{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:P5LQ2YC6LH4NXFX6CBQKJNE6JQ","short_pith_number":"pith:P5LQ2YC6","canonical_record":{"source":{"id":"1305.1672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-05-07T22:43:48Z","cross_cats_sorted":[],"title_canon_sha256":"bd87f4bf55652977d8cef7f79306bdca0fbc9c1ffe2e9ba28581187c66970ad9","abstract_canon_sha256":"b3094f22ebf278b6eaca9a0e565676077b1145a84dbd729dfe90cefcbf59c408"},"schema_version":"1.0"},"canonical_sha256":"7f570d605e59f8db96fe1060a4b49e4c0ad1df317e99d8641723dfe104626445","source":{"kind":"arxiv","id":"1305.1672","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.1672","created_at":"2026-05-18T03:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1305.1672v1","created_at":"2026-05-18T03:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.1672","created_at":"2026-05-18T03:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"P5LQ2YC6LH4N","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"P5LQ2YC6LH4NXFX6","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"P5LQ2YC6","created_at":"2026-05-18T12:27:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:P5LQ2YC6LH4NXFX6CBQKJNE6JQ","target":"record","payload":{"canonical_record":{"source":{"id":"1305.1672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-05-07T22:43:48Z","cross_cats_sorted":[],"title_canon_sha256":"bd87f4bf55652977d8cef7f79306bdca0fbc9c1ffe2e9ba28581187c66970ad9","abstract_canon_sha256":"b3094f22ebf278b6eaca9a0e565676077b1145a84dbd729dfe90cefcbf59c408"},"schema_version":"1.0"},"canonical_sha256":"7f570d605e59f8db96fe1060a4b49e4c0ad1df317e99d8641723dfe104626445","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:00.175736Z","signature_b64":"iKerU1ocM/ViINDMz/pslT7ghFWNtzlsFW088u39YWGVb065fuqfRjFQ05S4xfC0VHsyhz8gbN7RFPhsLRLyCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f570d605e59f8db96fe1060a4b49e4c0ad1df317e99d8641723dfe104626445","last_reissued_at":"2026-05-18T03:21:00.174909Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:00.174909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.1672","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":"2026-05-18T03:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IK0Y+8OCKXQ7D5kGTmstbQDk8NGkFOkW2aMvYoiJiANUyGOHXeEUGRGOjIyVWbqUDQb1jIzJPz8qTcod3o4wBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T03:21:19.942649Z"},"content_sha256":"b3077b748a028a1c75fae9ac77e52e9444e36ff93e6c6f158d4da81e56617811","schema_version":"1.0","event_id":"sha256:b3077b748a028a1c75fae9ac77e52e9444e36ff93e6c6f158d4da81e56617811"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:P5LQ2YC6LH4NXFX6CBQKJNE6JQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kervaire invariants and selfcoincidences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Duane Randall, Ulrich Koschorke","submitted_at":"2013-05-07T22:43:48Z","abstract_excerpt":"Minimum numbers decide e.g. whether a given map f: S^m --> S^n/G from a sphere into a spherical space form can be deformed to a map f' such that f(x) not equal f'(x) for all x in S^m. In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In the stable dimension range these numbers coincide. But already in the first nonstable range (when m=2n-2) the Kervaire invariant appears as a decisive additional obstruction which detects interesting geometric coincidence phenomena. Similar results (involving e.g. Hopf invariants, taken mod 4) are o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1672","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":"2026-05-18T03:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0dgxiUCQgJ79vtLZCWkttBkhbZGQ1bqqLpeKkXkKSbiGxPWvlTswEV7ktHrw5BL6b/nlBHTy0/ubgunEINp3Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T03:21:19.942993Z"},"content_sha256":"f6b6bd7deb04f34c956bb01d6783438e1066f437d88747d6876924af793f1917","schema_version":"1.0","event_id":"sha256:f6b6bd7deb04f34c956bb01d6783438e1066f437d88747d6876924af793f1917"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P5LQ2YC6LH4NXFX6CBQKJNE6JQ/bundle.json","state_url":"https://pith.science/pith/P5LQ2YC6LH4NXFX6CBQKJNE6JQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P5LQ2YC6LH4NXFX6CBQKJNE6JQ/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":"2026-06-29T03:21:19Z","links":{"resolver":"https://pith.science/pith/P5LQ2YC6LH4NXFX6CBQKJNE6JQ","bundle":"https://pith.science/pith/P5LQ2YC6LH4NXFX6CBQKJNE6JQ/bundle.json","state":"https://pith.science/pith/P5LQ2YC6LH4NXFX6CBQKJNE6JQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P5LQ2YC6LH4NXFX6CBQKJNE6JQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:P5LQ2YC6LH4NXFX6CBQKJNE6JQ","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":"b3094f22ebf278b6eaca9a0e565676077b1145a84dbd729dfe90cefcbf59c408","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-05-07T22:43:48Z","title_canon_sha256":"bd87f4bf55652977d8cef7f79306bdca0fbc9c1ffe2e9ba28581187c66970ad9"},"schema_version":"1.0","source":{"id":"1305.1672","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.1672","created_at":"2026-05-18T03:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1305.1672v1","created_at":"2026-05-18T03:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.1672","created_at":"2026-05-18T03:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"P5LQ2YC6LH4N","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"P5LQ2YC6LH4NXFX6","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"P5LQ2YC6","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:f6b6bd7deb04f34c956bb01d6783438e1066f437d88747d6876924af793f1917","target":"graph","created_at":"2026-05-18T03:21:00Z","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":"Minimum numbers decide e.g. whether a given map f: S^m --> S^n/G from a sphere into a spherical space form can be deformed to a map f' such that f(x) not equal f'(x) for all x in S^m. In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In the stable dimension range these numbers coincide. But already in the first nonstable range (when m=2n-2) the Kervaire invariant appears as a decisive additional obstruction which detects interesting geometric coincidence phenomena. Similar results (involving e.g. Hopf invariants, taken mod 4) are o","authors_text":"Duane Randall, Ulrich Koschorke","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-05-07T22:43:48Z","title":"Kervaire invariants and selfcoincidences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1672","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:b3077b748a028a1c75fae9ac77e52e9444e36ff93e6c6f158d4da81e56617811","target":"record","created_at":"2026-05-18T03:21:00Z","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":"b3094f22ebf278b6eaca9a0e565676077b1145a84dbd729dfe90cefcbf59c408","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-05-07T22:43:48Z","title_canon_sha256":"bd87f4bf55652977d8cef7f79306bdca0fbc9c1ffe2e9ba28581187c66970ad9"},"schema_version":"1.0","source":{"id":"1305.1672","kind":"arxiv","version":1}},"canonical_sha256":"7f570d605e59f8db96fe1060a4b49e4c0ad1df317e99d8641723dfe104626445","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7f570d605e59f8db96fe1060a4b49e4c0ad1df317e99d8641723dfe104626445","first_computed_at":"2026-05-18T03:21:00.174909Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:00.174909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iKerU1ocM/ViINDMz/pslT7ghFWNtzlsFW088u39YWGVb065fuqfRjFQ05S4xfC0VHsyhz8gbN7RFPhsLRLyCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:00.175736Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.1672","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b3077b748a028a1c75fae9ac77e52e9444e36ff93e6c6f158d4da81e56617811","sha256:f6b6bd7deb04f34c956bb01d6783438e1066f437d88747d6876924af793f1917"],"state_sha256":"99ea934032ea7f03f4044486a2f64154be5a8aa8e9cb131e6b5e908a5b213ca4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mXEsy13ul4sF1PlyAYtiYBdDzMGlez5obpVGU6pIC0Ptz8DuZ8qTjUoiqj5EENcQ2xuADhxNIib8u7VToOAeCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T03:21:19.944810Z","bundle_sha256":"f76a584fcaa02528d54842daf6f9fbd10e2fcfd3116c8674145b981b041e782f"}}