{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:PDC5ROMQBOKXUSMQJOKBFWNGQB","short_pith_number":"pith:PDC5ROMQ","canonical_record":{"source":{"id":"2506.04067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-04T15:32:40Z","cross_cats_sorted":[],"title_canon_sha256":"b039b1ac65b1a90c67b65afe0be2d5edbe0c6f2e609ae23fb553caca6b910603","abstract_canon_sha256":"e6ea95e8e6a8befaa3020438746465435651d19fe463fa59950355cbe2ebaaef"},"schema_version":"1.0"},"canonical_sha256":"78c5d8b9900b957a49904b9412d9a68055f7bad97c2e20674abf674e7a37665f","source":{"kind":"arxiv","id":"2506.04067","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.04067","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"arxiv_version","alias_value":"2506.04067v1","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.04067","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"pith_short_12","alias_value":"PDC5ROMQBOKX","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"pith_short_16","alias_value":"PDC5ROMQBOKXUSMQ","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"pith_short_8","alias_value":"PDC5ROMQ","created_at":"2026-07-05T11:15:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:PDC5ROMQBOKXUSMQJOKBFWNGQB","target":"record","payload":{"canonical_record":{"source":{"id":"2506.04067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-04T15:32:40Z","cross_cats_sorted":[],"title_canon_sha256":"b039b1ac65b1a90c67b65afe0be2d5edbe0c6f2e609ae23fb553caca6b910603","abstract_canon_sha256":"e6ea95e8e6a8befaa3020438746465435651d19fe463fa59950355cbe2ebaaef"},"schema_version":"1.0"},"canonical_sha256":"78c5d8b9900b957a49904b9412d9a68055f7bad97c2e20674abf674e7a37665f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:15:56.507840Z","signature_b64":"VCq8PX0ZYAjYGrmwsP2DlU1IcHgkp7mYmSJxwvIsTMqEosj1e9sDp+a2Gao+2JCwhLmf4IHpF2QdyHfuESzCBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78c5d8b9900b957a49904b9412d9a68055f7bad97c2e20674abf674e7a37665f","last_reissued_at":"2026-07-05T11:15:56.507305Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:15:56.507305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.04067","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":"2026-07-05T11:15:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pB4B7YJ/xe0tw5yoS1gy4+1ZJwCG0fg35/KgpyvgpRDsTEbWsgy2E5nZChyyPFAezPUsXZ6PmM2FOtANyeeoAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T02:11:05.400138Z"},"content_sha256":"1cefca19bf14071774d64c95b796a448b89eb8aafbcd6be052fd0bbd5bcd82a3","schema_version":"1.0","event_id":"sha256:1cefca19bf14071774d64c95b796a448b89eb8aafbcd6be052fd0bbd5bcd82a3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:PDC5ROMQBOKXUSMQJOKBFWNGQB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Free actions on products of real projective spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Ergun Yalcin","submitted_at":"2025-06-04T15:32:40Z","abstract_excerpt":"We prove that if $G=(\\mathbb{Z}/2)^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\\mathbb{R}P ^{n_1} \\times \\cdots \\times \\mathbb{R} P ^{n_k}$ with trivial action on the mod-$2$ cohomology, then $r \\leq \\mu (n_1)+ \\cdots + \\mu(n_k )$ where for each integer $n\\geq 0$, $\\mu (n)=0$ if $n$ is even, $\\mu(n)=1$ if $n\\equiv 1$ mod 4, and $\\mu(n)=2$ if $n\\equiv 3$ mod 4. This proves a homotopy-theoretic version of a conjecture of Cusick."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04067","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.04067/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":"2026-07-05T11:15:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z48FpOLX9HqDMkcXUV9pkJqpGmjYPVIbsLZMZt3pS/pi2U3zcciDfpdAKRuzJgVyoSEBwQMzlXaaZ1VPGM55BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T02:11:05.400635Z"},"content_sha256":"5fe5c65265a98b92352f9b853829850be54470aed3d2a4903b6531888ebe74fc","schema_version":"1.0","event_id":"sha256:5fe5c65265a98b92352f9b853829850be54470aed3d2a4903b6531888ebe74fc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PDC5ROMQBOKXUSMQJOKBFWNGQB/bundle.json","state_url":"https://pith