{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:THG5EX66B74GLVBB3OEJ6SCFIX","short_pith_number":"pith:THG5EX66","canonical_record":{"source":{"id":"1504.06301","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-23T19:17:50Z","cross_cats_sorted":["math.GN","math.OC"],"title_canon_sha256":"bec9ac422c7d0a459ed16a3ec239e902438ed69cb147f1b5c95ed1f39a7397b8","abstract_canon_sha256":"70ffcf1437dae7a76e2651130d4183d6310a038a753933f2e82056109b81220f"},"schema_version":"1.0"},"canonical_sha256":"99cdd25fde0ff865d421db889f484545f8ea0053051ce0733d1f88b5c1355201","source":{"kind":"arxiv","id":"1504.06301","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.06301","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"arxiv_version","alias_value":"1504.06301v4","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06301","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"pith_short_12","alias_value":"THG5EX66B74G","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"THG5EX66B74GLVBB","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"THG5EX66","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:THG5EX66B74GLVBB3OEJ6SCFIX","target":"record","payload":{"canonical_record":{"source":{"id":"1504.06301","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-23T19:17:50Z","cross_cats_sorted":["math.GN","math.OC"],"title_canon_sha256":"bec9ac422c7d0a459ed16a3ec239e902438ed69cb147f1b5c95ed1f39a7397b8","abstract_canon_sha256":"70ffcf1437dae7a76e2651130d4183d6310a038a753933f2e82056109b81220f"},"schema_version":"1.0"},"canonical_sha256":"99cdd25fde0ff865d421db889f484545f8ea0053051ce0733d1f88b5c1355201","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:12.820401Z","signature_b64":"HAXecxkSmUw+QZfrXZqMU9hrPii+nxJbetH1kGQ51LgjAHTvde/PihIa1NuENHewk16iBz9DHmemiXOC4czRDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99cdd25fde0ff865d421db889f484545f8ea0053051ce0733d1f88b5c1355201","last_reissued_at":"2026-05-18T01:17:12.819682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:12.819682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.06301","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":"2026-05-18T01:17:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Co7D6sezijuyeA6bGA2EnD6gdyPv3JdzspeMhaTXfeI7/CohJagk+AcuFNSPhNH5JO+sx1Q6++s1DAMagSi6Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:18:00.572362Z"},"content_sha256":"5e2615c098241898794127f528f0d5895c7834e42ded8131520fb94fe05e2d0b","schema_version":"1.0","event_id":"sha256:5e2615c098241898794127f528f0d5895c7834e42ded8131520fb94fe05e2d0b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:THG5EX66B74GLVBB3OEJ6SCFIX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-archimedean transportation problems and Kantorovich ultra-norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN","math.OC"],"primary_cat":"math.FA","authors_text":"Menachem Shlossberg, Michael Megrelishvili","submitted_at":"2015-04-23T19:17:50Z","abstract_excerpt":"We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field $\\mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06301","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":"2026-05-18T01:17:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uizMxS2vGf0A1AT3HQGSVyIObzg2o3BAIALPFuYKmi9jsHW+65D4N4sZLezo89K/JjB39EPFVIQjxCkUE9FvBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:18:00.572714Z"},"content_sha256":"81e5c7f7e51cb8def4f8f0cbe5061fcbf4234a458649a280760f591d50ee2372","schema_version":"1.0","event_id":"sha256:81e5c7f7e51cb8def4f8f0cbe5061fcbf4234a458649a280760f591d50ee2372"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/THG5EX66B74GLVBB3OEJ6SCFIX/bundle.json","state_url":"https://pith.science/pith/THG5EX66B74GLVBB3OEJ6SCFIX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/THG5EX66B74GLVBB3OEJ6SCFIX/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":"2026-06-22T10:18:00Z","links":{"resolver":"https://pith.science/pith/THG5EX66B74GLVBB3OEJ6SCFIX","bundle":"https://pith.science/pith/THG5EX66B74GLVBB3OEJ6SCFIX/bundle.json","state":"https://pith.science/pith/THG5EX66B74GLVBB3OEJ6SCFIX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/THG5EX66B74GLVBB3OEJ6SCFIX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:THG5EX66B74GLVBB3OEJ6SCFIX","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":"70ffcf1437dae7a76e2651130d4183d6310a038a753933f2e82056109b81220f","cross_cats_sorted":["math.GN","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-23T19:17:50Z","title_canon_sha256":"bec9ac422c7d0a459ed16a3ec239e902438ed69cb147f1b5c95ed1f39a7397b8"},"schema_version":"1.0","source":{"id":"1504.06301","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.06301","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"arxiv_version","alias_value":"1504.06301v4","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06301","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"pith_short_12","alias_value":"THG5EX66B74G","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"THG5EX66B74GLVBB","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"THG5EX66","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:81e5c7f7e51cb8def4f8f0cbe5061fcbf4234a458649a280760f591d50ee2372","target":"graph","created_at":"2026-05-18T01:17:12Z","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":"We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field $\\mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which","authors_text":"Menachem Shlossberg, Michael Megrelishvili","cross_cats":["math.GN","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-23T19:17:50Z","title":"Non-archimedean transportation problems and Kantorovich ultra-norms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06301","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":"sha256:5e2615c098241898794127f528f0d5895c7834e42ded8131520fb94fe05e2d0b","target":"record","created_at":"2026-05-18T01:17:12Z","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":"70ffcf1437dae7a76e2651130d4183d6310a038a753933f2e82056109b81220f","cross_cats_sorted":["math.GN","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-23T19:17:50Z","title_canon_sha256":"bec9ac422c7d0a459ed16a3ec239e902438ed69cb147f1b5c95ed1f39a7397b8"},"schema_version":"1.0","source":{"id":"1504.06301","kind":"arxiv","version":4}},"canonical_sha256":"99cdd25fde0ff865d421db889f484545f8ea0053051ce0733d1f88b5c1355201","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99cdd25fde0ff865d421db889f484545f8ea0053051ce0733d1f88b5c1355201","first_computed_at":"2026-05-18T01:17:12.819682Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:12.819682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HAXecxkSmUw+QZfrXZqMU9hrPii+nxJbetH1kGQ51LgjAHTvde/PihIa1NuENHewk16iBz9DHmemiXOC4czRDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:12.820401Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.06301","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e2615c098241898794127f528f0d5895c7834e42ded8131520fb94fe05e2d0b","sha256:81e5c7f7e51cb8def4f8f0cbe5061fcbf4234a458649a280760f591d50ee2372"],"state_sha256":"10fcec8172e3923d34983768fe57258477d7feb41b86bfc8331090e7907a2cbd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h1UT1gbldiN7KHJ0/CIxkNTiDZlMdn8koUdXKfKEF1XddUImDUJVc9GiEzVurywmVXzL3O9tCQZou2gVS2LQBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T10:18:00.574743Z","bundle_sha256":"c750c2c37f4666b872a51b124ab36ddee1e3de08789043d5f5dc31213e8f8f20"}}