulus metric (also called the capacity metric) on a domain $D\\subset \\mathbb{R}^n$ can be defined as $\\mu_D(x,y)=\\inf\\{{\\mbox{cap}}\\,(D,\\gamma)\\}$, where ${\\mbox{cap}}\\,(D,\\gamma)$ stands for the capacity of the condenser $(D,\\gamma)$ and the infimum is taken over all continua $\\gamma\\subset D$ containing the points $x$ and $y$. It was conjectured by J. Ferrand, G. Martin and M. Vuorinen in 1991 that every isometry in the modulus metric is a conformal mapping. In this note, we confirm this conjecture and prove new geometric properties of surfaces that are spheres in the metric space $(D,","authors_text":"Alexander Yu. Solynin, Stamatis Pouliasis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-12-11T19:07:49Z","title":"Infinitesimally small spheres and conformally invariant metrics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04651","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:f366c524797e327d9f02dc9af31b4d366454a69ad89696e067fb015157e63af2","target":"record","created_at":"2026-05-17T23:58:29Z","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":"51969b00597449d4173c888caded8544e6764b8a1519ae121721b0b119c694ab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-12-11T19:07:49Z","title_canon_sha256":"1cb954a12316d97875c001fac0c0862bf92a09e16b8fec678b5bdcc297484c2f"},"schema_version":"1.0","source":{"id":"1812.04651","kind":"arxiv","version":1}},"canonical_sha256":"ae1daf703228b4857eff2b599f3bd567db0dd1340ae1ee42203a13ff1e6c0315","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ae1daf703228b4857eff2b599f3bd567db0dd1340ae1ee42203a13ff1e6c0315","first_computed_at":"2026-05-17T23:58:29.914943Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:29.914943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cse01t3+SqvoR0wVkYM1YJ1LgvxuJs3d/wGPrLTAd47U2qXOgfdSIFk6cnwv+8xPhdnF68z92wLGTfZ+0n3BDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:29.915546Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.04651","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f366c524797e327d9f02dc9af31b4d366454a69ad89696e067fb015157e63af2","sha256:fde0edf32c698492479f207e8ec8847be4f67f2c50199a0592ded19cad2fe97f"],"state_sha256":"bb3f285b45988d5c11dbddbd8722a2f5f3a70a41878516248fc85b07241069d4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"08+8gYR5PoAm5qmot1k5mfUNfiPH6HFjeBK7yGzNwryO61LnNlWcQos4ewLi+aOkgqR3CFbsQGEcJ25Whp4zBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T23:20:28.192330Z","bundle_sha256":"27f3fe2d0ae5bb967cc6972cdf3091c024b9e5b838e0ebb12cfe7e1e35bd3934"}}