{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:O257SOOK4MQTPDICBABYJGOKAR","short_pith_number":"pith:O257SOOK","canonical_record":{"source":{"id":"1610.04886","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-10-16T16:46:38Z","cross_cats_sorted":[],"title_canon_sha256":"cd52def2784411bab9e0cc6e8d1792b87bdae146e0ea2e604f3360291aa043ae","abstract_canon_sha256":"1b012ef0e33aa4d49e372f3f1b9667ac23a34c8488e96475cda6d4f7ea5b7d34"},"schema_version":"1.0"},"canonical_sha256":"76bbf939cae321378d0208038499ca046f60dd142aeb3669942aae5817f6a562","source":{"kind":"arxiv","id":"1610.04886","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04886","created_at":"2026-05-18T00:52:25Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04886v4","created_at":"2026-05-18T00:52:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04886","created_at":"2026-05-18T00:52:25Z"},{"alias_kind":"pith_short_12","alias_value":"O257SOOK4MQT","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"O257SOOK4MQTPDIC","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"O257SOOK","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:O257SOOK4MQTPDICBABYJGOKAR","target":"record","payload":{"canonical_record":{"source":{"id":"1610.04886","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-10-16T16:46:38Z","cross_cats_sorted":[],"title_canon_sha256":"cd52def2784411bab9e0cc6e8d1792b87bdae146e0ea2e604f3360291aa043ae","abstract_canon_sha256":"1b012ef0e33aa4d49e372f3f1b9667ac23a34c8488e96475cda6d4f7ea5b7d34"},"schema_version":"1.0"},"canonical_sha256":"76bbf939cae321378d0208038499ca046f60dd142aeb3669942aae5817f6a562","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:25.293041Z","signature_b64":"kpUxK+6XHpeIKADr4jJS1HO9jyKel4Ysf0js/IiAnzXsmhvfbbNIvqzmD7TS8njHTzxihFdJP5IjfPeDlJ7ZDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76bbf939cae321378d0208038499ca046f60dd142aeb3669942aae5817f6a562","last_reissued_at":"2026-05-18T00:52:25.292387Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:25.292387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.04886","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:52:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E82SzLNB4j6SAoBObHaZaJNYd832cbSTGqazwQ20BCdMPxWlHG+H1WhPFPN3d6ejDF/BEfVfcWXyzZPmageGDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T12:45:43.575432Z"},"content_sha256":"b5b32a65c090478e3ea0dfbdd54665ef828bd2d545021121a6539e46be85dd63","schema_version":"1.0","event_id":"sha256:b5b32a65c090478e3ea0dfbdd54665ef828bd2d545021121a6539e46be85dd63"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:O257SOOK4MQTPDICBABYJGOKAR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Markov Type constants, flat tori and Wasserstein spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Vladimir Zolotov","submitted_at":"2016-10-16T16:46:38Z","abstract_excerpt":"Let $M_p(X,T)$ denote the Markov type $p$ constant at time $T$ of a metric space $X$, where $p \\ge 1$. We show that $M_p(Y,T) \\le M_p(X,T)$ in each of the following cases: (a)$X$ and $Y$ are geodesic spaces and $Y$ is covered by $X$ via a finite-sheeted locally isometric covering, (b)$Y$ is the quotient of $X$ by a finite group of isometries, (c) $Y$ is the $L^p$-Wasserstein space over $X$.\n  As an application of (a) we show that all compact flat manifolds have Markov type $2$ with constant $1$. In particular the circle with its intrinsic metric has Markov type $2$ with constant $1$. This answ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04886","kind":"arxiv","version":4},"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":"2026-05-18T00:52:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"26MB3N9RpKqJ4Omy6BMZusG5xjI8aXIYK2LjAitobRVTD5RyZ8jB/h7ywmzo5CgKgNhst8dSCdKhVs8Xhr0AAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T12:45:43.575793Z"},"content_sha256":"220bc774560be6283c4599b96ce6ed1ad487e19d047ea41a84c132f8b3133f78","schema_version":"1.0","event_id":"sha256:220bc774560be6283c4599b96ce6ed1ad487e19d047ea41a84c132f8b3133f78"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O257SOOK4MQTPDICBABYJGOKAR/bundle.json","state_url":"https://pith.science/pith