{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ILMALQVXMY4HJEDQX2WIVI4WJQ","short_pith_number":"pith:ILMALQVX","canonical_record":{"source":{"id":"2607.08621","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-09T15:54:14Z","cross_cats_sorted":[],"title_canon_sha256":"b3139b7c6e6a2f101b8ce3c91de484811a65d1bed601c24e5918d9ba691c6c5f","abstract_canon_sha256":"057c9d029e03affe793ff5672c9bad7e4e510052f86f436584d5b0a1111107a4"},"schema_version":"1.0"},"canonical_sha256":"42d805c2b76638749070beac8aa3964c2fd3272b7efc354c357b3ab0e4a33dc3","source":{"kind":"arxiv","id":"2607.08621","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.08621","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"arxiv_version","alias_value":"2607.08621v1","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.08621","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"pith_short_12","alias_value":"ILMALQVXMY4H","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"pith_short_16","alias_value":"ILMALQVXMY4HJEDQ","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"pith_short_8","alias_value":"ILMALQVX","created_at":"2026-07-10T01:19:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ILMALQVXMY4HJEDQX2WIVI4WJQ","target":"record","payload":{"canonical_record":{"source":{"id":"2607.08621","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-09T15:54:14Z","cross_cats_sorted":[],"title_canon_sha256":"b3139b7c6e6a2f101b8ce3c91de484811a65d1bed601c24e5918d9ba691c6c5f","abstract_canon_sha256":"057c9d029e03affe793ff5672c9bad7e4e510052f86f436584d5b0a1111107a4"},"schema_version":"1.0"},"canonical_sha256":"42d805c2b76638749070beac8aa3964c2fd3272b7efc354c357b3ab0e4a33dc3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-10T01:19:55.714367Z","signature_b64":"5w+R3McOmUaDKc4SE/JD7qZwG3y7DJ7285YMKdoT39Q3RKs4/MTU2r2S/vYeGbzdSNzX7eWljZ7Ycah4/U/hCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42d805c2b76638749070beac8aa3964c2fd3272b7efc354c357b3ab0e4a33dc3","last_reissued_at":"2026-07-10T01:19:55.713889Z","signature_status":"signed_v1","first_computed_at":"2026-07-10T01:19:55.713889Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.08621","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":"2026-07-10T01:19:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"obZmSBJBZXvyi7O7m6V1LuT3myPWdM4nhrnA1tiP7Opf8aTo72zEsOwTbXC9e8q0GjAdVj/ohqDYaSB2ylzBCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T22:47:00.514287Z"},"content_sha256":"4776939d12680237f738556fb68ec347427fac6f3628ccdff84abd482c79382f","schema_version":"1.0","event_id":"sha256:4776939d12680237f738556fb68ec347427fac6f3628ccdff84abd482c79382f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ILMALQVXMY4HJEDQX2WIVI4WJQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Rosenberg $ \\mathbb{S}^{1} $-Stability Conjecture for $ \\chi(X) = 0 $","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jie Xu","submitted_at":"2026-07-09T15:54:14Z","abstract_excerpt":"Let $ X $ be a closed, oriented manifold with $ \\dim X \\geqslant 5 $. In this article, we show that 2006 Rosenberg's $ \\mathbb{S}^{1} $-stability holds when $ X $ has zero Euler characteristic. The 2006 Rosenberg-Stolz Conjecture for $ X \\times \\mathbb{R} $ also follows under the same assumption, provided that the Riemannian metric $ g $ on $ X \\times \\mathbb{R} $ is complete, is of bounded curvature, and whose smallest eigenvalue is uniformly bounded below by some positive constant. We then show a $ \\mathbb{T}^{n} $-stability theorem with the same hypothesis of $ X $."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08621","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/2607.08621/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":"2026-07-10T01:19:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JukRGO50f9TewQ+DosiDjxkuL/N9Tmx2A5Wz9L5iz3mk7WmkV5L/5Lk8n4CE1EZQeO7Wzw1/rMQZhrnmYGFeCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T22:47:00.515288Z"},"content_sha256":"0236fc28cfce9bfd93f7ff9eed80b5eff9be37ec8925b5f6bdb313b482ee91f2","schema_version":"1.0","event_id":"sha256:0236fc28cfce9bfd93f7ff9eed80b5eff9be37ec8925b5f6bdb313b482ee91f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ILMALQVXMY4HJEDQX2WIVI4WJQ/bundle.json","state_url":"https://pith.science