{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:TO772IVNF5QJFUAYQDDPP6MQER","short_pith_number":"pith:TO772IVN","canonical_record":{"source":{"id":"2507.03782","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-04T19:16:02Z","cross_cats_sorted":["math.AP","math.DG","math.FA"],"title_canon_sha256":"70135c3bb9a803e53050958b9d4853e7b6128d8fab1ee83e41e0137003748d3c","abstract_canon_sha256":"144aad6da9a13b48039e807a96678712c2eb15b6f1a471997c8b2d77a51e6e67"},"schema_version":"1.0"},"canonical_sha256":"9bbffd22ad2f6092d01880c6f7f99024663d073a171d3f527680678039cf85b0","source":{"kind":"arxiv","id":"2507.03782","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.03782","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"arxiv_version","alias_value":"2507.03782v1","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.03782","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"pith_short_12","alias_value":"TO772IVNF5QJ","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"pith_short_16","alias_value":"TO772IVNF5QJFUAY","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"pith_short_8","alias_value":"TO772IVN","created_at":"2026-07-05T11:32:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:TO772IVNF5QJFUAYQDDPP6MQER","target":"record","payload":{"canonical_record":{"source":{"id":"2507.03782","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-04T19:16:02Z","cross_cats_sorted":["math.AP","math.DG","math.FA"],"title_canon_sha256":"70135c3bb9a803e53050958b9d4853e7b6128d8fab1ee83e41e0137003748d3c","abstract_canon_sha256":"144aad6da9a13b48039e807a96678712c2eb15b6f1a471997c8b2d77a51e6e67"},"schema_version":"1.0"},"canonical_sha256":"9bbffd22ad2f6092d01880c6f7f99024663d073a171d3f527680678039cf85b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:32:34.331226Z","signature_b64":"iqf6LjTEuXMHDTnbQ9/utxAbZSKdKPoTUfUB6MjebqPF9bTFgVZ+11aushlqEF204KTNhNHq3Tfl3au0b6y8AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bbffd22ad2f6092d01880c6f7f99024663d073a171d3f527680678039cf85b0","last_reissued_at":"2026-07-05T11:32:34.330656Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:32:34.330656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.03782","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":"2026-07-05T11:32:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4eTp98XZHJoT+GiG1x5fUSwrBOkzhBMUphE6rJO2DQH/hNS4XowjljZtI8izoRAEwBDqoihFZX0nzMpO1vx7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T15:49:45.119929Z"},"content_sha256":"eca8283e0761411ba74e46a4e296b50c504b05f3c004be8d409830a050de0c3f","schema_version":"1.0","event_id":"sha256:eca8283e0761411ba74e46a4e296b50c504b05f3c004be8d409830a050de0c3f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:TO772IVNF5QJFUAYQDDPP6MQER","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hilbert manifold structures on path spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG","math.FA"],"primary_cat":"math.SG","authors_text":"Joa Weber, Urs Frauenfelder","submitted_at":"2025-07-04T19:16:02Z","abstract_excerpt":"In Floer theory one has to deal with two-level manifolds like for instance the space of $W^{2,2}$ loops and the space of $W^{1,2}$ loops. Gradient flow lines in Floer theory are then trajectories in a two-level manifold. Inspired by our endeavor to find a general setup to construct Floer homology we therefore address in this paper the question if the space of paths on a two-level manifold has itself the structure of a Hilbert manifold. In view of the two topologies on a two-level manifold it is unclear how to define the exponential map on a general two-level manifold. We therefore study a diff"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.03782","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/2507.03782/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":"2026-07-05T11:32:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pqtvwcheL4drRAcC15hKnDWKn8V0+80uKkqGcGvCcLReZOmNSY+Lu5l6w0WrE2xli2Z80xNh31Xhu3DttnH8Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T15:49:45.127010Z"},"content_sha256":"7632ceca6b85f9c57d4d8b40848cfaf54298719762b3a690399a571cf58e2e08","schema_version":"1.0","event_id":"sha256:7632ceca6b85f9c57d4d8b40848cfaf54298719762b3a690399a571cf58e2e08"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TO772IVNF5QJFUAYQDDPP6MQER/bundle.json","state_url":"https://pith