{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:QKSQUXREJ6QHFLRZKR7LEM6Z77","short_pith_number":"pith:QKSQUXRE","canonical_record":{"source":{"id":"1707.09271","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-07-28T15:05:17Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"638a104625008eea890203e0d4be2d54ae82b81dea2968e5f8436d8415274c56","abstract_canon_sha256":"a7c06f662db9974a7ab2cdfba307c2252c9ad616e43cfb1905b87fa234165a02"},"schema_version":"1.0"},"canonical_sha256":"82a50a5e244fa072ae39547eb233d9ffc72d93c15f3b367b9b29587aa06be46f","source":{"kind":"arxiv","id":"1707.09271","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.09271","created_at":"2026-05-18T00:22:39Z"},{"alias_kind":"arxiv_version","alias_value":"1707.09271v2","created_at":"2026-05-18T00:22:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.09271","created_at":"2026-05-18T00:22:39Z"},{"alias_kind":"pith_short_12","alias_value":"QKSQUXREJ6QH","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"QKSQUXREJ6QHFLRZ","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"QKSQUXRE","created_at":"2026-05-18T12:31:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:QKSQUXREJ6QHFLRZKR7LEM6Z77","target":"record","payload":{"canonical_record":{"source":{"id":"1707.09271","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-07-28T15:05:17Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"638a104625008eea890203e0d4be2d54ae82b81dea2968e5f8436d8415274c56","abstract_canon_sha256":"a7c06f662db9974a7ab2cdfba307c2252c9ad616e43cfb1905b87fa234165a02"},"schema_version":"1.0"},"canonical_sha256":"82a50a5e244fa072ae39547eb233d9ffc72d93c15f3b367b9b29587aa06be46f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:39.687750Z","signature_b64":"VggEZf/Ob05YRBcjKJPh0WrlYJXmN4xKtI+Uc3pCxLgaXeA6jbQybK01e+FnbdOSoS6a+wWjTCh+l4CeWjfMBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82a50a5e244fa072ae39547eb233d9ffc72d93c15f3b367b9b29587aa06be46f","last_reissued_at":"2026-05-18T00:22:39.687292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:39.687292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.09271","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":"2026-05-18T00:22:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/VHUPQZpGP+W8bxrqiiiye+6ks8K01EuGtpQ3ILDdUVg8bvhkrZSPo9C9UtiD9/CVDZip/rrAzMoI4F+MQ4NDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:10:34.604472Z"},"content_sha256":"b82443845df0f5b2d854558d9fe8d1f0e37407d44d8480e1c04041663c3ee475","schema_version":"1.0","event_id":"sha256:b82443845df0f5b2d854558d9fe8d1f0e37407d44d8480e1c04041663c3ee475"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:QKSQUXREJ6QHFLRZKR7LEM6Z77","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Small simplicial complexes with prescribed torsion in homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.AT","authors_text":"Andrew Newman","submitted_at":"2017-07-28T15:05:17Z","abstract_excerpt":"For $d \\geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be the minimum number of vertices $n$ so that there exists a simplicial complex $X$ on $n$ vertices which has the torsion part of $H_{d - 1}(X)$ isomorphic to $G$. Here we establish an upper bound on $T_d(G)$ which matches the known lower bound up to a constant factor. That is, we prove that for every $d \\geq 2$ there exist constants $c_d$ and $C_d$ so that for any finite abelian group $c_d(\\log |G|)^{1/d} \\leq T_d(G) \\leq C_d(\\log |G|)^{1/d}.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09271","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":"2026-05-18T00:22:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v7eZsVtX5AjRnEJV0O9+Bxz7RwwckAqUc0QFxWNBZMauyHl0EnX/SC7jS9zZF6opU/RZqCyKIY462AnkQfvEAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:10:34.605175Z"},"content_sha256":"22c2108806539c19b2371db3f6fd9d830c60a206401f0b00122f354c2b1f7261","schema_version":"1.0","event_id":"sha256:22c2108806539c19b2371db3f6fd9d830c60a206401f0b00122f354c2b1f7261"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QKSQUXREJ6QHFLRZKR7LEM6Z77/bundle.json","state_url":"https