{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:QLPHKLEWVFA2FKVUEY7346AV3V","short_pith_number":"pith:QLPHKLEW","canonical_record":{"source":{"id":"1206.4540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-20T15:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"af3264b8bbe60652a85c0478d32e6227b1f5b7004b4fc8e50a7bd2a296a2bc97","abstract_canon_sha256":"c28f2f7bd4a53bc911f28bba034b4d6a8e5615bb8ca2a53586c28062c0769880"},"schema_version":"1.0"},"canonical_sha256":"82de752c96a941a2aab4263fbe7815dd4c1cf579d7b0975d6cdb064d8d473dea","source":{"kind":"arxiv","id":"1206.4540","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4540","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4540v1","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4540","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"pith_short_12","alias_value":"QLPHKLEWVFA2","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QLPHKLEWVFA2FKVU","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QLPHKLEW","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:QLPHKLEWVFA2FKVUEY7346AV3V","target":"record","payload":{"canonical_record":{"source":{"id":"1206.4540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-20T15:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"af3264b8bbe60652a85c0478d32e6227b1f5b7004b4fc8e50a7bd2a296a2bc97","abstract_canon_sha256":"c28f2f7bd4a53bc911f28bba034b4d6a8e5615bb8ca2a53586c28062c0769880"},"schema_version":"1.0"},"canonical_sha256":"82de752c96a941a2aab4263fbe7815dd4c1cf579d7b0975d6cdb064d8d473dea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:05.477391Z","signature_b64":"lpj64wXRrRdE1FvJlRlcxXBMYedVKU/XBpBhcCxTFPwm3HeZREyspM++x+eEGmcg6r96Xg7x6lqhA4CcKi4lBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82de752c96a941a2aab4263fbe7815dd4c1cf579d7b0975d6cdb064d8d473dea","last_reissued_at":"2026-05-18T03:53:05.476624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:05.476624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.4540","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":"2026-05-18T03:53:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q1NxDehVGsjsPGmshIDxNef2WPWzPuZYW+nxh2vxx//75HnYATzebS9Rtxt+UJ0BztEiQZ1J1KVDb3jw0o50Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T00:19:32.384232Z"},"content_sha256":"dc1e8497f3f6578f8ce4885f8ff8538809af9449b7d2d0c64c7d57b1fc238093","schema_version":"1.0","event_id":"sha256:dc1e8497f3f6578f8ce4885f8ff8538809af9449b7d2d0c64c7d57b1fc238093"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:QLPHKLEWVFA2FKVUEY7346AV3V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Analysis of the Hodge Laplacian on the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Detlef M\\\"uller, Fulvio Ricci, Marco M. Peloso","submitted_at":"2012-06-20T15:37:23Z","abstract_excerpt":"We consider the Hodge Laplacian $\\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant and U(n)-invariant Riemannian metric. For $0\\le k\\le 2n+1$, let $\\Delta_k$ denote the Hodge Laplacian restricted to $k$-forms.\n  Our first main result shows that $L^2\\Lambda^k(H_n)$ decomposes into finitely many mutually orthogonal subspaces $\\V_\\nu$ with the properties: {itemize} $\\dom \\Delta_k$ splits along the $\\V_\\nu$'s as $\\sum_\\nu(\\dom\\Delta_k\\cap \\V_\\nu)$; $\\Delta_k:(\\dom\\Delta_k\\cap \\V_\\nu)\\longrightarrow \\V_\\nu$ for every $\\nu$; for each $\\nu$, there is a Hilbert space $\\cH_\\nu$ of $L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4540","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":""},"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":"2026-05-18T03:53:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nCyxSh6U1cvQAs6TKZ1P0XWRAe5w/T5DKIQJlOdiQu8FEH9xZB6O7YVQRHNUHEmmhwRn9wtA4BbieXH0hrzBCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T00:19:32.384921Z"},"content_sha256":"55d8b808c9b0ad1bcd1e16390d8c6d57eda2b7ffa93fb2f5838b5f9c0c794a31","schema_version":"1.0","event_id":"sha256:55d8b808c9b0ad1bcd1e16390d8c6d57eda2b7ffa93fb2f5838b5f9c0c794a31"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QLPHKLEWVFA2FKVUEY7346AV3V/bundle.json","state_url":"https://pith.science/pith/QLPHKLEWVFA2FKVUEY7346AV3V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QLPHKLEWVFA2FKVUEY7346AV3V/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":"2026-06-07T00:19:32Z","links":{"resolver":"https://pith.science/pith/QLPHKLEWVFA2FKVUEY7346AV3V","bundle":"https://pith.science/pith/QLPHKLEWVFA2FKVUEY7346AV3V/bundle.json","state":"https://pith.science/pith/QLPHKLEWVFA2FKVUEY7346AV3V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QLPHKLEWVFA2FKVUEY7346AV3V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QLPHKLEWVFA2FKVUEY7346AV3V","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":"c28f2f7bd4a53bc911f28bba034b4d6a8e5615bb8ca2a53586c28062c0769880","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-20T15:37:23Z","title_canon_sha256":"af3264b8bbe60652a85c0478d32e6227b1f5b7004b4fc8e50a7bd2a296a2bc97"},"schema_version":"1.0","source":{"id":"1206.4540","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4540","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4540v1","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4540","created_at":"2026-05-18T03:53:05Z"},{"alias_kind":"pith_short_12","alias_value":"QLPHKLEWVFA2","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QLPHKLEWVFA2FKVU","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QLPHKLEW","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:55d8b808c9b0ad1bcd1e16390d8c6d57eda2b7ffa93fb2f5838b5f9c0c794a31","target":"graph","created_at":"2026-05-18T03:53:05Z","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"},"paper":{"abstract_excerpt":"We consider the Hodge Laplacian $\\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant and U(n)-invariant Riemannian metric. For $0\\le k\\le 2n+1$, let $\\Delta_k$ denote the Hodge Laplacian restricted to $k$-forms.\n  Our first main result shows that $L^2\\Lambda^k(H_n)$ decomposes into finitely many mutually orthogonal subspaces $\\V_\\nu$ with the properties: {itemize} $\\dom \\Delta_k$ splits along the $\\V_\\nu$'s as $\\sum_\\nu(\\dom\\Delta_k\\cap \\V_\\nu)$; $\\Delta_k:(\\dom\\Delta_k\\cap \\V_\\nu)\\longrightarrow \\V_\\nu$ for every $\\nu$; for each $\\nu$, there is a Hilbert space $\\cH_\\nu$ of $L","authors_text":"Detlef M\\\"uller, Fulvio Ricci, Marco M. Peloso","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-20T15:37:23Z","title":"Analysis of the Hodge Laplacian on the Heisenberg group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4540","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:dc1e8497f3f6578f8ce4885f8ff8538809af9449b7d2d0c64c7d57b1fc238093","target":"record","created_at":"2026-05-18T03:53:05Z","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":"c28f2f7bd4a53bc911f28bba034b4d6a8e5615bb8ca2a53586c28062c0769880","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-20T15:37:23Z","title_canon_sha256":"af3264b8bbe60652a85c0478d32e6227b1f5b7004b4fc8e50a7bd2a296a2bc97"},"schema_version":"1.0","source":{"id":"1206.4540","kind":"arxiv","version":1}},"canonical_sha256":"82de752c96a941a2aab4263fbe7815dd4c1cf579d7b0975d6cdb064d8d473dea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82de752c96a941a2aab4263fbe7815dd4c1cf579d7b0975d6cdb064d8d473dea","first_computed_at":"2026-05-18T03:53:05.476624Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:05.476624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lpj64wXRrRdE1FvJlRlcxXBMYedVKU/XBpBhcCxTFPwm3HeZREyspM++x+eEGmcg6r96Xg7x6lqhA4CcKi4lBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:05.477391Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.4540","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc1e8497f3f6578f8ce4885f8ff8538809af9449b7d2d0c64c7d57b1fc238093","sha256:55d8b808c9b0ad1bcd1e16390d8c6d57eda2b7ffa93fb2f5838b5f9c0c794a31"],"state_sha256":"f0573db6262308608079cdff27340d5951b7e3fe37e249719aa5c057f652997e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZX1E6XFRpbbtWvgnhUY3C2cqKcYdvFXH/y8FtFdjecIziCi8QH9hQw0QI/M4qdbzqhRP9gzOWhU/GSB+Mh62CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T00:19:32.388966Z","bundle_sha256":"2e7b078044955840f74c5ef61802bbaa9a420c5e638906ff1d09c1a0d8d5233c"}}