{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:SP4JAFFHHRQJKAAULRJR5HJYQR","short_pith_number":"pith:SP4JAFFH","canonical_record":{"source":{"id":"0908.0688","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-05T15:31:38Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d1bc2b976d35031f7b3cdebe6c55b507d0722344dddfd49d1ed0498393b7d76d","abstract_canon_sha256":"69eaf461e0adbc46d2062bb9eb98aa8a5c1d96c75f03fe878539831186e40a17"},"schema_version":"1.0"},"canonical_sha256":"93f89014a73c609500145c531e9d388460b8d813211ad570cf7d485afc03e1b8","source":{"kind":"arxiv","id":"0908.0688","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.0688","created_at":"2026-05-18T03:35:26Z"},{"alias_kind":"arxiv_version","alias_value":"0908.0688v1","created_at":"2026-05-18T03:35:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.0688","created_at":"2026-05-18T03:35:26Z"},{"alias_kind":"pith_short_12","alias_value":"SP4JAFFHHRQJ","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SP4JAFFHHRQJKAAU","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SP4JAFFH","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:SP4JAFFHHRQJKAAULRJR5HJYQR","target":"record","payload":{"canonical_record":{"source":{"id":"0908.0688","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-05T15:31:38Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d1bc2b976d35031f7b3cdebe6c55b507d0722344dddfd49d1ed0498393b7d76d","abstract_canon_sha256":"69eaf461e0adbc46d2062bb9eb98aa8a5c1d96c75f03fe878539831186e40a17"},"schema_version":"1.0"},"canonical_sha256":"93f89014a73c609500145c531e9d388460b8d813211ad570cf7d485afc03e1b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:26.706630Z","signature_b64":"aCYgFF3EX7YIeyZWlkQxT/hrdFpjRfsUg7vW+fXatXljvN2InVmrC2TKrJvGjRUDRaDfKqYevOJXXbb2pdBKDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93f89014a73c609500145c531e9d388460b8d813211ad570cf7d485afc03e1b8","last_reissued_at":"2026-05-18T03:35:26.705788Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:26.705788Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0908.0688","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":"2026-05-18T03:35:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5r5wihR6rIyH1tjioyX/kjyOvirrmfv1RjrLUGsmnUJL4p7Dzsad4RCmXBcrnnY9M/oqr3ajiEzicV8hbPFqBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T04:38:25.088741Z"},"content_sha256":"71295202243b73aac3a29431c0017c07c0a476d7dc961a17b07089fa13bf3010","schema_version":"1.0","event_id":"sha256:71295202243b73aac3a29431c0017c07c0a476d7dc961a17b07089fa13bf3010"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:SP4JAFFHHRQJKAAULRJR5HJYQR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"About the blowup of quasimodes on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Christopher D. Sogge, John A. Toth, Steve Zelditch","submitted_at":"2009-08-05T15:31:38Z","abstract_excerpt":"On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\\phi_{\\lambda}}$ satisfy $||\\phi_{\\lambda}||_{\\infty} \\leq C \\lambda^{\\frac{n-1}{2}}$ where $-\\Delta \\phi_{\\lambda} = \\lambda^2 \\phi_{\\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\\R^n/\\Gamma$. We say that $S^n$, but not $\\R^n/\\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.0688","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":"2026-05-18T03:35:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FIRltrJvnivVjsDXt3YgrHaBZcW3XndNQGjg5sTDAYh+ZMt6kk3sLdXen6+JD8RxOI8qf/7TyvUKR/yIs60UAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T04:38:25.089096Z"},"content_sha256":"6c01b98f452e89353389161bad5a78965f26b3572e0da471f697d64a31181fcd","schema_version":"1.0","event_id":"sha256:6c01b98f452e89353389161bad5a78965f26b3572e0da471f697d64a31181fcd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SP4JAFFHHRQJKAAULRJR5HJYQR/bundle.json","state_url":"https://pith.science/pith/SP4JAFFHHRQJKAAULRJR5HJYQR