{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:BPTJUGJXZCJSNRALMCEXT3WRUD","short_pith_number":"pith:BPTJUGJX","canonical_record":{"source":{"id":"1309.7018","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-26T19:12:32Z","cross_cats_sorted":[],"title_canon_sha256":"f0a4af4b8ad0ace8f5dfc408d5a793897c24bc34d52dfcc5bda4875278474688","abstract_canon_sha256":"cc895aa6aed6049ac8eb1f2572e8bc1ebb06a987c74b43074096c6ccfdfda03c"},"schema_version":"1.0"},"canonical_sha256":"0be69a1937c89326c40b608979eed1a0e331a99761ccc6d7a12477ccb79dbe76","source":{"kind":"arxiv","id":"1309.7018","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7018","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7018v3","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7018","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"pith_short_12","alias_value":"BPTJUGJXZCJS","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BPTJUGJXZCJSNRAL","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BPTJUGJX","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:BPTJUGJXZCJSNRALMCEXT3WRUD","target":"record","payload":{"canonical_record":{"source":{"id":"1309.7018","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-26T19:12:32Z","cross_cats_sorted":[],"title_canon_sha256":"f0a4af4b8ad0ace8f5dfc408d5a793897c24bc34d52dfcc5bda4875278474688","abstract_canon_sha256":"cc895aa6aed6049ac8eb1f2572e8bc1ebb06a987c74b43074096c6ccfdfda03c"},"schema_version":"1.0"},"canonical_sha256":"0be69a1937c89326c40b608979eed1a0e331a99761ccc6d7a12477ccb79dbe76","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:24.875689Z","signature_b64":"vAeDSDZd3t3OMWt0qCCWjGyJuanTMt4xgIP+28TxB1SW+m/yfukXFT4GglT4+OfRZmoWZ2tfhGZL7TU2BWiDDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0be69a1937c89326c40b608979eed1a0e331a99761ccc6d7a12477ccb79dbe76","last_reissued_at":"2026-05-18T01:22:24.875115Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:24.875115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.7018","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:22:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SmBlRQMAH1CEsW05yA4Z8leUYSaS23tzLgBk9ee8iuX2zzLzLXi+0LokwHPqf/a45Jp+0Yr6UyPbE1nD0EY9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T11:21:10.839615Z"},"content_sha256":"960b8bafbd0d69bbaa84bbdc637367bba6e7e1ad73d981a4b5df238779ada3b4","schema_version":"1.0","event_id":"sha256:960b8bafbd0d69bbaa84bbdc637367bba6e7e1ad73d981a4b5df238779ada3b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:BPTJUGJXZCJSNRALMCEXT3WRUD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Eulerian cube complexes and reciprocity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Richard Scott","submitted_at":"2013-09-26T19:12:32Z","abstract_excerpt":"Let $G$ be the fundamental group of a compact nonpositively curved cube complex $Y$. With respect to a basepoint $x$, one obtains an integer-valued length function on $G$ by counting the number of edges in a minimal length edge-path representing each group element. The growth series of $G$ with respect to $x$ is then defined to be the power series $G_x(t)=\\sum_g t^{|g|}$ where $|g|$ denotes the length of $g$. Using the fact that $G$ admits a suitable automatic structure, $G_x(t)$ can be shown to be a rational function. We prove that if $Y$ is a manifold of dimension $n$, then this rational fun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7018","kind":"arxiv","version":3},"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":"2026-05-18T01:22:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XRsxXaK92e5UkiBucf8D7iWinJ0lqImPtRncBVJI5qJtN2tnattBX1sFxt+UHYBQ2In6xlBMLQzWyrfxbp1+Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T11:21:10.839964Z"},"content_sha256":"083f9dacc34cdf2e6948824fba874126be0a4397c95cb917491d0226d49f1804","schema_version":"1.0","event_id":"sha256:083f9dacc34cdf2e6948824fba874126be0a4397c95cb917491d0226d49f1804"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BPTJUGJXZCJSNRALMCEXT3WRUD/bundle.json","state_url":"https://pith.science/pith