{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:C5O4C2AD2MU2BGVZOYOIIWL6ES","short_pith_number":"pith:C5O4C2AD","canonical_record":{"source":{"id":"2607.25615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-07-28T11:45:39Z","cross_cats_sorted":[],"title_canon_sha256":"21f9a96670194b4ff6657195f5febd6f14de4db21b7867f04f4ba89b9f1b55d9","abstract_canon_sha256":"2068e4113f723b6b20e442d6b75eb3e5870c3bd93fcb0f7df3e05c3ec93e1c9d"},"schema_version":"1.0"},"canonical_sha256":"175dc16803d329a09ab9761c84597e2492ed6fcf8d2ec109b84dbb3aa6863e9c","source":{"kind":"arxiv","id":"2607.25615","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25615","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25615v1","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25615","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"pith_short_12","alias_value":"C5O4C2AD2MU2","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"C5O4C2AD2MU2BGVZ","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"C5O4C2AD","created_at":"2026-07-29T01:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:C5O4C2AD2MU2BGVZOYOIIWL6ES","target":"record","payload":{"canonical_record":{"source":{"id":"2607.25615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-07-28T11:45:39Z","cross_cats_sorted":[],"title_canon_sha256":"21f9a96670194b4ff6657195f5febd6f14de4db21b7867f04f4ba89b9f1b55d9","abstract_canon_sha256":"2068e4113f723b6b20e442d6b75eb3e5870c3bd93fcb0f7df3e05c3ec93e1c9d"},"schema_version":"1.0"},"canonical_sha256":"175dc16803d329a09ab9761c84597e2492ed6fcf8d2ec109b84dbb3aa6863e9c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-29T01:25:53.263553Z","signature_b64":"Gkbj7GaPBusqc/rQeykQUNc9tAqmcpHsfc7c7+ZXK7UmaVo/H2ZKOsi96fqd7PGIxM5VLLq56Exl+sDARE/PDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"175dc16803d329a09ab9761c84597e2492ed6fcf8d2ec109b84dbb3aa6863e9c","last_reissued_at":"2026-07-29T01:25:53.262722Z","signature_status":"signed_v1","first_computed_at":"2026-07-29T01:25:53.262722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.25615","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":"2026-07-29T01:25:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"orHJL/sYjAi0hvQD+AMbpkd9g3n9Z0k5OM+HsjyseVyiDLeiPqtbmTvJ+ra8VvmM+EUBN89bOnZxh41SLEnECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T10:14:57.884686Z"},"content_sha256":"5d0a55fc4f8a1f775b5dbcdbff66d8827d995b9ad100363c10af2b793db2c6c5","schema_version":"1.0","event_id":"sha256:5d0a55fc4f8a1f775b5dbcdbff66d8827d995b9ad100363c10af2b793db2c6c5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:C5O4C2AD2MU2BGVZOYOIIWL6ES","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a twisted Singer conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jacopo G. Chen","submitted_at":"2026-07-28T11:45:39Z","abstract_excerpt":"According to the Singer conjecture, the $L^2$-Betti numbers of a closed aspherical manifold are expected to vanish outside the middle dimension. In this paper, we study a natural analog of the Singer conjecture concerning the vanishing of twisted $L^2$-Betti numbers outside the lower middle dimension. These invariants can be thought of as $L^2$-Betti numbers of an infinite cyclic cover, depending on a first cohomology class. This is related to a previous conjecture by the author, which connects the twisted $L^2$-Euler characteristic to the Thurston norm, and which we prove for a class of close"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25615","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/2607.25615/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":"2026-07-29T01:25:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FY2Z0bfLYnEuA5TgzKhvH78IY0lPzgJtBy0BUdUxsbujp/ianNRiySqsDY8E6pJRlctgLw+8nPJLPcK5C/7pDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T10:14:57.885208Z"},"content_sha256":"fc5fd375be85f6a2b96aa356a967e2ae4737afbe384c97f697cf9e6fe5637252","schema_version":"1.0","event_id":"sha256:fc5fd375be85f6a2b96aa356a967e2ae4737afbe384c97f697cf9e6fe5637252"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C5O4C2AD2MU2BGVZOYOIIWL6ES/bundle.json","state_url":"https://pith.science/pith/