.science/pith/TO772IVNF5QJFUAYQDDPP6MQER/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TO772IVNF5QJFUAYQDDPP6MQER/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":"2026-08-10T15:49:45Z","links":{"resolver":"https://pith.science/pith/TO772IVNF5QJFUAYQDDPP6MQER","bundle":"https://pith.science/pith/TO772IVNF5QJFUAYQDDPP6MQER/bundle.json","state":"https://pith.science/pith/TO772IVNF5QJFUAYQDDPP6MQER/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TO772IVNF5QJFUAYQDDPP6MQER/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:TO772IVNF5QJFUAYQDDPP6MQER","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":"144aad6da9a13b48039e807a96678712c2eb15b6f1a471997c8b2d77a51e6e67","cross_cats_sorted":["math.AP","math.DG","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-04T19:16:02Z","title_canon_sha256":"70135c3bb9a803e53050958b9d4853e7b6128d8fab1ee83e41e0137003748d3c"},"schema_version":"1.0","source":{"id":"2507.03782","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.03782","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"arxiv_version","alias_value":"2507.03782v1","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.03782","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"pith_short_12","alias_value":"TO772IVNF5QJ","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"pith_short_16","alias_value":"TO772IVNF5QJFUAY","created_at":"2026-07-05T11:32:34Z"},{"alias_kind":"pith_short_8","alias_value":"TO772IVN","created_at":"2026-07-05T11:32:34Z"}],"graph_snapshots":[{"event_id":"sha256:7632ceca6b85f9c57d4d8b40848cfaf54298719762b3a690399a571cf58e2e08","target":"graph","created_at":"2026-07-05T11:32:34Z","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/2507.03782/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In Floer theory one has to deal with two-level manifolds like for instance the space of $W^{2,2}$ loops and the space of $W^{1,2}$ loops. Gradient flow lines in Floer theory are then trajectories in a two-level manifold. Inspired by our endeavor to find a general setup to construct Floer homology we therefore address in this paper the question if the space of paths on a two-level manifold has itself the structure of a Hilbert manifold. In view of the two topologies on a two-level manifold it is unclear how to define the exponential map on a general two-level manifold. We therefore study a diff","authors_text":"Joa Weber, Urs Frauenfelder","cross_cats":["math.AP","math.DG","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-04T19:16:02Z","title":"Hilbert manifold structures on path spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.03782","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:eca8283e0761411ba74e46a4e296b50c504b05f3c004be8d409830a050de0c3f","target":"record","created_at":"2026-07-05T11:32:34Z","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":"144aad6da9a13b48039e807a96678712c2eb15b6f1a471997c8b2d77a51e6e67","cross_cats_sorted":["math.AP","math.DG","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-04T19:16:02Z","title_canon_sha256":"70135c3bb9a803e53050958b9d4853e7b6128d8fab1ee83e41e0137003748d3c"},"schema_version":"1.0","source":{"id":"2507.03782","kind":"arxiv","version":1}},"canonical_sha256":"9bbffd22ad2f6092d01880c6f7f99024663d073a171d3f527680678039cf85b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9bbffd22ad2f6092d01880c6f7f99024663d073a171d3f527680678039cf85b0","first_computed_at":"2026-07-05T11:32:34.330656Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:32:34.330656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iqf6LjTEuXMHDTnbQ9/utxAbZSKdKPoTUfUB6MjebqPF9bTFgVZ+11aushlqEF204KTNhNHq3Tfl3au0b6y8AA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:32:34.331226Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.03782","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eca8283e0761411ba74e46a4e296b50c504b05f3c004be8d409830a050de0c3f","sha256:7632ceca6b85f9c57d4d8b40848cfaf54298719762b3a690399a571cf58e2e08"],"state_sha256":"4baca0275a676bb8d4358c9508a7a682770550d9bbdff90c0d460833aaef92ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KxFbB+dsPuvBjxzIJHUjjKuOYi3rceF31WXE2ohLoHIKKKWA3AGEo14C/3G8O0VUYsEgzV2rYCVOVAD6GTE6AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T15:49:45.181547Z","bundle_sha256":"866736bcbe4fa2109ff507f2310d754711056fcdac4f0eee90c030ce37583b4e"}}