{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:JKP53EQWHDITOQF5FQGUMNVGMY","short_pith_number":"pith:JKP53EQW","canonical_record":{"source":{"id":"math/0606155","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RT","submitted_at":"2006-06-07T14:05:45Z","cross_cats_sorted":["math.GR","math.GT","math.OA"],"title_canon_sha256":"46114ba2e7e008f3e59aacdce5dd7ad67ffc3a951e337b3e2081fd05db4722e4","abstract_canon_sha256":"1238127d3d2ab75603ba803538f168ff32143223ae3f99d46118a37ce4b402c5"},"schema_version":"1.0"},"canonical_sha256":"4a9fdd921638d13740bd2c0d4636a66639648e626dad03e0ff93cc439ca20324","source":{"kind":"arxiv","id":"math/0606155","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0606155","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0606155v1","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0606155","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"pith_short_12","alias_value":"JKP53EQWHDIT","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"JKP53EQWHDITOQF5","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"JKP53EQW","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:JKP53EQWHDITOQF5FQGUMNVGMY","target":"record","payload":{"canonical_record":{"source":{"id":"math/0606155","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RT","submitted_at":"2006-06-07T14:05:45Z","cross_cats_sorted":["math.GR","math.GT","math.OA"],"title_canon_sha256":"46114ba2e7e008f3e59aacdce5dd7ad67ffc3a951e337b3e2081fd05db4722e4","abstract_canon_sha256":"1238127d3d2ab75603ba803538f168ff32143223ae3f99d46118a37ce4b402c5"},"schema_version":"1.0"},"canonical_sha256":"4a9fdd921638d13740bd2c0d4636a66639648e626dad03e0ff93cc439ca20324","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:22.665535Z","signature_b64":"zcw37zlo79Qlh9EKAmFMwtEqRNd1ZvUlQU9RVWmDFdWTKi0h8jZwYFYtonf/AQiA4YgzUB/tYSdWYLU/9LgEAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a9fdd921638d13740bd2c0d4636a66639648e626dad03e0ff93cc439ca20324","last_reissued_at":"2026-05-18T01:05:22.665017Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:22.665017Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0606155","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":"2026-05-18T01:05:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iVSaL7EsJHs5yCLeIthiTg1YCJ1u2DCyUgGo6rGR5eOY0/uqKB3nPwJr6Hcpl72C5Izd47Njg3K602ECros9BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T06:16:48.080372Z"},"content_sha256":"6521269e7ddd1fa7276e5b3a46ed4045b7b3e79d58f060b0b74899fed660597b","schema_version":"1.0","event_id":"sha256:6521269e7ddd1fa7276e5b3a46ed4045b7b3e79d58f060b0b74899fed660597b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:JKP53EQWHDITOQF5FQGUMNVGMY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A twisted Burnside theorem for countable groups and Reidemeister numbers","license":"","headline":"","cross_cats":["math.GR","math.GT","math.OA"],"primary_cat":"math.RT","authors_text":"Alexander Fel'shtyn, Evgenij Troitsky","submitted_at":"2006-06-07T14:05:45Z","abstract_excerpt":"The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem.\n Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other.\n This conjecture plays an important role in the theory of twisted conjugacy classes and has very important consequences in D"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606155","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":"2026-05-18T01:05:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5PDhqEIY4dsjAM+lejZj3euJ6KY1wBrogiMfBo98Y8OT9rlC0Cwa1qnE/y0rZ5Jh/J/E2IVxEazd1tyjnnzHBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T06:16:48.081066Z"},"content_sha256":"38b94222e3149b881aa728f6ca7650f0eedbfb7253738de8c984679c9a7feb6e","schema_version":"1.0","event_id":"sha256:38b94222e3149b881aa728f6ca7650f0eedbfb7253738de8c984679c9a7feb6e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JKP53EQWHDITOQF5FQGUMNVGMY/bundle.json","state_url":"https://pith.science/pith/JKP53EQWHDITOQF5FQGUMNVGMY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JKP53EQWHDITOQF5FQGUMNVGMY/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":"2026-06-08T06:16:48Z","links":{"resolver":"https://pith.science/pith/JKP53EQWHDITOQF5FQGUMNVGMY","bundle":"https://pith.science/pith/JKP53EQWHDITOQF5FQGUMNVGMY/bundle.json","state":"https://pith.science/pith/JKP53EQWHDITOQF5FQGUMNVGMY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JKP53EQWHDITOQF5FQGUMNVGMY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:JKP53EQWHDITOQF5FQGUMNVGMY","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":"1238127d3d2ab75603ba803538f168ff32143223ae3f99d46118a37ce4b402c5","cross_cats_sorted":["math.GR","math.GT","math.OA"],"license":"","primary_cat":"math.RT","submitted_at":"2006-06-07T14:05:45Z","title_canon_sha256":"46114ba2e7e008f3e59aacdce5dd7ad67ffc3a951e337b3e2081fd05db4722e4"},"schema_version":"1.0","source":{"id":"math/0606155","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0606155","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0606155v1","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0606155","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"pith_short_12","alias_value":"JKP53EQWHDIT","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"JKP53EQWHDITOQF5","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"JKP53EQW","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:38b94222e3149b881aa728f6ca7650f0eedbfb7253738de8c984679c9a7feb6e","target":"graph","created_at":"2026-05-18T01:05:22Z","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":"The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem.\n Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other.\n This conjecture plays an important role in the theory of twisted conjugacy classes and has very important consequences in D","authors_text":"Alexander Fel'shtyn, Evgenij Troitsky","cross_cats":["math.GR","math.GT","math.OA"],"headline":"","license":"","primary_cat":"math.RT","submitted_at":"2006-06-07T14:05:45Z","title":"A twisted Burnside theorem for countable groups and Reidemeister numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606155","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":"sha256:6521269e7ddd1fa7276e5b3a46ed4045b7b3e79d58f060b0b74899fed660597b","target":"record","created_at":"2026-05-18T01:05:22Z","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":"1238127d3d2ab75603ba803538f168ff32143223ae3f99d46118a37ce4b402c5","cross_cats_sorted":["math.GR","math.GT","math.OA"],"license":"","primary_cat":"math.RT","submitted_at":"2006-06-07T14:05:45Z","title_canon_sha256":"46114ba2e7e008f3e59aacdce5dd7ad67ffc3a951e337b3e2081fd05db4722e4"},"schema_version":"1.0","source":{"id":"math/0606155","kind":"arxiv","version":1}},"canonical_sha256":"4a9fdd921638d13740bd2c0d4636a66639648e626dad03e0ff93cc439ca20324","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a9fdd921638d13740bd2c0d4636a66639648e626dad03e0ff93cc439ca20324","first_computed_at":"2026-05-18T01:05:22.665017Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:22.665017Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zcw37zlo79Qlh9EKAmFMwtEqRNd1ZvUlQU9RVWmDFdWTKi0h8jZwYFYtonf/AQiA4YgzUB/tYSdWYLU/9LgEAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:22.665535Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0606155","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6521269e7ddd1fa7276e5b3a46ed4045b7b3e79d58f060b0b74899fed660597b","sha256:38b94222e3149b881aa728f6ca7650f0eedbfb7253738de8c984679c9a7feb6e"],"state_sha256":"77f2dd03d3ee3240b9926a4a4cb98253aed97c6bd82f555279163d2c8e827bf0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VbhQWs0tPV/jTrV4EQEP9NTK2WTlbCcX+/7bRrEZCjMpwqnUhKGoseDgfhnfIN5bxbIi2kYjBwpTeKcvN6RrDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T06:16:48.085488Z","bundle_sha256":"df5e86247fb7a9dcc434c24cb4c053c846092354361cf7eb2be59914105f9195"}}