{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:64XXDKFHHNKQHMQ4FJ7CIRIJJL","short_pith_number":"pith:64XXDKFH","canonical_record":{"source":{"id":"1008.3002","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-18T03:31:07Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"fca6d9d916a67f3f3a63a93408285cbf64c1cb5f8cd069787590a37a9ca72c29","abstract_canon_sha256":"392754130b8275f64e0dade4e0ab16f815a71b6585c985a301534c39a84d61d1"},"schema_version":"1.0"},"canonical_sha256":"f72f71a8a73b5503b21c2a7e2445094af75d8f78c6d4182fb742d145f39c4d4f","source":{"kind":"arxiv","id":"1008.3002","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3002","created_at":"2026-05-18T04:42:03Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3002v1","created_at":"2026-05-18T04:42:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3002","created_at":"2026-05-18T04:42:03Z"},{"alias_kind":"pith_short_12","alias_value":"64XXDKFHHNKQ","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"64XXDKFHHNKQHMQ4","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"64XXDKFH","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:64XXDKFHHNKQHMQ4FJ7CIRIJJL","target":"record","payload":{"canonical_record":{"source":{"id":"1008.3002","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-18T03:31:07Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"fca6d9d916a67f3f3a63a93408285cbf64c1cb5f8cd069787590a37a9ca72c29","abstract_canon_sha256":"392754130b8275f64e0dade4e0ab16f815a71b6585c985a301534c39a84d61d1"},"schema_version":"1.0"},"canonical_sha256":"f72f71a8a73b5503b21c2a7e2445094af75d8f78c6d4182fb742d145f39c4d4f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:03.752074Z","signature_b64":"9zFXZjjz5UmsrQVH6W2CngbktDrbiF7ORYdxyh/J8kbJqaWWJggaIn4ml3WSJRT9dFSAqo1lsR0FG65IkLLjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f72f71a8a73b5503b21c2a7e2445094af75d8f78c6d4182fb742d145f39c4d4f","last_reissued_at":"2026-05-18T04:42:03.751644Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:03.751644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.3002","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":"2026-05-18T04:42:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QfzKFTzbtiuBRoMm7tQ6pgFq5MI/QNer2GnJL+jV6SoUFrgEdCYQuE65rrNSn9Q1QGyck0gNtyM1tIIvJm3nCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:33:31.845781Z"},"content_sha256":"4c3ddce77e371faf2d05047eb66a243d9b737b9d7ea1f3d32cc2f14e1e5094ba","schema_version":"1.0","event_id":"sha256:4c3ddce77e371faf2d05047eb66a243d9b737b9d7ea1f3d32cc2f14e1e5094ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:64XXDKFHHNKQHMQ4FJ7CIRIJJL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Golod-Shafarevich Equality and p-Tower Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Cam McLeman","submitted_at":"2010-08-18T03:31:07Z","abstract_excerpt":"All current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod-Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can occur as Galois groups of finite p-class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod-Shafarevich type inequalities, we prove a Golod-Shafare"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3002","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":"2026-05-18T04:42:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sJSUF6Zj0yUgKrCGsYSjefOCJ6d3RKSRCOyaivcnKjZHywbOrS7g1n16U5cW6T/6VXMNyK5LXJk0vleVWTa3DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:33:31.846126Z"},"content_sha256":"47d79306b28f3d1786d02c53752aadfc3cee66726a7b4930ec01d57aa2678b98","schema_version":"1.0","event_id":"sha256:47d79306b28f3d1786d02c53752aadfc3cee66726a7b4930ec01d57aa2678b98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/64XXDKFHHNKQHMQ4FJ7CIRIJJL/bundle.json","state_url":"https://pith.science/pith/64XXDKFHHNKQHMQ4FJ7CIRIJJL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/64XXDKFHHNKQHMQ4FJ7CIRIJJL/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":"2026-06-25T08:33:31Z","links":{"resolver":"https://pith.science/pith/64XXDKFHHNKQHMQ4FJ7CIRIJJL","bundle":"https://pith.science/pith/64XXDKFHHNKQHMQ4FJ7CIRIJJL/bundle.json","state":"https://pith.science/pith/64XXDKFHHNKQHMQ4FJ7CIRIJJL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/64XXDKFHHNKQHMQ4FJ7CIRIJJL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:64XXDKFHHNKQHMQ4FJ7CIRIJJL","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":"392754130b8275f64e0dade4e0ab16f815a71b6585c985a301534c39a84d61d1","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-18T03:31:07Z","title_canon_sha256":"fca6d9d916a67f3f3a63a93408285cbf64c1cb5f8cd069787590a37a9ca72c29"},"schema_version":"1.0","source":{"id":"1008.3002","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3002","created_at":"2026-05-18T04:42:03Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3002v1","created_at":"2026-05-18T04:42:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3002","created_at":"2026-05-18T04:42:03Z"},{"alias_kind":"pith_short_12","alias_value":"64XXDKFHHNKQ","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"64XXDKFHHNKQHMQ4","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"64XXDKFH","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:47d79306b28f3d1786d02c53752aadfc3cee66726a7b4930ec01d57aa2678b98","target":"graph","created_at":"2026-05-18T04:42:03Z","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":"All current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod-Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can occur as Galois groups of finite p-class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod-Shafarevich type inequalities, we prove a Golod-Shafare","authors_text":"Cam McLeman","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-18T03:31:07Z","title":"A Golod-Shafarevich Equality and p-Tower Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3002","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":"sha256:4c3ddce77e371faf2d05047eb66a243d9b737b9d7ea1f3d32cc2f14e1e5094ba","target":"record","created_at":"2026-05-18T04:42:03Z","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":"392754130b8275f64e0dade4e0ab16f815a71b6585c985a301534c39a84d61d1","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-18T03:31:07Z","title_canon_sha256":"fca6d9d916a67f3f3a63a93408285cbf64c1cb5f8cd069787590a37a9ca72c29"},"schema_version":"1.0","source":{"id":"1008.3002","kind":"arxiv","version":1}},"canonical_sha256":"f72f71a8a73b5503b21c2a7e2445094af75d8f78c6d4182fb742d145f39c4d4f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f72f71a8a73b5503b21c2a7e2445094af75d8f78c6d4182fb742d145f39c4d4f","first_computed_at":"2026-05-18T04:42:03.751644Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:03.751644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9zFXZjjz5UmsrQVH6W2CngbktDrbiF7ORYdxyh/J8kbJqaWWJggaIn4ml3WSJRT9dFSAqo1lsR0FG65IkLLjAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:03.752074Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.3002","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c3ddce77e371faf2d05047eb66a243d9b737b9d7ea1f3d32cc2f14e1e5094ba","sha256:47d79306b28f3d1786d02c53752aadfc3cee66726a7b4930ec01d57aa2678b98"],"state_sha256":"2426fcb224b670919547aa23bef6937577d2ef2cd18b50ba67f809420237b2d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mHkcAg7L6t3fxJO53lMFmoD9cSWu0gVTmdg3ywiMEQbejfNleFXCI+Cfx51a7JH936HisusAVY1zG4ysUdQ3Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T08:33:31.847991Z","bundle_sha256":"b6987c3b7b0cdb5db567febde7fe7e5aef2d9579281ae05a56f055e752863638"}}