{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:IHPQ2CIBDBYUA5J5L3JPC7IPVC","short_pith_number":"pith:IHPQ2CIB","canonical_record":{"source":{"id":"1410.5134","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GR","submitted_at":"2014-10-20T01:22:20Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"e0fc52ea8b7aa8cbef289c0996b928675b79784a31c2705049b330f02d51b4c8","abstract_canon_sha256":"894cec91885d02029e5857d1d9f4d04d31b37f35b59588022066f9d4ed7d80a6"},"schema_version":"1.0"},"canonical_sha256":"41df0d0901187140753d5ed2f17d0fa89f0ac18430a11236e9f7da5730d149ef","source":{"kind":"arxiv","id":"1410.5134","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5134","created_at":"2026-05-18T02:05:24Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5134v2","created_at":"2026-05-18T02:05:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5134","created_at":"2026-05-18T02:05:24Z"},{"alias_kind":"pith_short_12","alias_value":"IHPQ2CIBDBYU","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IHPQ2CIBDBYUA5J5","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IHPQ2CIB","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:IHPQ2CIBDBYUA5J5L3JPC7IPVC","target":"record","payload":{"canonical_record":{"source":{"id":"1410.5134","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GR","submitted_at":"2014-10-20T01:22:20Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"e0fc52ea8b7aa8cbef289c0996b928675b79784a31c2705049b330f02d51b4c8","abstract_canon_sha256":"894cec91885d02029e5857d1d9f4d04d31b37f35b59588022066f9d4ed7d80a6"},"schema_version":"1.0"},"canonical_sha256":"41df0d0901187140753d5ed2f17d0fa89f0ac18430a11236e9f7da5730d149ef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:05:24.247794Z","signature_b64":"n4gMG5/epDkT8IXbw45Qs2fWsMf62QA9o0Vm6pvrBSU6APwt0qg9RMicDueL3QWY0ODgeHiyDW6zbWVHkIbFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41df0d0901187140753d5ed2f17d0fa89f0ac18430a11236e9f7da5730d149ef","last_reissued_at":"2026-05-18T02:05:24.247004Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:05:24.247004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.5134","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":"2026-05-18T02:05:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fkvJczLmXK3eVLGNz8HxQlreTgWygSbqP4CvVYNpmrX3+fcR8L8cc5mPXw6NN0r/BGfPqBvv51X7iyif0FyLAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:30:47.061458Z"},"content_sha256":"21f83fc6fb14349c4dcd5c0f44f826e206d10aca15ec527ab199d90696f67288","schema_version":"1.0","event_id":"sha256:21f83fc6fb14349c4dcd5c0f44f826e206d10aca15ec527ab199d90696f67288"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:IHPQ2CIBDBYUA5J5L3JPC7IPVC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A gap theorem for the ZL-amenability constant of a finite group","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GR","authors_text":"Yemon Choi","submitted_at":"2014-10-20T01:22:20Z","abstract_excerpt":"It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 2009; arxiv 0805.3685] that the ZL-amenability constant of a finite group is always at least 1, with equality if and only if the group is abelian. It was also shown in the same paper that for any finite non-abelian group this invariant is at least 301/300, but the proof relies crucially on a deep result of D. A. Rider on norms of central idempotents in group algebras.\n  Here we show that if G is finite and non-abelian then its ZL-amenability constant is at least 7/4, which is known to be best possible. We avoid use of Rider's result, by a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5134","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":"2026-05-18T02:05:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6C7hEnJ/EY7pSkMp+eT8jbUb/YQzCYRfWsn9ZMIxTMV5rrZubS14r46dGc+IG05BsWHVksLcefjjfcqLQeMxDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:30:47.061812Z"},"content_sha256":"3c8d1a1335e444765657808d20bba204e5e222e15b0d57f3b92242bf0fd9e825","schema_version":"1.0","event_id":"sha256:3c8d1a1335e444765657808d20bba204e5e222e15b0d57f3b92242bf0fd9e825"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IHPQ2CIBDBYUA5J5L3JPC7IPVC/bundle.json","state_url":"https://pith.science