/O257SOOK4MQTPDICBABYJGOKAR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O257SOOK4MQTPDICBABYJGOKAR/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":"2026-06-26T12:45:43Z","links":{"resolver":"https://pith.science/pith/O257SOOK4MQTPDICBABYJGOKAR","bundle":"https://pith.science/pith/O257SOOK4MQTPDICBABYJGOKAR/bundle.json","state":"https://pith.science/pith/O257SOOK4MQTPDICBABYJGOKAR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O257SOOK4MQTPDICBABYJGOKAR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:O257SOOK4MQTPDICBABYJGOKAR","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":"1b012ef0e33aa4d49e372f3f1b9667ac23a34c8488e96475cda6d4f7ea5b7d34","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-10-16T16:46:38Z","title_canon_sha256":"cd52def2784411bab9e0cc6e8d1792b87bdae146e0ea2e604f3360291aa043ae"},"schema_version":"1.0","source":{"id":"1610.04886","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04886","created_at":"2026-05-18T00:52:25Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04886v4","created_at":"2026-05-18T00:52:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04886","created_at":"2026-05-18T00:52:25Z"},{"alias_kind":"pith_short_12","alias_value":"O257SOOK4MQT","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"O257SOOK4MQTPDIC","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"O257SOOK","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:220bc774560be6283c4599b96ce6ed1ad487e19d047ea41a84c132f8b3133f78","target":"graph","created_at":"2026-05-18T00:52:25Z","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 $M_p(X,T)$ denote the Markov type $p$ constant at time $T$ of a metric space $X$, where $p \\ge 1$. We show that $M_p(Y,T) \\le M_p(X,T)$ in each of the following cases: (a)$X$ and $Y$ are geodesic spaces and $Y$ is covered by $X$ via a finite-sheeted locally isometric covering, (b)$Y$ is the quotient of $X$ by a finite group of isometries, (c) $Y$ is the $L^p$-Wasserstein space over $X$.\n  As an application of (a) we show that all compact flat manifolds have Markov type $2$ with constant $1$. In particular the circle with its intrinsic metric has Markov type $2$ with constant $1$. This answ","authors_text":"Vladimir Zolotov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-10-16T16:46:38Z","title":"Markov Type constants, flat tori and Wasserstein spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04886","kind":"arxiv","version":4},"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:b5b32a65c090478e3ea0dfbdd54665ef828bd2d545021121a6539e46be85dd63","target":"record","created_at":"2026-05-18T00:52:25Z","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":"1b012ef0e33aa4d49e372f3f1b9667ac23a34c8488e96475cda6d4f7ea5b7d34","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-10-16T16:46:38Z","title_canon_sha256":"cd52def2784411bab9e0cc6e8d1792b87bdae146e0ea2e604f3360291aa043ae"},"schema_version":"1.0","source":{"id":"1610.04886","kind":"arxiv","version":4}},"canonical_sha256":"76bbf939cae321378d0208038499ca046f60dd142aeb3669942aae5817f6a562","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76bbf939cae321378d0208038499ca046f60dd142aeb3669942aae5817f6a562","first_computed_at":"2026-05-18T00:52:25.292387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:52:25.292387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kpUxK+6XHpeIKADr4jJS1HO9jyKel4Ysf0js/IiAnzXsmhvfbbNIvqzmD7TS8njHTzxihFdJP5IjfPeDlJ7ZDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:52:25.293041Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.04886","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b5b32a65c090478e3ea0dfbdd54665ef828bd2d545021121a6539e46be85dd63","sha256:220bc774560be6283c4599b96ce6ed1ad487e19d047ea41a84c132f8b3133f78"],"state_sha256":"735509b0bf3417604c8d767943e415751daa2a5fbbceada28f22bbb2003ac1e7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kHezJ1LdxK5SGDkG3A1PGwG7EZABjKUHDkA6G8zHTdc2qP6Hazuhx5A6avkDf3Hy1UnW/w/rINZp1PBAlnQ6DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T12:45:43.577684Z","bundle_sha256":"1ab1cf8b96657dcfe0c287a59c18372d28d59f7ee75a4dc342aa67d096a45824"}}