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SP4JAFFHHRQJKAAULRJR5HJYQR/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":"2026-05-28T04:38:25Z","links":{"resolver":"https://pith.science/pith/SP4JAFFHHRQJKAAULRJR5HJYQR","bundle":"https://pith.science/pith/SP4JAFFHHRQJKAAULRJR5HJYQR/bundle.json","state":"https://pith.science/pith/SP4JAFFHHRQJKAAULRJR5HJYQR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SP4JAFFHHRQJKAAULRJR5HJYQR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:SP4JAFFHHRQJKAAULRJR5HJYQR","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":"69eaf461e0adbc46d2062bb9eb98aa8a5c1d96c75f03fe878539831186e40a17","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-05T15:31:38Z","title_canon_sha256":"d1bc2b976d35031f7b3cdebe6c55b507d0722344dddfd49d1ed0498393b7d76d"},"schema_version":"1.0","source":{"id":"0908.0688","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.0688","created_at":"2026-05-18T03:35:26Z"},{"alias_kind":"arxiv_version","alias_value":"0908.0688v1","created_at":"2026-05-18T03:35:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.0688","created_at":"2026-05-18T03:35:26Z"},{"alias_kind":"pith_short_12","alias_value":"SP4JAFFHHRQJ","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SP4JAFFHHRQJKAAU","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SP4JAFFH","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:6c01b98f452e89353389161bad5a78965f26b3572e0da471f697d64a31181fcd","target":"graph","created_at":"2026-05-18T03:35:26Z","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":"On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\\phi_{\\lambda}}$ satisfy $||\\phi_{\\lambda}||_{\\infty} \\leq C \\lambda^{\\frac{n-1}{2}}$ where $-\\Delta \\phi_{\\lambda} = \\lambda^2 \\phi_{\\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\\R^n/\\Gamma$. We say that $S^n$, but not $\\R^n/\\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates t","authors_text":"Christopher D. Sogge, John A. Toth, Steve Zelditch","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-05T15:31:38Z","title":"About the blowup of quasimodes on Riemannian manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.0688","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":"sha256:71295202243b73aac3a29431c0017c07c0a476d7dc961a17b07089fa13bf3010","target":"record","created_at":"2026-05-18T03:35:26Z","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":"69eaf461e0adbc46d2062bb9eb98aa8a5c1d96c75f03fe878539831186e40a17","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-08-05T15:31:38Z","title_canon_sha256":"d1bc2b976d35031f7b3cdebe6c55b507d0722344dddfd49d1ed0498393b7d76d"},"schema_version":"1.0","source":{"id":"0908.0688","kind":"arxiv","version":1}},"canonical_sha256":"93f89014a73c609500145c531e9d388460b8d813211ad570cf7d485afc03e1b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"93f89014a73c609500145c531e9d388460b8d813211ad570cf7d485afc03e1b8","first_computed_at":"2026-05-18T03:35:26.705788Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:35:26.705788Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aCYgFF3EX7YIeyZWlkQxT/hrdFpjRfsUg7vW+fXatXljvN2InVmrC2TKrJvGjRUDRaDfKqYevOJXXbb2pdBKDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:35:26.706630Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.0688","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71295202243b73aac3a29431c0017c07c0a476d7dc961a17b07089fa13bf3010","sha256:6c01b98f452e89353389161bad5a78965f26b3572e0da471f697d64a31181fcd"],"state_sha256":"a0d487e52b6167e7aa9b41c75a537199b89a49cae67415ec3ce3946f74b9b786"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9TNqspsZqTge5PwwOW2QTOD2bolpIJWvfDOvqsJR2z3qilRU0NUh6xufaLfK0uhXWjcusn5Diq8HvoziND/NBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T04:38:25.091159Z","bundle_sha256":"ab0bbc03f3b476ace3e0ff900dd7135d5d1dd883bee60e6466ac822b6d01c148"}}