{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:WAI352XCWUBAEMR7VYOBMEWFKJ","short_pith_number":"pith:WAI352XC","canonical_record":{"source":{"id":"1707.01026","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-04T14:59:34Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"9a5b72eaaec79e54faf670d4e8f582af76f40e8bdacce39624018cc76573d312","abstract_canon_sha256":"ec201e539e2e56f988d6fb5de75989655c18c5857fabf38f0b847f186a654891"},"schema_version":"1.0"},"canonical_sha256":"b011beeae2b50202323fae1c1612c5524580ba9498b206a401d0cb1ae622c082","source":{"kind":"arxiv","id":"1707.01026","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.01026","created_at":"2026-05-18T00:38:50Z"},{"alias_kind":"arxiv_version","alias_value":"1707.01026v2","created_at":"2026-05-18T00:38:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.01026","created_at":"2026-05-18T00:38:50Z"},{"alias_kind":"pith_short_12","alias_value":"WAI352XCWUBA","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WAI352XCWUBAEMR7","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WAI352XC","created_at":"2026-05-18T12:31:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:WAI352XCWUBAEMR7VYOBMEWFKJ","target":"record","payload":{"canonical_record":{"source":{"id":"1707.01026","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-04T14:59:34Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"9a5b72eaaec79e54faf670d4e8f582af76f40e8bdacce39624018cc76573d312","abstract_canon_sha256":"ec201e539e2e56f988d6fb5de75989655c18c5857fabf38f0b847f186a654891"},"schema_version":"1.0"},"canonical_sha256":"b011beeae2b50202323fae1c1612c5524580ba9498b206a401d0cb1ae622c082","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:50.212894Z","signature_b64":"+DvQTVHUmpMyY0SAke4kBXg6C97NNyFHZzpsw/kbHcwq82vd+ztYjZdqNMRNkwSxLMcoHlVzhIW49mXIC79rBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b011beeae2b50202323fae1c1612c5524580ba9498b206a401d0cb1ae622c082","last_reissued_at":"2026-05-18T00:38:50.212199Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:50.212199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.01026","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":"2026-05-18T00:38:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4RyK/zEFVMk+UcNYVwjm3j2QStFl26Vdo7X607BBJ5oEKt5Kn9kAPt8pBcsaxEWD4OvJFOSlYdzznsuvHd7qAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T12:41:41.096172Z"},"content_sha256":"7413a6dc7971d280dde2b2a15f3492b7fe83504218e0f2d0386156363f2915dc","schema_version":"1.0","event_id":"sha256:7413a6dc7971d280dde2b2a15f3492b7fe83504218e0f2d0386156363f2915dc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:WAI352XCWUBAEMR7VYOBMEWFKJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polyhedra and parameter spaces for matroids over valuation rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Alex Fink, Luca Moci","submitted_at":"2017-07-04T14:59:34Z","abstract_excerpt":"In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a parameter space for matroids over a valuation ring, which turns out to be a tropical version of the Bott-Samelson varieties for the full flag variety. Thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a.k.a. valuated matroids."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01026","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":"2026-05-18T00:38:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yn0yzUslKv9/nc21F0BoORlTAyl7Cp3ahHnCjKmHzJ1p3bBF+oxFXKKhM7FR6IV2GyBy/peHixJislCcQkw2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T12:41:41.096884Z"},"content_sha256":"eaa760e54017b6a6669ad00944e7a4ccc30c26f3811a10fad531598e8eb28935","schema_version":"1.0","event_id":"sha256:eaa760e54017b6a6669ad00944e7a4ccc30c26f3811a10fad531598e8eb28935"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WAI352XCWUBAEMR7VYOBMEWFKJ/bundle.json","state_url":"https://pith.science/pith/WAI352XCWUBAEMR7VYOBMEWFKJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WAI352XCWUBAEMR7VYOBMEWFKJ/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":"2026-05-25T12:41:41Z","links":{"resolver":"https://pith.science/pith/WAI352XCWUBAEMR7VYOBMEWFKJ","bundle":"https://pith.science/pith/WAI352XCWUBAEMR7VYOBMEWFKJ/bundle.json","state":"https://pith.science/pith/WAI352XCWUBAEMR7VYOBMEWFKJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WAI352XCWUBAEMR7VYOBMEWFKJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:WAI352XCWUBAEMR7VYOBMEWFKJ","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":"ec201e539e2e56f988d6fb5de75989655c18c5857fabf38f0b847f186a654891","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-04T14:59:34Z","title_canon_sha256":"9a5b72eaaec79e54faf670d4e8f582af76f40e8bdacce39624018cc76573d312"},"schema_version":"1.0","source":{"id":"1707.01026","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.01026","created_at":"2026-05-18T00:38:50Z"},{"alias_kind":"arxiv_version","alias_value":"1707.01026v2","created_at":"2026-05-18T00:38:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.01026","created_at":"2026-05-18T00:38:50Z"},{"alias_kind":"pith_short_12","alias_value":"WAI352XCWUBA","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WAI352XCWUBAEMR7","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WAI352XC","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:eaa760e54017b6a6669ad00944e7a4ccc30c26f3811a10fad531598e8eb28935","target":"graph","created_at":"2026-05-18T00:38:50Z","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":"In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a parameter space for matroids over a valuation ring, which turns out to be a tropical version of the Bott-Samelson varieties for the full flag variety. Thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a.k.a. valuated matroids.","authors_text":"Alex Fink, Luca Moci","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-04T14:59:34Z","title":"Polyhedra and parameter spaces for matroids over valuation rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01026","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":"sha256:7413a6dc7971d280dde2b2a15f3492b7fe83504218e0f2d0386156363f2915dc","target":"record","created_at":"2026-05-18T00:38:50Z","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":"ec201e539e2e56f988d6fb5de75989655c18c5857fabf38f0b847f186a654891","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-04T14:59:34Z","title_canon_sha256":"9a5b72eaaec79e54faf670d4e8f582af76f40e8bdacce39624018cc76573d312"},"schema_version":"1.0","source":{"id":"1707.01026","kind":"arxiv","version":2}},"canonical_sha256":"b011beeae2b50202323fae1c1612c5524580ba9498b206a401d0cb1ae622c082","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b011beeae2b50202323fae1c1612c5524580ba9498b206a401d0cb1ae622c082","first_computed_at":"2026-05-18T00:38:50.212199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:50.212199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+DvQTVHUmpMyY0SAke4kBXg6C97NNyFHZzpsw/kbHcwq82vd+ztYjZdqNMRNkwSxLMcoHlVzhIW49mXIC79rBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:50.212894Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.01026","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7413a6dc7971d280dde2b2a15f3492b7fe83504218e0f2d0386156363f2915dc","sha256:eaa760e54017b6a6669ad00944e7a4ccc30c26f3811a10fad531598e8eb28935"],"state_sha256":"8decdb43671d856a3f11f4ff125006f1024bf679db1b0468bb8b841a82300db1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GSbS5HPQcYy8YCioD27KtxANP1gqTNlYUH95du/QsA5gk2oIK0/v//rOJ/GAHo7Ju2Le5wmv8HzU4sPs/L5bAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T12:41:41.100494Z","bundle_sha256":"08457743d9bffeced86ff07d159fbf5dc154f4fc6b10510b1be19aa666461d48"}}