/pith/ILMALQVXMY4HJEDQX2WIVI4WJQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ILMALQVXMY4HJEDQX2WIVI4WJQ/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":"2026-08-10T22:47:00Z","links":{"resolver":"https://pith.science/pith/ILMALQVXMY4HJEDQX2WIVI4WJQ","bundle":"https://pith.science/pith/ILMALQVXMY4HJEDQX2WIVI4WJQ/bundle.json","state":"https://pith.science/pith/ILMALQVXMY4HJEDQX2WIVI4WJQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ILMALQVXMY4HJEDQX2WIVI4WJQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ILMALQVXMY4HJEDQX2WIVI4WJQ","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":"057c9d029e03affe793ff5672c9bad7e4e510052f86f436584d5b0a1111107a4","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-09T15:54:14Z","title_canon_sha256":"b3139b7c6e6a2f101b8ce3c91de484811a65d1bed601c24e5918d9ba691c6c5f"},"schema_version":"1.0","source":{"id":"2607.08621","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.08621","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"arxiv_version","alias_value":"2607.08621v1","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.08621","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"pith_short_12","alias_value":"ILMALQVXMY4H","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"pith_short_16","alias_value":"ILMALQVXMY4HJEDQ","created_at":"2026-07-10T01:19:55Z"},{"alias_kind":"pith_short_8","alias_value":"ILMALQVX","created_at":"2026-07-10T01:19:55Z"}],"graph_snapshots":[{"event_id":"sha256:0236fc28cfce9bfd93f7ff9eed80b5eff9be37ec8925b5f6bdb313b482ee91f2","target":"graph","created_at":"2026-07-10T01:19:55Z","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/2607.08621/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $ X $ be a closed, oriented manifold with $ \\dim X \\geqslant 5 $. In this article, we show that 2006 Rosenberg's $ \\mathbb{S}^{1} $-stability holds when $ X $ has zero Euler characteristic. The 2006 Rosenberg-Stolz Conjecture for $ X \\times \\mathbb{R} $ also follows under the same assumption, provided that the Riemannian metric $ g $ on $ X \\times \\mathbb{R} $ is complete, is of bounded curvature, and whose smallest eigenvalue is uniformly bounded below by some positive constant. We then show a $ \\mathbb{T}^{n} $-stability theorem with the same hypothesis of $ X $.","authors_text":"Jie Xu","cross_cats":[],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-09T15:54:14Z","title":"The Rosenberg $ \\mathbb{S}^{1} $-Stability Conjecture for $ \\chi(X) = 0 $"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08621","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":"sha256:4776939d12680237f738556fb68ec347427fac6f3628ccdff84abd482c79382f","target":"record","created_at":"2026-07-10T01:19:55Z","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":"057c9d029e03affe793ff5672c9bad7e4e510052f86f436584d5b0a1111107a4","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-09T15:54:14Z","title_canon_sha256":"b3139b7c6e6a2f101b8ce3c91de484811a65d1bed601c24e5918d9ba691c6c5f"},"schema_version":"1.0","source":{"id":"2607.08621","kind":"arxiv","version":1}},"canonical_sha256":"42d805c2b76638749070beac8aa3964c2fd3272b7efc354c357b3ab0e4a33dc3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"42d805c2b76638749070beac8aa3964c2fd3272b7efc354c357b3ab0e4a33dc3","first_computed_at":"2026-07-10T01:19:55.713889Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-10T01:19:55.713889Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5w+R3McOmUaDKc4SE/JD7qZwG3y7DJ7285YMKdoT39Q3RKs4/MTU2r2S/vYeGbzdSNzX7eWljZ7Ycah4/U/hCg==","signature_status":"signed_v1","signed_at":"2026-07-10T01:19:55.714367Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.08621","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4776939d12680237f738556fb68ec347427fac6f3628ccdff84abd482c79382f","sha256:0236fc28cfce9bfd93f7ff9eed80b5eff9be37ec8925b5f6bdb313b482ee91f2"],"state_sha256":"6af8fe02e6ad403cc2a3fc9e5821eeff09b2c67aa76e83ba24ffcf469a0c88fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KyowIuMOAkPEYHu3ZEdeoAHEKcUW3pe+Ac6jyfKPqzUL9Eo6IDRIpVUZl3tfdb2HqcqsH1beSPRxKrn0Ev1DDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T22:47:00.527471Z","bundle_sha256":"c9e15201a7da8f7ae328a32afcfb5bf3d2a76827ea0ca7ce55efbdf765b8ded2"}}