{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:WJZLLSLDMHZXSSDQFITRXYA7A6","short_pith_number":"pith:WJZLLSLD","canonical_record":{"source":{"id":"2509.08754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-09-10T16:43:26Z","cross_cats_sorted":["math.FA","math.OA"],"title_canon_sha256":"baf60b237a6770fbca2d92ee26c0755cb57999327338c92cfc7b25a3f0ecbeb8","abstract_canon_sha256":"cf01ad91a79a5625d2a1ccf987bbed46ee60e169f2ec4dab2b506a85df879cca"},"schema_version":"1.0"},"canonical_sha256":"b272b5c96361f37948702a271be01f078cec4a9b18719d2ccc496b73a059901a","source":{"kind":"arxiv","id":"2509.08754","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08754","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08754v1","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08754","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"pith_short_12","alias_value":"WJZLLSLDMHZX","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"pith_short_16","alias_value":"WJZLLSLDMHZXSSDQ","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"pith_short_8","alias_value":"WJZLLSLD","created_at":"2026-07-05T12:08:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:WJZLLSLDMHZXSSDQFITRXYA7A6","target":"record","payload":{"canonical_record":{"source":{"id":"2509.08754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-09-10T16:43:26Z","cross_cats_sorted":["math.FA","math.OA"],"title_canon_sha256":"baf60b237a6770fbca2d92ee26c0755cb57999327338c92cfc7b25a3f0ecbeb8","abstract_canon_sha256":"cf01ad91a79a5625d2a1ccf987bbed46ee60e169f2ec4dab2b506a85df879cca"},"schema_version":"1.0"},"canonical_sha256":"b272b5c96361f37948702a271be01f078cec4a9b18719d2ccc496b73a059901a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:08:35.478006Z","signature_b64":"Ux0I7xx32qGJEPCaqeO6r131JcFNF6XPOHsiJj6Fy+nd3ljE/Pagta+vIz0WwJs2x4VubgFDQO/tbjv+jYHrBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b272b5c96361f37948702a271be01f078cec4a9b18719d2ccc496b73a059901a","last_reissued_at":"2026-07-05T12:08:35.477390Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:08:35.477390Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.08754","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":"2026-07-05T12:08:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nf8RIwpnRuOt4ycJN1nvF+WWiJ9v1M98C8Nc3NtWfLehSDODrEaQNBcRNanSW2xG2uABucrJvOzxmEy8QMTRDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T13:30:12.841626Z"},"content_sha256":"34182659bbf9277f6ee97cccd1b9046e4344f3cb9c7b3ef7598be53a0a00a951","schema_version":"1.0","event_id":"sha256:34182659bbf9277f6ee97cccd1b9046e4344f3cb9c7b3ef7598be53a0a00a951"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:WJZLLSLDMHZXSSDQFITRXYA7A6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some remarks on decay in countable groups and amalgamated free products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OA"],"primary_cat":"math.GR","authors_text":"Gregory Patchell, Lizzy Teryoshin, Srivatsav Kunnawalkam Elayavalli","submitted_at":"2025-09-10T16:43:26Z","abstract_excerpt":"In this note, we first study the notion of subexponential decay (SD) for countable groups with respect to a length function, which generalizes the well-known rapid decay (RD) property, first discovered by Haagerup in 1979. Several natural properties and examples are studied, especially including groups that have SD, but not RD. This consideration naturally has applications in $C^*$-algebras. We also consider in this setting a permanence theorem for decay in amalgamated free products (proved also recently by Chatterji--Gautero), and demonstrate that it is in a precise sense optimal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08754","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.08754/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T12:08:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0wOEc3OCujVKjKEaonJyMIcEOJhYZVFYbdkUx3iYZg+Sr85G07QOfYmv9057kjEL0qRE+Dt6A3lbX+wQWVFyCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T13:30:12.842146Z"},"content_sha256":"d2b0d1b73c799cf8467e2260f844e9ecbc5880eef59544155d832637f30b3fbf","schema_version":"1.0","event_id":"sha256:d2b0d1b73c799cf8467e2260f844e9ecbc5880eef59544155d832637f30b3fbf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WJZLLSLDMHZXSSDQFITRXYA7A6/bundle.json","state_url":"https://pith.science/pith/