.science/pith/PDC5ROMQBOKXUSMQJOKBFWNGQB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PDC5ROMQBOKXUSMQJOKBFWNGQB/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":"2026-08-08T02:11:05Z","links":{"resolver":"https://pith.science/pith/PDC5ROMQBOKXUSMQJOKBFWNGQB","bundle":"https://pith.science/pith/PDC5ROMQBOKXUSMQJOKBFWNGQB/bundle.json","state":"https://pith.science/pith/PDC5ROMQBOKXUSMQJOKBFWNGQB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PDC5ROMQBOKXUSMQJOKBFWNGQB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:PDC5ROMQBOKXUSMQJOKBFWNGQB","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":"e6ea95e8e6a8befaa3020438746465435651d19fe463fa59950355cbe2ebaaef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-04T15:32:40Z","title_canon_sha256":"b039b1ac65b1a90c67b65afe0be2d5edbe0c6f2e609ae23fb553caca6b910603"},"schema_version":"1.0","source":{"id":"2506.04067","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.04067","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"arxiv_version","alias_value":"2506.04067v1","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.04067","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"pith_short_12","alias_value":"PDC5ROMQBOKX","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"pith_short_16","alias_value":"PDC5ROMQBOKXUSMQ","created_at":"2026-07-05T11:15:56Z"},{"alias_kind":"pith_short_8","alias_value":"PDC5ROMQ","created_at":"2026-07-05T11:15:56Z"}],"graph_snapshots":[{"event_id":"sha256:5fe5c65265a98b92352f9b853829850be54470aed3d2a4903b6531888ebe74fc","target":"graph","created_at":"2026-07-05T11:15:56Z","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/2506.04067/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that if $G=(\\mathbb{Z}/2)^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\\mathbb{R}P ^{n_1} \\times \\cdots \\times \\mathbb{R} P ^{n_k}$ with trivial action on the mod-$2$ cohomology, then $r \\leq \\mu (n_1)+ \\cdots + \\mu(n_k )$ where for each integer $n\\geq 0$, $\\mu (n)=0$ if $n$ is even, $\\mu(n)=1$ if $n\\equiv 1$ mod 4, and $\\mu(n)=2$ if $n\\equiv 3$ mod 4. This proves a homotopy-theoretic version of a conjecture of Cusick.","authors_text":"Ergun Yalcin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-04T15:32:40Z","title":"Free actions on products of real projective spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04067","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":"sha256:1cefca19bf14071774d64c95b796a448b89eb8aafbcd6be052fd0bbd5bcd82a3","target":"record","created_at":"2026-07-05T11:15:56Z","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":"e6ea95e8e6a8befaa3020438746465435651d19fe463fa59950355cbe2ebaaef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-04T15:32:40Z","title_canon_sha256":"b039b1ac65b1a90c67b65afe0be2d5edbe0c6f2e609ae23fb553caca6b910603"},"schema_version":"1.0","source":{"id":"2506.04067","kind":"arxiv","version":1}},"canonical_sha256":"78c5d8b9900b957a49904b9412d9a68055f7bad97c2e20674abf674e7a37665f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"78c5d8b9900b957a49904b9412d9a68055f7bad97c2e20674abf674e7a37665f","first_computed_at":"2026-07-05T11:15:56.507305Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:15:56.507305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VCq8PX0ZYAjYGrmwsP2DlU1IcHgkp7mYmSJxwvIsTMqEosj1e9sDp+a2Gao+2JCwhLmf4IHpF2QdyHfuESzCBw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:15:56.507840Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.04067","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1cefca19bf14071774d64c95b796a448b89eb8aafbcd6be052fd0bbd5bcd82a3","sha256:5fe5c65265a98b92352f9b853829850be54470aed3d2a4903b6531888ebe74fc"],"state_sha256":"09caf854e3e99ca60911704e14e64002292a5d886f77ac4dc9b14254d46ae76e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bYcbZn4R4CWRoJvGl0sEmp3xMYoXRcMTXBfXIoPKTxRocSfgY0rTBDSAuntuOhxiGhbihsu1VMmqTtq/baRfCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T02:11:05.404513Z","bundle_sha256":"c7451588c4d7c085fb016c3ac3bb9b80111c3a48d16cffdc0df2a8fdabab7be3"}}