{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:AWEM635IK3Q2ZZDR3ROML4ZGBV","short_pith_number":"pith:AWEM635I","canonical_record":{"source":{"id":"math/0509546","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2005-09-23T10:54:10Z","cross_cats_sorted":[],"title_canon_sha256":"510aab6c57abd10d0b6e60f1c8862c01652ccae095cafc9aa8972b21bae59ea5","abstract_canon_sha256":"550d1dc979fd8df3c81a866100669dd3fd52eb80d53cb8c71daf013dd013189b"},"schema_version":"1.0"},"canonical_sha256":"0588cf6fa856e1ace471dc5cc5f3260d4fbf35c4cdbd2877ed2407d9e2831763","source":{"kind":"arxiv","id":"math/0509546","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0509546","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/0509546v1","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0509546","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"pith_short_12","alias_value":"AWEM635IK3Q2","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"AWEM635IK3Q2ZZDR","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"AWEM635I","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:AWEM635IK3Q2ZZDR3ROML4ZGBV","target":"record","payload":{"canonical_record":{"source":{"id":"math/0509546","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2005-09-23T10:54:10Z","cross_cats_sorted":[],"title_canon_sha256":"510aab6c57abd10d0b6e60f1c8862c01652ccae095cafc9aa8972b21bae59ea5","abstract_canon_sha256":"550d1dc979fd8df3c81a866100669dd3fd52eb80d53cb8c71daf013dd013189b"},"schema_version":"1.0"},"canonical_sha256":"0588cf6fa856e1ace471dc5cc5f3260d4fbf35c4cdbd2877ed2407d9e2831763","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:50.903682Z","signature_b64":"MXa8j8QRA6/vFp8HMPPIxTYsa20C6GVHACqzfBVTHOXBbf1d8+TGZzItwMQT48+lG5G+4UNL3k4pct96JbyFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0588cf6fa856e1ace471dc5cc5f3260d4fbf35c4cdbd2877ed2407d9e2831763","last_reissued_at":"2026-05-18T01:08:50.903179Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:50.903179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0509546","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"URmfe+4szvxmDoaqmE333/81NdDMGT45n66wcHgewZbB5nqyiLtCbSTyVLaAQtXziR4Fgcebtk4urOuMXib5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:19:38.005523Z"},"content_sha256":"302f41595f29b652181cc8bf076c567969e3386ca63153ed5f683bed4dac659b","schema_version":"1.0","event_id":"sha256:302f41595f29b652181cc8bf076c567969e3386ca63153ed5f683bed4dac659b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:AWEM635IK3Q2ZZDR3ROML4ZGBV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exponential growth of Lie algebras of finite global dimension","license":"","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Jean-Claude Thomas (LAREMA), Steve Halperin, Yves F\\'elix","submitted_at":"2005-09-23T10:54:10Z","abstract_excerpt":"Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H\\_*(\\Omega X;\\mathbb Q)$ is the universal enveloping algebra of a graded Lie algebra $L\\_X$ isomorphic with $ pi\\_{*-1} (X)\\otimes \\mathbb Q$. Let $Q\\_X \\subset L\\_X$ be a minimal generating subspace, and set $\\alpha = \\limsup\\_i \\frac{\\log{\\scriptsize rk} \\pi\\_i(X)}{i}$. Theorem: If ${dim} L\\_X = \\infty$ and $\\limsup ({dim} (Q\\_X)\\_k)^{1/k} < \\limsup ({dim} (L\\_X)\\_k)^{1/k}$ then $$\\sum\\_{i=1}^{n-1} {rk} \\pi\\_{k+i}(X) = e^{(\\alpha + \\epsilon\\_k)k} \\hspace{1cm} {where} \\epsilon\\_k \\to 0 {as} k\\to \\infty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0509546","kind":"arxiv","version":1},"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":"2026-05-18T01:08:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RqSqcWoeAs/ftN8vy4nczzGSxOjA92H7kCOVAuoYAyqy8NfDwFkwzcLl3KuA8BrS32Zj24DTeJqOAYWQy5FlBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:19:38.005887Z"},"content_sha256":"29de789158f2b9f66efcb8bacf3289451c3f6114ffc252081e0e0bc392c1653b","schema_version":"1.0","event_id":"sha256:29de789158f2b9f66efcb8bacf3289451c3f6114ffc252081e0e0bc392c1653b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AWEM635IK3Q2ZZDR3ROML4ZGBV/bundle.json","state_url":"https://pith.science/pith/AWEM635IK3Q2ZZDR3ROML4ZGBV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AWEM635IK3Q2ZZDR3ROML4ZGBV/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":"2026-06-02T15:19:38Z","links":{"resolver":"https://pith.science/pith/AWEM635IK3Q2ZZDR3ROML4ZGBV","bundle":"https://pith.science/pith/AWEM635IK3Q2ZZDR3ROML4ZGBV/bundle.json","state":"https://pith.science/pith/AWEM635IK3Q2ZZDR3ROML4ZGBV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AWEM635IK3Q2ZZDR3ROML4ZGBV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:AWEM635IK3Q2ZZDR3ROML4ZGBV","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":"550d1dc979fd8df3c81a866100669dd3fd52eb80d53cb8c71daf013dd013189b","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2005-09-23T10:54:10Z","title_canon_sha256":"510aab6c57abd10d0b6e60f1c8862c01652ccae095cafc9aa8972b21bae59ea5"},"schema_version":"1.0","source":{"id":"math/0509546","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0509546","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/0509546v1","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0509546","created_at":"2026-05-18T01:08:50Z"},{"alias_kind":"pith_short_12","alias_value":"AWEM635IK3Q2","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"AWEM635IK3Q2ZZDR","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"AWEM635I","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:29de789158f2b9f66efcb8bacf3289451c3f6114ffc252081e0e0bc392c1653b","target":"graph","created_at":"2026-05-18T01:08:50Z","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":"Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H\\_*(\\Omega X;\\mathbb Q)$ is the universal enveloping algebra of a graded Lie algebra $L\\_X$ isomorphic with $ pi\\_{*-1} (X)\\otimes \\mathbb Q$. Let $Q\\_X \\subset L\\_X$ be a minimal generating subspace, and set $\\alpha = \\limsup\\_i \\frac{\\log{\\scriptsize rk} \\pi\\_i(X)}{i}$. Theorem: If ${dim} L\\_X = \\infty$ and $\\limsup ({dim} (Q\\_X)\\_k)^{1/k} < \\limsup ({dim} (L\\_X)\\_k)^{1/k}$ then $$\\sum\\_{i=1}^{n-1} {rk} \\pi\\_{k+i}(X) = e^{(\\alpha + \\epsilon\\_k)k} \\hspace{1cm} {where} \\epsilon\\_k \\to 0 {as} k\\to \\infty","authors_text":"Jean-Claude Thomas (LAREMA), Steve Halperin, Yves F\\'elix","cross_cats":[],"headline":"","license":"","primary_cat":"math.AT","submitted_at":"2005-09-23T10:54:10Z","title":"Exponential growth of Lie algebras of finite global dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0509546","kind":"arxiv","version":1},"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":"sha256:302f41595f29b652181cc8bf076c567969e3386ca63153ed5f683bed4dac659b","target":"record","created_at":"2026-05-18T01:08:50Z","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":"550d1dc979fd8df3c81a866100669dd3fd52eb80d53cb8c71daf013dd013189b","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2005-09-23T10:54:10Z","title_canon_sha256":"510aab6c57abd10d0b6e60f1c8862c01652ccae095cafc9aa8972b21bae59ea5"},"schema_version":"1.0","source":{"id":"math/0509546","kind":"arxiv","version":1}},"canonical_sha256":"0588cf6fa856e1ace471dc5cc5f3260d4fbf35c4cdbd2877ed2407d9e2831763","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0588cf6fa856e1ace471dc5cc5f3260d4fbf35c4cdbd2877ed2407d9e2831763","first_computed_at":"2026-05-18T01:08:50.903179Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:50.903179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MXa8j8QRA6/vFp8HMPPIxTYsa20C6GVHACqzfBVTHOXBbf1d8+TGZzItwMQT48+lG5G+4UNL3k4pct96JbyFBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:50.903682Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0509546","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:302f41595f29b652181cc8bf076c567969e3386ca63153ed5f683bed4dac659b","sha256:29de789158f2b9f66efcb8bacf3289451c3f6114ffc252081e0e0bc392c1653b"],"state_sha256":"da59db9d68f39cf2b83d007dc25dc8cd6e83ec827c56adca8022a162ba70ce08"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hdUjwRSU6J69oBz66wUMOzXZp4nSoKNpzb5Lh3mG7kUYqnMPpG9pxBD/+R2bi+ynOypI4TeB7kRd4PUXjQfWCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T15:19:38.007915Z","bundle_sha256":"b336582128d67e69674aae8f6d79cd310b8c00426373eac85633b4d2440cad6d"}}