C5O4C2AD2MU2BGVZOYOIIWL6ES/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C5O4C2AD2MU2BGVZOYOIIWL6ES/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":"2026-08-04T10:14:57Z","links":{"resolver":"https://pith.science/pith/C5O4C2AD2MU2BGVZOYOIIWL6ES","bundle":"https://pith.science/pith/C5O4C2AD2MU2BGVZOYOIIWL6ES/bundle.json","state":"https://pith.science/pith/C5O4C2AD2MU2BGVZOYOIIWL6ES/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C5O4C2AD2MU2BGVZOYOIIWL6ES/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:C5O4C2AD2MU2BGVZOYOIIWL6ES","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":"2068e4113f723b6b20e442d6b75eb3e5870c3bd93fcb0f7df3e05c3ec93e1c9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-07-28T11:45:39Z","title_canon_sha256":"21f9a96670194b4ff6657195f5febd6f14de4db21b7867f04f4ba89b9f1b55d9"},"schema_version":"1.0","source":{"id":"2607.25615","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25615","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25615v1","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25615","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"pith_short_12","alias_value":"C5O4C2AD2MU2","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"C5O4C2AD2MU2BGVZ","created_at":"2026-07-29T01:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"C5O4C2AD","created_at":"2026-07-29T01:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:fc5fd375be85f6a2b96aa356a967e2ae4737afbe384c97f697cf9e6fe5637252","target":"graph","created_at":"2026-07-29T01:25:53Z","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/2607.25615/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"According to the Singer conjecture, the $L^2$-Betti numbers of a closed aspherical manifold are expected to vanish outside the middle dimension. In this paper, we study a natural analog of the Singer conjecture concerning the vanishing of twisted $L^2$-Betti numbers outside the lower middle dimension. These invariants can be thought of as $L^2$-Betti numbers of an infinite cyclic cover, depending on a first cohomology class. This is related to a previous conjecture by the author, which connects the twisted $L^2$-Euler characteristic to the Thurston norm, and which we prove for a class of close","authors_text":"Jacopo G. Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-07-28T11:45:39Z","title":"On a twisted Singer conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25615","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":"sha256:5d0a55fc4f8a1f775b5dbcdbff66d8827d995b9ad100363c10af2b793db2c6c5","target":"record","created_at":"2026-07-29T01:25:53Z","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":"2068e4113f723b6b20e442d6b75eb3e5870c3bd93fcb0f7df3e05c3ec93e1c9d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-07-28T11:45:39Z","title_canon_sha256":"21f9a96670194b4ff6657195f5febd6f14de4db21b7867f04f4ba89b9f1b55d9"},"schema_version":"1.0","source":{"id":"2607.25615","kind":"arxiv","version":1}},"canonical_sha256":"175dc16803d329a09ab9761c84597e2492ed6fcf8d2ec109b84dbb3aa6863e9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"175dc16803d329a09ab9761c84597e2492ed6fcf8d2ec109b84dbb3aa6863e9c","first_computed_at":"2026-07-29T01:25:53.262722Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T01:25:53.262722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Gkbj7GaPBusqc/rQeykQUNc9tAqmcpHsfc7c7+ZXK7UmaVo/H2ZKOsi96fqd7PGIxM5VLLq56Exl+sDARE/PDg==","signature_status":"signed_v1","signed_at":"2026-07-29T01:25:53.263553Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.25615","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d0a55fc4f8a1f775b5dbcdbff66d8827d995b9ad100363c10af2b793db2c6c5","sha256:fc5fd375be85f6a2b96aa356a967e2ae4737afbe384c97f697cf9e6fe5637252"],"state_sha256":"b2595f5d95271435bde76c5fecd9b7b6939c89b7734cb7f25f71d857535ca4d2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"77HRAQbd2EgLh1D++z/MKDEZtRBmcuGKpGmhe1KFf45A2wvcbfM74W5lQfZZ1OVx5kwdAQ1kGjuu8XurLsTODA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T10:14:57.889956Z","bundle_sha256":"574681b46862f82b476f9af95a18275097e8a99d88797ed47a4121925c199009"}}