/BPTJUGJXZCJSNRALMCEXT3WRUD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BPTJUGJXZCJSNRALMCEXT3WRUD/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":"2026-06-25T11:21:10Z","links":{"resolver":"https://pith.science/pith/BPTJUGJXZCJSNRALMCEXT3WRUD","bundle":"https://pith.science/pith/BPTJUGJXZCJSNRALMCEXT3WRUD/bundle.json","state":"https://pith.science/pith/BPTJUGJXZCJSNRALMCEXT3WRUD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BPTJUGJXZCJSNRALMCEXT3WRUD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:BPTJUGJXZCJSNRALMCEXT3WRUD","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":"cc895aa6aed6049ac8eb1f2572e8bc1ebb06a987c74b43074096c6ccfdfda03c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-26T19:12:32Z","title_canon_sha256":"f0a4af4b8ad0ace8f5dfc408d5a793897c24bc34d52dfcc5bda4875278474688"},"schema_version":"1.0","source":{"id":"1309.7018","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7018","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7018v3","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7018","created_at":"2026-05-18T01:22:24Z"},{"alias_kind":"pith_short_12","alias_value":"BPTJUGJXZCJS","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BPTJUGJXZCJSNRAL","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BPTJUGJX","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:083f9dacc34cdf2e6948824fba874126be0a4397c95cb917491d0226d49f1804","target":"graph","created_at":"2026-05-18T01:22:24Z","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 $G$ be the fundamental group of a compact nonpositively curved cube complex $Y$. With respect to a basepoint $x$, one obtains an integer-valued length function on $G$ by counting the number of edges in a minimal length edge-path representing each group element. The growth series of $G$ with respect to $x$ is then defined to be the power series $G_x(t)=\\sum_g t^{|g|}$ where $|g|$ denotes the length of $g$. Using the fact that $G$ admits a suitable automatic structure, $G_x(t)$ can be shown to be a rational function. We prove that if $Y$ is a manifold of dimension $n$, then this rational fun","authors_text":"Richard Scott","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-26T19:12:32Z","title":"Eulerian cube complexes and reciprocity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7018","kind":"arxiv","version":3},"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":"sha256:960b8bafbd0d69bbaa84bbdc637367bba6e7e1ad73d981a4b5df238779ada3b4","target":"record","created_at":"2026-05-18T01:22:24Z","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":"cc895aa6aed6049ac8eb1f2572e8bc1ebb06a987c74b43074096c6ccfdfda03c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-26T19:12:32Z","title_canon_sha256":"f0a4af4b8ad0ace8f5dfc408d5a793897c24bc34d52dfcc5bda4875278474688"},"schema_version":"1.0","source":{"id":"1309.7018","kind":"arxiv","version":3}},"canonical_sha256":"0be69a1937c89326c40b608979eed1a0e331a99761ccc6d7a12477ccb79dbe76","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0be69a1937c89326c40b608979eed1a0e331a99761ccc6d7a12477ccb79dbe76","first_computed_at":"2026-05-18T01:22:24.875115Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:24.875115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vAeDSDZd3t3OMWt0qCCWjGyJuanTMt4xgIP+28TxB1SW+m/yfukXFT4GglT4+OfRZmoWZ2tfhGZL7TU2BWiDDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:24.875689Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.7018","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:960b8bafbd0d69bbaa84bbdc637367bba6e7e1ad73d981a4b5df238779ada3b4","sha256:083f9dacc34cdf2e6948824fba874126be0a4397c95cb917491d0226d49f1804"],"state_sha256":"99e5841d8125d1ac6a763122408552ab0104c32019d848f6e55b1f7b410d8dfc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ERNLqkFS4SoxMfdyXXDewVh/ntqgHgRErJaq2/GYgPYsL4/w7Xzua8Z1rj6eCsfF+jVXA1hS+0YCk9B7uClJCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T11:21:10.841897Z","bundle_sha256":"9e8476d5f1234845d4a6ccaf9bd5e6ba53ac7598e121d23a7dfddaa760801c84"}}