{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:7JKCBRPOUP4JUBHAUNMBB5KVNP","short_pith_number":"pith:7JKCBRPO","canonical_record":{"source":{"id":"1505.02164","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-05-08T20:01:55Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"0de280fb2d45e94d3a8bf73c9d4c1477c5f0bcf5a6fb5e2af7651fd5365db3b9","abstract_canon_sha256":"db26c015353876e92073f5ea2c36a692d48d122ad0ab3736cb4f43e6351db252"},"schema_version":"1.0"},"canonical_sha256":"fa5420c5eea3f89a04e0a35810f5556be23fda0d6f26d436d0e011014f11851a","source":{"kind":"arxiv","id":"1505.02164","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.02164","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"arxiv_version","alias_value":"1505.02164v1","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.02164","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"pith_short_12","alias_value":"7JKCBRPOUP4J","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7JKCBRPOUP4JUBHA","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7JKCBRPO","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:7JKCBRPOUP4JUBHAUNMBB5KVNP","target":"record","payload":{"canonical_record":{"source":{"id":"1505.02164","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-05-08T20:01:55Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"0de280fb2d45e94d3a8bf73c9d4c1477c5f0bcf5a6fb5e2af7651fd5365db3b9","abstract_canon_sha256":"db26c015353876e92073f5ea2c36a692d48d122ad0ab3736cb4f43e6351db252"},"schema_version":"1.0"},"canonical_sha256":"fa5420c5eea3f89a04e0a35810f5556be23fda0d6f26d436d0e011014f11851a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:43.856156Z","signature_b64":"D4MY9SzwJcSI65RJpgjNS7vMXAwSRfwE7r6xzpQ44+wr4j59T9ikxfhYAcFXrXZ3Iz0ltwj5/E0lxTisS9qjAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa5420c5eea3f89a04e0a35810f5556be23fda0d6f26d436d0e011014f11851a","last_reissued_at":"2026-05-18T01:11:43.855832Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:43.855832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.02164","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":"2026-05-18T01:11:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/Mv+qtoHJ20A/XTN44/xVrXXAoLTFmrMhYL9XONrKiUMSOMrHUave8xwkFUtPpPbboIWaz5I7UfRolw/buXbCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T03:45:22.741699Z"},"content_sha256":"b8f4fde61e83e579d7d9f12345b91f46b0af0e7e5fbf17b81167bb2a00d8ceb6","schema_version":"1.0","event_id":"sha256:b8f4fde61e83e579d7d9f12345b91f46b0af0e7e5fbf17b81167bb2a00d8ceb6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:7JKCBRPOUP4JUBHAUNMBB5KVNP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Diastatic entropy and rigidity of hyperbolic manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DG","authors_text":"Roberto Mossa","submitted_at":"2015-05-08T20:01:55Z","abstract_excerpt":"Let $f: Y \\rightarrow X$ be a continuous map between a compact real analytic K\\\"ahler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary, when $X=Y$, we show that the minimal diastatic entropy is achieved if and only if $g$ is ho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.02164","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":"2026-05-18T01:11:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OwuJWoyfENYjbiQCA1W3BaCRdhgLiLDf+O9gMd5lVsJTfy5BLlTscbVpWNuANZcalB8uJ3OemOpQa6gzeV06DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T03:45:22.742033Z"},"content_sha256":"40f63ffe9f389e9b949ece98847a72af0b27607ed43aa0a4156b575b396d0eaf","schema_version":"1.0","event_id":"sha256:40f63ffe9f389e9b949ece98847a72af0b27607ed43aa0a4156b575b396d0eaf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7JKCBRPOUP4JUBHAUNMBB5KVNP/bundle.json","state_url":"https://pith.science/pith/7JKCBRPOUP4JUBHAUNMBB5KVNP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7JKCBRPOUP4JUBHAUNMBB5KVNP/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":"2026-06-21T03:45:22Z","links":{"resolver":"https://pith.science/pith/7JKCBRPOUP4JUBHAUNMBB5KVNP","bundle":"https://pith.science/pith/7JKCBRPOUP4JUBHAUNMBB5KVNP/bundle.json","state":"https://pith.science/pith/7JKCBRPOUP4JUBHAUNMBB5KVNP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7JKCBRPOUP4JUBHAUNMBB5KVNP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7JKCBRPOUP4JUBHAUNMBB5KVNP","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":"db26c015353876e92073f5ea2c36a692d48d122ad0ab3736cb4f43e6351db252","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-05-08T20:01:55Z","title_canon_sha256":"0de280fb2d45e94d3a8bf73c9d4c1477c5f0bcf5a6fb5e2af7651fd5365db3b9"},"schema_version":"1.0","source":{"id":"1505.02164","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.02164","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"arxiv_version","alias_value":"1505.02164v1","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.02164","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"pith_short_12","alias_value":"7JKCBRPOUP4J","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7JKCBRPOUP4JUBHA","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7JKCBRPO","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:40f63ffe9f389e9b949ece98847a72af0b27607ed43aa0a4156b575b396d0eaf","target":"graph","created_at":"2026-05-18T01:11:43Z","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 $f: Y \\rightarrow X$ be a continuous map between a compact real analytic K\\\"ahler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary, when $X=Y$, we show that the minimal diastatic entropy is achieved if and only if $g$ is ho","authors_text":"Roberto Mossa","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-05-08T20:01:55Z","title":"Diastatic entropy and rigidity of hyperbolic manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.02164","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:b8f4fde61e83e579d7d9f12345b91f46b0af0e7e5fbf17b81167bb2a00d8ceb6","target":"record","created_at":"2026-05-18T01:11:43Z","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":"db26c015353876e92073f5ea2c36a692d48d122ad0ab3736cb4f43e6351db252","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-05-08T20:01:55Z","title_canon_sha256":"0de280fb2d45e94d3a8bf73c9d4c1477c5f0bcf5a6fb5e2af7651fd5365db3b9"},"schema_version":"1.0","source":{"id":"1505.02164","kind":"arxiv","version":1}},"canonical_sha256":"fa5420c5eea3f89a04e0a35810f5556be23fda0d6f26d436d0e011014f11851a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa5420c5eea3f89a04e0a35810f5556be23fda0d6f26d436d0e011014f11851a","first_computed_at":"2026-05-18T01:11:43.855832Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:43.855832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D4MY9SzwJcSI65RJpgjNS7vMXAwSRfwE7r6xzpQ44+wr4j59T9ikxfhYAcFXrXZ3Iz0ltwj5/E0lxTisS9qjAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:43.856156Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.02164","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8f4fde61e83e579d7d9f12345b91f46b0af0e7e5fbf17b81167bb2a00d8ceb6","sha256:40f63ffe9f389e9b949ece98847a72af0b27607ed43aa0a4156b575b396d0eaf"],"state_sha256":"88e89951b9b0168972c936113f9dd929fa5cd9d8b89d5ecdb3d6b87b7ff03556"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XKdGMWvFyVreQdoPUCReK+Af6wSacBuaA9h+qT9WaFKOqAOhV9BoWI4oL/6tMr0GKbUNWdZWgtgzCDXtQW43Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T03:45:22.743862Z","bundle_sha256":"52540aeb5b7c103c1b0d73ad9c29e46e6fc461849449cdf7c79c21784db2602f"}}