{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:BRVKKCPHNZWRBYI3XZABL4QMDW","short_pith_number":"pith:BRVKKCPH","canonical_record":{"source":{"id":"2606.03778","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-02T15:32:15Z","cross_cats_sorted":[],"title_canon_sha256":"9ea5d74bd7fd9318574c8acf4fe44b9a174e89b9753b3a5c0f3f928f8a96fb5a","abstract_canon_sha256":"c75f865721c5504e0a949d118110f93dd076e417c8c96707e800107e83177111"},"schema_version":"1.0"},"canonical_sha256":"0c6aa509e76e6d10e11bbe4015f20c1d9454ea2109b643a021536beea0e6ba06","source":{"kind":"arxiv","id":"2606.03778","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.03778","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"arxiv_version","alias_value":"2606.03778v1","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.03778","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"pith_short_12","alias_value":"BRVKKCPHNZWR","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"pith_short_16","alias_value":"BRVKKCPHNZWRBYI3","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"pith_short_8","alias_value":"BRVKKCPH","created_at":"2026-06-03T02:06:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:BRVKKCPHNZWRBYI3XZABL4QMDW","target":"record","payload":{"canonical_record":{"source":{"id":"2606.03778","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-02T15:32:15Z","cross_cats_sorted":[],"title_canon_sha256":"9ea5d74bd7fd9318574c8acf4fe44b9a174e89b9753b3a5c0f3f928f8a96fb5a","abstract_canon_sha256":"c75f865721c5504e0a949d118110f93dd076e417c8c96707e800107e83177111"},"schema_version":"1.0"},"canonical_sha256":"0c6aa509e76e6d10e11bbe4015f20c1d9454ea2109b643a021536beea0e6ba06","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T02:06:02.203270Z","signature_b64":"y4a8pxa8/L+B5X6TfJ9DpswPW1+bpvqOf/Nem4l6eWDGkMMCwt6ven0kVZITIiEYpJkjoFfeblzKzxkOa7pPAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c6aa509e76e6d10e11bbe4015f20c1d9454ea2109b643a021536beea0e6ba06","last_reissued_at":"2026-06-03T02:06:02.202849Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T02:06:02.202849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.03778","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":"2026-06-03T02:06:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tFzniO/ZAEQUWCWFH+YTwvZgCzI1cbIgZpRsF5WpN3jOn9l3ui6LKMpBLsAPyXbv9fUsrp/r9AWa2n1VaaudBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T13:25:49.009846Z"},"content_sha256":"52210149564f2233ace34d101d9934b946147547a91f2c59677f34dab65a231f","schema_version":"1.0","event_id":"sha256:52210149564f2233ace34d101d9934b946147547a91f2c59677f34dab65a231f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:BRVKKCPHNZWRBYI3XZABL4QMDW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Fontaine operator at cusps of modular curves at infinite level","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tian Qiu","submitted_at":"2026-06-02T15:32:15Z","abstract_excerpt":"We explicitly calculate Pan's geometric intertwining operator and the Fontaine operator on modular curves at infinite level via $q$-expansions, using Heuer's theory of cusps at infinite level. We prove that these two operators coincide on such expansions up to an explicit constant. As an application, we combine this result with $q$-expansion principles to provide a new proof of Pan's theorem that these operators are equal on the locally analytic vectors of completed cohomology of modular curves."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03778","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/2606.03778/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":"2026-06-03T02:06:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iBOhkzAcmTkIfjGM1m7ilPtIAUJ/3pWFUXYZyQhfQlrqkhE8URwcVrZunl1lhuUJBUrrdPoMgpF0cJvDSoJtDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T13:25:49.010234Z"},"content_sha256":"6a996b1b244ef785b521c73af8adc5a2ba1ab418ee725e06d29b6bc8ea523820","schema_version":"1.0","event_id":"sha256:6a996b1b244ef785b521c73af8adc5a2ba1ab418ee725e06d29b6bc8ea523820"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BRVKKCPHNZWRBYI3XZABL4QMDW/bundle.json","state_url":"https://pith.science