{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:4QCTMV2EPKM7RSV6KFM7WD4B73","short_pith_number":"pith:4QCTMV2E","canonical_record":{"source":{"id":"2606.17492","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-16T04:03:39Z","cross_cats_sorted":[],"title_canon_sha256":"53e4c12bd492f86e2d7d46795c6e791813dcb6dba625eb1fada321ce92fc7b7b","abstract_canon_sha256":"d5222a7159963f7992577f612af0dd9832592b1a21c16c01d2e5e2de78179bb3"},"schema_version":"1.0"},"canonical_sha256":"e4053657447a99f8cabe5159fb0f81feccd965bf349de78475006766f4a4306f","source":{"kind":"arxiv","id":"2606.17492","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.17492","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"2606.17492v1","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.17492","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"4QCTMV2EPKM7","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"pith_short_16","alias_value":"4QCTMV2EPKM7RSV6","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"pith_short_8","alias_value":"4QCTMV2E","created_at":"2026-06-19T16:10:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:4QCTMV2EPKM7RSV6KFM7WD4B73","target":"record","payload":{"canonical_record":{"source":{"id":"2606.17492","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-16T04:03:39Z","cross_cats_sorted":[],"title_canon_sha256":"53e4c12bd492f86e2d7d46795c6e791813dcb6dba625eb1fada321ce92fc7b7b","abstract_canon_sha256":"d5222a7159963f7992577f612af0dd9832592b1a21c16c01d2e5e2de78179bb3"},"schema_version":"1.0"},"canonical_sha256":"e4053657447a99f8cabe5159fb0f81feccd965bf349de78475006766f4a4306f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:14.348887Z","signature_b64":"YcZDWA5KWcggswQwajzVqHD9Y9WKC5j5qUIDW3MUbvC66w1oxk8+cD0QnprTHbL1f+3SHpvhBhGPuuYTksk/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4053657447a99f8cabe5159fb0f81feccd965bf349de78475006766f4a4306f","last_reissued_at":"2026-06-19T16:10:14.348520Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:14.348520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.17492","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":"2026-06-19T16:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I+PCrKBWyy+jSavX02j7fgFUEij52eXj4HS/y+ENekSC2DTAVWN575R654Rkm24nYKSa4lG1KlR+XYYWKlu2DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T07:57:30.489245Z"},"content_sha256":"c8fabeeaf1e1712edad85d7da9c3199f4c552dc46effe534a39f1daba566d919","schema_version":"1.0","event_id":"sha256:c8fabeeaf1e1712edad85d7da9c3199f4c552dc46effe534a39f1daba566d919"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:4QCTMV2EPKM7RSV6KFM7WD4B73","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Tatsuya Tate","submitted_at":"2026-06-16T04:03:39Z","abstract_excerpt":"The purpose of this note is to study spectral properties of the horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution. We show that the horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion. We give an application of this interpretation to the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations introduced by Berard-Bergery and Bourguignon. Our approach e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17492","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/2606.17492/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":"2026-06-19T16:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nCbjnGgWUIbKpd+3s0SgADtLFZbKwOS+7dEZ7dyewtj0lLS2exr+8anWUnlTE1+AOpJyCuupyJNyLSQwtpgwBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T07:57:30.489635Z"},"content_sha256":"052a8006b17fd245915a4396cdbf889647ed88e888d7a5c894013707e8025cbf","schema_version":"1.0","event_id":"sha256:052a8006b17fd245915a4396cdbf889647ed88e888d7a5c894013707e8025cbf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4QCTMV2EPKM7RSV6KFM7WD4B73/bundle.json","state_url":"https://pith.science/pith/4QCTMV2EPKM7RSV6KFM7WD4B73/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4QCTMV2EPKM7RSV6KFM7WD4B73/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":"2026-07-04T07:57:30Z","links":{"resolver":"https://pith.science/pith/4QCTMV2EPKM7RSV6KFM7WD4B73","bundle":"https://pith.science/pith/4QCTMV2EPKM7RSV6KFM7WD4B73/bundle.json","state":"https://pith.science/pith/4QCTMV2EPKM7RSV6KFM7WD4B73/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4QCTMV2EPKM7RSV6KFM7WD4B73/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4QCTMV2EPKM7RSV6KFM7WD4B73","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":"d5222a7159963f7992577f612af0dd9832592b1a21c16c01d2e5e2de78179bb3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-16T04:03:39Z","title_canon_sha256":"53e4c12bd492f86e2d7d46795c6e791813dcb6dba625eb1fada321ce92fc7b7b"},"schema_version":"1.0","source":{"id":"2606.17492","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.17492","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"2606.17492v1","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.17492","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"4QCTMV2EPKM7","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"pith_short_16","alias_value":"4QCTMV2EPKM7RSV6","created_at":"2026-06-19T16:10:14Z"},{"alias_kind":"pith_short_8","alias_value":"4QCTMV2E","created_at":"2026-06-19T16:10:14Z"}],"graph_snapshots":[{"event_id":"sha256:052a8006b17fd245915a4396cdbf889647ed88e888d7a5c894013707e8025cbf","target":"graph","created_at":"2026-06-19T16:10:14Z","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/2606.17492/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The purpose of this note is to study spectral properties of the horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution. We show that the horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion. We give an application of this interpretation to the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations introduced by Berard-Bergery and Bourguignon. Our approach e","authors_text":"Tatsuya Tate","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-16T04:03:39Z","title":"The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17492","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:c8fabeeaf1e1712edad85d7da9c3199f4c552dc46effe534a39f1daba566d919","target":"record","created_at":"2026-06-19T16:10:14Z","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":"d5222a7159963f7992577f612af0dd9832592b1a21c16c01d2e5e2de78179bb3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-16T04:03:39Z","title_canon_sha256":"53e4c12bd492f86e2d7d46795c6e791813dcb6dba625eb1fada321ce92fc7b7b"},"schema_version":"1.0","source":{"id":"2606.17492","kind":"arxiv","version":1}},"canonical_sha256":"e4053657447a99f8cabe5159fb0f81feccd965bf349de78475006766f4a4306f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e4053657447a99f8cabe5159fb0f81feccd965bf349de78475006766f4a4306f","first_computed_at":"2026-06-19T16:10:14.348520Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:10:14.348520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YcZDWA5KWcggswQwajzVqHD9Y9WKC5j5qUIDW3MUbvC66w1oxk8+cD0QnprTHbL1f+3SHpvhBhGPuuYTksk/Dw==","signature_status":"signed_v1","signed_at":"2026-06-19T16:10:14.348887Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.17492","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8fabeeaf1e1712edad85d7da9c3199f4c552dc46effe534a39f1daba566d919","sha256:052a8006b17fd245915a4396cdbf889647ed88e888d7a5c894013707e8025cbf"],"state_sha256":"c14d2f48b60f6fa29d466dea8dc00cefc8652f19c72d6a6e1ff8bae501b5fd29"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PmYUIibeJge5OEkiumlUo57juDR4B9XZn7TdiPzhUW5bzLufhiw8vVKVw+Mtj6qSxmEl526yrhXoylcYw1h5AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T07:57:30.491706Z","bundle_sha256":"cf903bda9af04e3ec45fbcb2f32f74b27d0d3c1cd8d57aa3e43dc99597d0ab38"}}