{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:5WARSR6TU2B6LRKDSQFCG5GBTI","short_pith_number":"pith:5WARSR6T","canonical_record":{"source":{"id":"1109.0046","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-08-31T22:23:50Z","cross_cats_sorted":["math.AT","math.NT"],"title_canon_sha256":"860a030ea9e7cafce60c0c44aa377172283e76dc28026b260d41e6bf54cbce0a","abstract_canon_sha256":"0bac15f773cfaadf715a15d7a9a2a01b5339579f309c2c68a716ac361ccafde2"},"schema_version":"1.0"},"canonical_sha256":"ed811947d3a683e5c543940a2374c19a094c4227cda52ea96e35f5ea72c4ed56","source":{"kind":"arxiv","id":"1109.0046","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.0046","created_at":"2026-05-18T02:50:24Z"},{"alias_kind":"arxiv_version","alias_value":"1109.0046v3","created_at":"2026-05-18T02:50:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0046","created_at":"2026-05-18T02:50:24Z"},{"alias_kind":"pith_short_12","alias_value":"5WARSR6TU2B6","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5WARSR6TU2B6LRKD","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5WARSR6T","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:5WARSR6TU2B6LRKDSQFCG5GBTI","target":"record","payload":{"canonical_record":{"source":{"id":"1109.0046","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-08-31T22:23:50Z","cross_cats_sorted":["math.AT","math.NT"],"title_canon_sha256":"860a030ea9e7cafce60c0c44aa377172283e76dc28026b260d41e6bf54cbce0a","abstract_canon_sha256":"0bac15f773cfaadf715a15d7a9a2a01b5339579f309c2c68a716ac361ccafde2"},"schema_version":"1.0"},"canonical_sha256":"ed811947d3a683e5c543940a2374c19a094c4227cda52ea96e35f5ea72c4ed56","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:24.567015Z","signature_b64":"TwBoEENHNzdhQ93ysqqKnU8hR7ck/2UTm41EQhGCoLMPZTEpkg+mo4eIvOOj7sIwjX4X5IZRWCGIuG1cfsnWCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed811947d3a683e5c543940a2374c19a094c4227cda52ea96e35f5ea72c4ed56","last_reissued_at":"2026-05-18T02:50:24.566602Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:24.566602Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.0046","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:50:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZD7aQc6S/tn9Ya8x/fii8ETw8/kdUQGQfNpcD1GyhxlNN/q6WFrSHLSAEYQph0ZXe2ldNab4rC1agZMYpkQcAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:11:03.051558Z"},"content_sha256":"edf24f63318883bad1831764dcdf0b2d92d2433bc0946081a15a9aac44fcef87","schema_version":"1.0","event_id":"sha256:edf24f63318883bad1831764dcdf0b2d92d2433bc0946081a15a9aac44fcef87"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:5WARSR6TU2B6LRKDSQFCG5GBTI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Milnor K-theory and the graded representation ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.NT"],"primary_cat":"math.KT","authors_text":"Jan Minac, Pierre Guillot","submitted_at":"2011-08-31T22:23:50Z","abstract_excerpt":"Let F be a field, let G be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k_*(F) to the graded ring gr R(G, k) associated to Grothendieck's \\gamma-filtration. We study this map in particular cases, as well as a related map involving the W-group of F rather than G. The latter is an isomorphism in all cases considered.\n  Naturally this echoes the Milnor conjecture (now a theorem), which states that k_*(F) is isomorphic to the mod 2 cohomology of the absolute Galoi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0046","kind":"arxiv","version":3},"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":"2026-05-18T02:50:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZcHvn5W7iYnjmelA7+XC3Eg2oObu75ldK0aDuOAyCpGr0gFGYvJ9GYAnEVPgnzgTAPPen2CqeJEX4F4iodraDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:11:03.051928Z"},"content_sha256":"f4e009a5a11c634c1715cf50d7c6a109bfb70157313fbf06be997e59b553b20b","schema_version":"1.0","event_id":"sha256:f4e009a5a11c634c1715cf50d7c6a109bfb70157313fbf06be997e59b553b20b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5WARSR6TU2B6LRKDSQFCG5GBTI/bundle.json","state_url":"https://pith.science/pith/5WARSR6TU2B6LRKDSQFCG5GBTI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5WARSR6TU2B6LRKDSQFCG5GBTI/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":"2026-06-03T03:11:03Z","links":{"resolver":"https://pith.science/pith/5WARSR6TU2B6LRKDSQFCG5GBTI","bundle":"https://pith.science/pith/5WARSR6TU2B6LRKDSQFCG5GBTI/bundle.json","state":"https://pith.science/pith/5WARSR6TU2B6LRKDSQFCG5GBTI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5WARSR6TU2B6LRKDSQFCG5GBTI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5WARSR6TU2B6LRKDSQFCG5GBTI","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":"0bac15f773cfaadf715a15d7a9a2a01b5339579f309c2c68a716ac361ccafde2","cross_cats_sorted":["math.AT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-08-31T22:23:50Z","title_canon_sha256":"860a030ea9e7cafce60c0c44aa377172283e76dc28026b260d41e6bf54cbce0a"},"schema_version":"1.0","source":{"id":"1109.0046","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.0046","created_at":"2026-05-18T02:50:24Z"},{"alias_kind":"arxiv_version","alias_value":"1109.0046v3","created_at":"2026-05-18T02:50:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0046","created_at":"2026-05-18T02:50:24Z"},{"alias_kind":"pith_short_12","alias_value":"5WARSR6TU2B6","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5WARSR6TU2B6LRKD","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5WARSR6T","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:f4e009a5a11c634c1715cf50d7c6a109bfb70157313fbf06be997e59b553b20b","target":"graph","created_at":"2026-05-18T02:50:24Z","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":"Let F be a field, let G be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k_*(F) to the graded ring gr R(G, k) associated to Grothendieck's \\gamma-filtration. We study this map in particular cases, as well as a related map involving the W-group of F rather than G. The latter is an isomorphism in all cases considered.\n  Naturally this echoes the Milnor conjecture (now a theorem), which states that k_*(F) is isomorphic to the mod 2 cohomology of the absolute Galoi","authors_text":"Jan Minac, Pierre Guillot","cross_cats":["math.AT","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-08-31T22:23:50Z","title":"Milnor K-theory and the graded representation ring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0046","kind":"arxiv","version":3},"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:edf24f63318883bad1831764dcdf0b2d92d2433bc0946081a15a9aac44fcef87","target":"record","created_at":"2026-05-18T02:50:24Z","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":"0bac15f773cfaadf715a15d7a9a2a01b5339579f309c2c68a716ac361ccafde2","cross_cats_sorted":["math.AT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-08-31T22:23:50Z","title_canon_sha256":"860a030ea9e7cafce60c0c44aa377172283e76dc28026b260d41e6bf54cbce0a"},"schema_version":"1.0","source":{"id":"1109.0046","kind":"arxiv","version":3}},"canonical_sha256":"ed811947d3a683e5c543940a2374c19a094c4227cda52ea96e35f5ea72c4ed56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed811947d3a683e5c543940a2374c19a094c4227cda52ea96e35f5ea72c4ed56","first_computed_at":"2026-05-18T02:50:24.566602Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:24.566602Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TwBoEENHNzdhQ93ysqqKnU8hR7ck/2UTm41EQhGCoLMPZTEpkg+mo4eIvOOj7sIwjX4X5IZRWCGIuG1cfsnWCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:24.567015Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.0046","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:edf24f63318883bad1831764dcdf0b2d92d2433bc0946081a15a9aac44fcef87","sha256:f4e009a5a11c634c1715cf50d7c6a109bfb70157313fbf06be997e59b553b20b"],"state_sha256":"927843d9860c77bc67d2cfdb29f9a67260059f985d61755ba1497ad64228ee36"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HmAdI+Dqjg4rdZNCvLGwt4aV2V8YCT13N7NPFwSqpWGogLGYsUjhayJkvEuplRxvQGdfNuBepufXLA7CNaHaBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T03:11:03.053976Z","bundle_sha256":"c9984c778197519866165f38d7cd86b9391fdd7d31d3cbf9733b970401f1e122"}}