WJZLLSLDMHZXSSDQFITRXYA7A6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WJZLLSLDMHZXSSDQFITRXYA7A6/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":"2026-08-08T13:30:12Z","links":{"resolver":"https://pith.science/pith/WJZLLSLDMHZXSSDQFITRXYA7A6","bundle":"https://pith.science/pith/WJZLLSLDMHZXSSDQFITRXYA7A6/bundle.json","state":"https://pith.science/pith/WJZLLSLDMHZXSSDQFITRXYA7A6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WJZLLSLDMHZXSSDQFITRXYA7A6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:WJZLLSLDMHZXSSDQFITRXYA7A6","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":"cf01ad91a79a5625d2a1ccf987bbed46ee60e169f2ec4dab2b506a85df879cca","cross_cats_sorted":["math.FA","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-09-10T16:43:26Z","title_canon_sha256":"baf60b237a6770fbca2d92ee26c0755cb57999327338c92cfc7b25a3f0ecbeb8"},"schema_version":"1.0","source":{"id":"2509.08754","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08754","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08754v1","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08754","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"pith_short_12","alias_value":"WJZLLSLDMHZX","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"pith_short_16","alias_value":"WJZLLSLDMHZXSSDQ","created_at":"2026-07-05T12:08:35Z"},{"alias_kind":"pith_short_8","alias_value":"WJZLLSLD","created_at":"2026-07-05T12:08:35Z"}],"graph_snapshots":[{"event_id":"sha256:d2b0d1b73c799cf8467e2260f844e9ecbc5880eef59544155d832637f30b3fbf","target":"graph","created_at":"2026-07-05T12:08:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2509.08754/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note, we first study the notion of subexponential decay (SD) for countable groups with respect to a length function, which generalizes the well-known rapid decay (RD) property, first discovered by Haagerup in 1979. Several natural properties and examples are studied, especially including groups that have SD, but not RD. This consideration naturally has applications in $C^*$-algebras. We also consider in this setting a permanence theorem for decay in amalgamated free products (proved also recently by Chatterji--Gautero), and demonstrate that it is in a precise sense optimal.","authors_text":"Gregory Patchell, Lizzy Teryoshin, Srivatsav Kunnawalkam Elayavalli","cross_cats":["math.FA","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-09-10T16:43:26Z","title":"Some remarks on decay in countable groups and amalgamated free products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08754","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":"sha256:34182659bbf9277f6ee97cccd1b9046e4344f3cb9c7b3ef7598be53a0a00a951","target":"record","created_at":"2026-07-05T12:08:35Z","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":"cf01ad91a79a5625d2a1ccf987bbed46ee60e169f2ec4dab2b506a85df879cca","cross_cats_sorted":["math.FA","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-09-10T16:43:26Z","title_canon_sha256":"baf60b237a6770fbca2d92ee26c0755cb57999327338c92cfc7b25a3f0ecbeb8"},"schema_version":"1.0","source":{"id":"2509.08754","kind":"arxiv","version":1}},"canonical_sha256":"b272b5c96361f37948702a271be01f078cec4a9b18719d2ccc496b73a059901a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b272b5c96361f37948702a271be01f078cec4a9b18719d2ccc496b73a059901a","first_computed_at":"2026-07-05T12:08:35.477390Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:08:35.477390Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ux0I7xx32qGJEPCaqeO6r131JcFNF6XPOHsiJj6Fy+nd3ljE/Pagta+vIz0WwJs2x4VubgFDQO/tbjv+jYHrBw==","signature_status":"signed_v1","signed_at":"2026-07-05T12:08:35.478006Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.08754","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34182659bbf9277f6ee97cccd1b9046e4344f3cb9c7b3ef7598be53a0a00a951","sha256:d2b0d1b73c799cf8467e2260f844e9ecbc5880eef59544155d832637f30b3fbf"],"state_sha256":"64cc543f610563a3b0bcde2016081788b271a6bc2444d1f6c1d5d19ba379f184"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vnPW6Ao56ryhR4veA9Q+bVO6/5i9zLdlPsxhyI4m/pJIQyAjr9UBAqg73pfdjOQ3/yuF1pMpl9VQGWtmnUdpAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T13:30:12.847024Z","bundle_sha256":"862666ae9fd9ce5aa11b6e0bf4b521e317db7b619344fc91a3e08ce740f4ad8d"}}