/pith/IHPQ2CIBDBYUA5J5L3JPC7IPVC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IHPQ2CIBDBYUA5J5L3JPC7IPVC/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":"2026-06-01T06:30:47Z","links":{"resolver":"https://pith.science/pith/IHPQ2CIBDBYUA5J5L3JPC7IPVC","bundle":"https://pith.science/pith/IHPQ2CIBDBYUA5J5L3JPC7IPVC/bundle.json","state":"https://pith.science/pith/IHPQ2CIBDBYUA5J5L3JPC7IPVC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IHPQ2CIBDBYUA5J5L3JPC7IPVC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:IHPQ2CIBDBYUA5J5L3JPC7IPVC","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":"894cec91885d02029e5857d1d9f4d04d31b37f35b59588022066f9d4ed7d80a6","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GR","submitted_at":"2014-10-20T01:22:20Z","title_canon_sha256":"e0fc52ea8b7aa8cbef289c0996b928675b79784a31c2705049b330f02d51b4c8"},"schema_version":"1.0","source":{"id":"1410.5134","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.5134","created_at":"2026-05-18T02:05:24Z"},{"alias_kind":"arxiv_version","alias_value":"1410.5134v2","created_at":"2026-05-18T02:05:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5134","created_at":"2026-05-18T02:05:24Z"},{"alias_kind":"pith_short_12","alias_value":"IHPQ2CIBDBYU","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IHPQ2CIBDBYUA5J5","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IHPQ2CIB","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:3c8d1a1335e444765657808d20bba204e5e222e15b0d57f3b92242bf0fd9e825","target":"graph","created_at":"2026-05-18T02:05:24Z","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":"It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 2009; arxiv 0805.3685] that the ZL-amenability constant of a finite group is always at least 1, with equality if and only if the group is abelian. It was also shown in the same paper that for any finite non-abelian group this invariant is at least 301/300, but the proof relies crucially on a deep result of D. A. Rider on norms of central idempotents in group algebras.\n  Here we show that if G is finite and non-abelian then its ZL-amenability constant is at least 7/4, which is known to be best possible. We avoid use of Rider's result, by a","authors_text":"Yemon Choi","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GR","submitted_at":"2014-10-20T01:22:20Z","title":"A gap theorem for the ZL-amenability constant of a finite group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5134","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":"sha256:21f83fc6fb14349c4dcd5c0f44f826e206d10aca15ec527ab199d90696f67288","target":"record","created_at":"2026-05-18T02:05:24Z","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":"894cec91885d02029e5857d1d9f4d04d31b37f35b59588022066f9d4ed7d80a6","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GR","submitted_at":"2014-10-20T01:22:20Z","title_canon_sha256":"e0fc52ea8b7aa8cbef289c0996b928675b79784a31c2705049b330f02d51b4c8"},"schema_version":"1.0","source":{"id":"1410.5134","kind":"arxiv","version":2}},"canonical_sha256":"41df0d0901187140753d5ed2f17d0fa89f0ac18430a11236e9f7da5730d149ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"41df0d0901187140753d5ed2f17d0fa89f0ac18430a11236e9f7da5730d149ef","first_computed_at":"2026-05-18T02:05:24.247004Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:05:24.247004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n4gMG5/epDkT8IXbw45Qs2fWsMf62QA9o0Vm6pvrBSU6APwt0qg9RMicDueL3QWY0ODgeHiyDW6zbWVHkIbFBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:05:24.247794Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.5134","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21f83fc6fb14349c4dcd5c0f44f826e206d10aca15ec527ab199d90696f67288","sha256:3c8d1a1335e444765657808d20bba204e5e222e15b0d57f3b92242bf0fd9e825"],"state_sha256":"d426f6cd08a014e281bc60e723f68fa79921ab4e9b6dbc79c4fc01fb18a1a46b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hV7pfmIkb4wTFRN9azHQmtmLFD1cNzyBqeO5PjaRtpQDVGeOtfBK8nTtm2ZQxIGGW7xyE+2hOuod8oCpLqLGCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T06:30:47.063883Z","bundle_sha256":"408966b342be8b2861c6485fa68b842fd7bdbb5a2a297d0f45072c83aca04619"}}