{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ZFISQRNT3GE75NUPMMAKV5KKXF","short_pith_number":"pith:ZFISQRNT","canonical_record":{"source":{"id":"1910.00664","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-10-01T20:55:42Z","cross_cats_sorted":[],"title_canon_sha256":"df6703b801f01dcb700f18506a6249403a3faaf5145b2b80fdc4984e343af573","abstract_canon_sha256":"26a1d73a6bbe25e12a2c23a2905d5ec870dc09c0926ddb9b1f3fa271f22502a0"},"schema_version":"1.0"},"canonical_sha256":"c9512845b3d989feb68f6300aaf54ab945d2ebbb0f9cbddd47a8bb5b5be940f8","source":{"kind":"arxiv","id":"1910.00664","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.00664","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"arxiv_version","alias_value":"1910.00664v2","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.00664","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"pith_short_12","alias_value":"ZFISQRNT3GE7","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZFISQRNT3GE75NUP","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZFISQRNT","created_at":"2026-07-05T03:52:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ZFISQRNT3GE75NUPMMAKV5KKXF","target":"record","payload":{"canonical_record":{"source":{"id":"1910.00664","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-10-01T20:55:42Z","cross_cats_sorted":[],"title_canon_sha256":"df6703b801f01dcb700f18506a6249403a3faaf5145b2b80fdc4984e343af573","abstract_canon_sha256":"26a1d73a6bbe25e12a2c23a2905d5ec870dc09c0926ddb9b1f3fa271f22502a0"},"schema_version":"1.0"},"canonical_sha256":"c9512845b3d989feb68f6300aaf54ab945d2ebbb0f9cbddd47a8bb5b5be940f8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:52:53.075748Z","signature_b64":"oLNbOc7SkVfejGr/udAh0UFZ8F5DeFyOwbSaNSCncYKuHOXH4zLimuFAvo2nrQrZTds6y51yK7GxQMh1pt9GCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9512845b3d989feb68f6300aaf54ab945d2ebbb0f9cbddd47a8bb5b5be940f8","last_reissued_at":"2026-07-05T03:52:53.075309Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:52:53.075309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1910.00664","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:52:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IZwEb3gwyM8N+hhETONc6XMy0zF0c+/Wk3r+wsTNJWgMpb9pQb8dXV4s/b7TLjmusNI1HS68BUU3/Xooc14GCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T05:38:31.924835Z"},"content_sha256":"70584db02955eb4d0dbc8f7fe7efaadbdf386ab55642a4a591ee021276302a0b","schema_version":"1.0","event_id":"sha256:70584db02955eb4d0dbc8f7fe7efaadbdf386ab55642a4a591ee021276302a0b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ZFISQRNT3GE75NUPMMAKV5KKXF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Freeness and equivariant stable homotopy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Michael A. Hill","submitted_at":"2019-10-01T20:55:42Z","abstract_excerpt":"We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way.\n  We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill--Hopkins--Ravenel. For these spectra, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.00664","kind":"arxiv","version":2},"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/1910.00664/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":"2026-07-05T03:52:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fmv/Y1Q6rc7RTq4UmewJqh56OKpX20WbaV8lCALQ4OKkQwVQA1HwUf4CoCDXtSSfsSk3TyuASMjFmnrp0kjmAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T05:38:31.925895Z"},"content_sha256":"ddc2e7d7756ae22db274d51f4802a2dc9cd7d94eddafea64f0545b9a44e45531","schema_version":"1.0","event_id":"sha256:ddc2e7d7756ae22db274d51f4802a2dc9cd7d94eddafea64f0545b9a44e45531"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZFISQRNT3GE75NUPMMAKV5KKXF/bundle.json","state_url":"https://pith.science/pith/ZFISQRNT3GE75NUPMMAKV5KKXF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZFISQRNT3GE75NUPMMAKV5KKXF/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":"2026-08-17T05:38:31Z","links":{"resolver":"https://pith.science/pith/ZFISQRNT3GE75NUPMMAKV5KKXF","bundle":"https://pith.science/pith/ZFISQRNT3GE75NUPMMAKV5KKXF/bundle.json","state":"https://pith.science/pith/ZFISQRNT3GE75NUPMMAKV5KKXF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZFISQRNT3GE75NUPMMAKV5KKXF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZFISQRNT3GE75NUPMMAKV5KKXF","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":"26a1d73a6bbe25e12a2c23a2905d5ec870dc09c0926ddb9b1f3fa271f22502a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-10-01T20:55:42Z","title_canon_sha256":"df6703b801f01dcb700f18506a6249403a3faaf5145b2b80fdc4984e343af573"},"schema_version":"1.0","source":{"id":"1910.00664","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.00664","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"arxiv_version","alias_value":"1910.00664v2","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.00664","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"pith_short_12","alias_value":"ZFISQRNT3GE7","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZFISQRNT3GE75NUP","created_at":"2026-07-05T03:52:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZFISQRNT","created_at":"2026-07-05T03:52:53Z"}],"graph_snapshots":[{"event_id":"sha256:ddc2e7d7756ae22db274d51f4802a2dc9cd7d94eddafea64f0545b9a44e45531","target":"graph","created_at":"2026-07-05T03:52:53Z","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/1910.00664/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way.\n  We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill--Hopkins--Ravenel. For these spectra, th","authors_text":"Michael A. Hill","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-10-01T20:55:42Z","title":"Freeness and equivariant stable homotopy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.00664","kind":"arxiv","version":2},"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":"sha256:70584db02955eb4d0dbc8f7fe7efaadbdf386ab55642a4a591ee021276302a0b","target":"record","created_at":"2026-07-05T03:52:53Z","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":"26a1d73a6bbe25e12a2c23a2905d5ec870dc09c0926ddb9b1f3fa271f22502a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-10-01T20:55:42Z","title_canon_sha256":"df6703b801f01dcb700f18506a6249403a3faaf5145b2b80fdc4984e343af573"},"schema_version":"1.0","source":{"id":"1910.00664","kind":"arxiv","version":2}},"canonical_sha256":"c9512845b3d989feb68f6300aaf54ab945d2ebbb0f9cbddd47a8bb5b5be940f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9512845b3d989feb68f6300aaf54ab945d2ebbb0f9cbddd47a8bb5b5be940f8","first_computed_at":"2026-07-05T03:52:53.075309Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:52:53.075309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oLNbOc7SkVfejGr/udAh0UFZ8F5DeFyOwbSaNSCncYKuHOXH4zLimuFAvo2nrQrZTds6y51yK7GxQMh1pt9GCw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:52:53.075748Z","signed_message":"canonical_sha256_bytes"},"source_id":"1910.00664","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70584db02955eb4d0dbc8f7fe7efaadbdf386ab55642a4a591ee021276302a0b","sha256:ddc2e7d7756ae22db274d51f4802a2dc9cd7d94eddafea64f0545b9a44e45531"],"state_sha256":"7ce6ff7210e3e305d534065d3579c3d82a4bba1e406465d5fa23f5590cce5bbc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4wI735ajIoYvySTAYHtCA0uo8p22uEIR5+NSBdzQCCj+0e+tIfP5GcKWvVrptNkEbFejnBjXSqa0/huK/xouDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T05:38:31.931891Z","bundle_sha256":"e637086048f4197089139bd645093a11edc167e8dbd6c26de2fb70d81bccdd0d"}}