://pith.science/pith/QKSQUXREJ6QHFLRZKR7LEM6Z77/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QKSQUXREJ6QHFLRZKR7LEM6Z77/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":"2026-06-11T02:10:34Z","links":{"resolver":"https://pith.science/pith/QKSQUXREJ6QHFLRZKR7LEM6Z77","bundle":"https://pith.science/pith/QKSQUXREJ6QHFLRZKR7LEM6Z77/bundle.json","state":"https://pith.science/pith/QKSQUXREJ6QHFLRZKR7LEM6Z77/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QKSQUXREJ6QHFLRZKR7LEM6Z77/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:QKSQUXREJ6QHFLRZKR7LEM6Z77","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":"a7c06f662db9974a7ab2cdfba307c2252c9ad616e43cfb1905b87fa234165a02","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-07-28T15:05:17Z","title_canon_sha256":"638a104625008eea890203e0d4be2d54ae82b81dea2968e5f8436d8415274c56"},"schema_version":"1.0","source":{"id":"1707.09271","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.09271","created_at":"2026-05-18T00:22:39Z"},{"alias_kind":"arxiv_version","alias_value":"1707.09271v2","created_at":"2026-05-18T00:22:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.09271","created_at":"2026-05-18T00:22:39Z"},{"alias_kind":"pith_short_12","alias_value":"QKSQUXREJ6QH","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"QKSQUXREJ6QHFLRZ","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"QKSQUXRE","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:22c2108806539c19b2371db3f6fd9d830c60a206401f0b00122f354c2b1f7261","target":"graph","created_at":"2026-05-18T00:22:39Z","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":"For $d \\geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be the minimum number of vertices $n$ so that there exists a simplicial complex $X$ on $n$ vertices which has the torsion part of $H_{d - 1}(X)$ isomorphic to $G$. Here we establish an upper bound on $T_d(G)$ which matches the known lower bound up to a constant factor. That is, we prove that for every $d \\geq 2$ there exist constants $c_d$ and $C_d$ so that for any finite abelian group $c_d(\\log |G|)^{1/d} \\leq T_d(G) \\leq C_d(\\log |G|)^{1/d}.$","authors_text":"Andrew Newman","cross_cats":["math.CO","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-07-28T15:05:17Z","title":"Small simplicial complexes with prescribed torsion in homology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09271","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:b82443845df0f5b2d854558d9fe8d1f0e37407d44d8480e1c04041663c3ee475","target":"record","created_at":"2026-05-18T00:22:39Z","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":"a7c06f662db9974a7ab2cdfba307c2252c9ad616e43cfb1905b87fa234165a02","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-07-28T15:05:17Z","title_canon_sha256":"638a104625008eea890203e0d4be2d54ae82b81dea2968e5f8436d8415274c56"},"schema_version":"1.0","source":{"id":"1707.09271","kind":"arxiv","version":2}},"canonical_sha256":"82a50a5e244fa072ae39547eb233d9ffc72d93c15f3b367b9b29587aa06be46f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82a50a5e244fa072ae39547eb233d9ffc72d93c15f3b367b9b29587aa06be46f","first_computed_at":"2026-05-18T00:22:39.687292Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:39.687292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VggEZf/Ob05YRBcjKJPh0WrlYJXmN4xKtI+Uc3pCxLgaXeA6jbQybK01e+FnbdOSoS6a+wWjTCh+l4CeWjfMBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:39.687750Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.09271","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b82443845df0f5b2d854558d9fe8d1f0e37407d44d8480e1c04041663c3ee475","sha256:22c2108806539c19b2371db3f6fd9d830c60a206401f0b00122f354c2b1f7261"],"state_sha256":"5877d3603546e67c16566472e72ca917a67283dd73e64d4b541c9deca3f2eb88"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vvFtC7EkFWqOjB4liXLlDQ1/vDOg36eghPB3gGOeHu8MxzygIUVzljp7+vxlJVBIANNHKBToAP/mMTgCqDW/Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T02:10:34.609070Z","bundle_sha256":"00dde1d63d590e483db9316a866c40226955c3e6aa568604e0e3b5c18b579cd9"}}