/pith/BRVKKCPHNZWRBYI3XZABL4QMDW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BRVKKCPHNZWRBYI3XZABL4QMDW/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":"2026-06-11T13:25:49Z","links":{"resolver":"https://pith.science/pith/BRVKKCPHNZWRBYI3XZABL4QMDW","bundle":"https://pith.science/pith/BRVKKCPHNZWRBYI3XZABL4QMDW/bundle.json","state":"https://pith.science/pith/BRVKKCPHNZWRBYI3XZABL4QMDW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BRVKKCPHNZWRBYI3XZABL4QMDW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BRVKKCPHNZWRBYI3XZABL4QMDW","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":"c75f865721c5504e0a949d118110f93dd076e417c8c96707e800107e83177111","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-02T15:32:15Z","title_canon_sha256":"9ea5d74bd7fd9318574c8acf4fe44b9a174e89b9753b3a5c0f3f928f8a96fb5a"},"schema_version":"1.0","source":{"id":"2606.03778","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.03778","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"arxiv_version","alias_value":"2606.03778v1","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.03778","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"pith_short_12","alias_value":"BRVKKCPHNZWR","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"pith_short_16","alias_value":"BRVKKCPHNZWRBYI3","created_at":"2026-06-03T02:06:02Z"},{"alias_kind":"pith_short_8","alias_value":"BRVKKCPH","created_at":"2026-06-03T02:06:02Z"}],"graph_snapshots":[{"event_id":"sha256:6a996b1b244ef785b521c73af8adc5a2ba1ab418ee725e06d29b6bc8ea523820","target":"graph","created_at":"2026-06-03T02:06:02Z","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/2606.03778/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We explicitly calculate Pan's geometric intertwining operator and the Fontaine operator on modular curves at infinite level via $q$-expansions, using Heuer's theory of cusps at infinite level. We prove that these two operators coincide on such expansions up to an explicit constant. As an application, we combine this result with $q$-expansion principles to provide a new proof of Pan's theorem that these operators are equal on the locally analytic vectors of completed cohomology of modular curves.","authors_text":"Tian Qiu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-02T15:32:15Z","title":"The Fontaine operator at cusps of modular curves at infinite level"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03778","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":"sha256:52210149564f2233ace34d101d9934b946147547a91f2c59677f34dab65a231f","target":"record","created_at":"2026-06-03T02:06:02Z","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":"c75f865721c5504e0a949d118110f93dd076e417c8c96707e800107e83177111","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-02T15:32:15Z","title_canon_sha256":"9ea5d74bd7fd9318574c8acf4fe44b9a174e89b9753b3a5c0f3f928f8a96fb5a"},"schema_version":"1.0","source":{"id":"2606.03778","kind":"arxiv","version":1}},"canonical_sha256":"0c6aa509e76e6d10e11bbe4015f20c1d9454ea2109b643a021536beea0e6ba06","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c6aa509e76e6d10e11bbe4015f20c1d9454ea2109b643a021536beea0e6ba06","first_computed_at":"2026-06-03T02:06:02.202849Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T02:06:02.202849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y4a8pxa8/L+B5X6TfJ9DpswPW1+bpvqOf/Nem4l6eWDGkMMCwt6ven0kVZITIiEYpJkjoFfeblzKzxkOa7pPAQ==","signature_status":"signed_v1","signed_at":"2026-06-03T02:06:02.203270Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.03778","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52210149564f2233ace34d101d9934b946147547a91f2c59677f34dab65a231f","sha256:6a996b1b244ef785b521c73af8adc5a2ba1ab418ee725e06d29b6bc8ea523820"],"state_sha256":"49128596ca24e827a939a8d3e9c17fa96ad7605b8a4836811cbcaf924b15e06b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NnZqRX5YJLNfsh4TV+2CJuEwGbM/7h5cWfBDwrUpaOYxqY6gXbDgRdZZ93CkPCqrNLAbyCigw3QshutSYscoCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T13:25:49.012425Z","bundle_sha256":"5d06ee685bca17049df35d6db70838a094d2a92fe85949ebff3a4e937f7c3ca5"}}