{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GNMTWPERPUGVN5LIVXWRL2QVH3","short_pith_number":"pith:GNMTWPER","canonical_record":{"source":{"id":"2606.24532","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T13:01:58Z","cross_cats_sorted":["math.AG","math.KT"],"title_canon_sha256":"6e2e94daf431a9d2a7d1c2685338abb3fecbab40f7f287b630d6dd654ca1d46c","abstract_canon_sha256":"82852a75cd1b675ed9c58ff9bfd66dbbee61e6e8abd52ef32f9e741c7b61dfbd"},"schema_version":"1.0"},"canonical_sha256":"33593b3c917d0d56f568aded15ea153ee9da3116817650e81156ae19b0560bd2","source":{"kind":"arxiv","id":"2606.24532","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24532","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24532v1","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24532","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_12","alias_value":"GNMTWPERPUGV","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_16","alias_value":"GNMTWPERPUGVN5LI","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_8","alias_value":"GNMTWPER","created_at":"2026-06-24T01:15:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GNMTWPERPUGVN5LIVXWRL2QVH3","target":"record","payload":{"canonical_record":{"source":{"id":"2606.24532","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T13:01:58Z","cross_cats_sorted":["math.AG","math.KT"],"title_canon_sha256":"6e2e94daf431a9d2a7d1c2685338abb3fecbab40f7f287b630d6dd654ca1d46c","abstract_canon_sha256":"82852a75cd1b675ed9c58ff9bfd66dbbee61e6e8abd52ef32f9e741c7b61dfbd"},"schema_version":"1.0"},"canonical_sha256":"33593b3c917d0d56f568aded15ea153ee9da3116817650e81156ae19b0560bd2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:15:33.179865Z","signature_b64":"uja99k/5vVk/lrrsa6Lqk5dcc5OjFdFT0w9FSWVZV3ATwgAhK6hGJgOGGP5nZPz/1VQLMMlRnwTkFqmjhGOmDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33593b3c917d0d56f568aded15ea153ee9da3116817650e81156ae19b0560bd2","last_reissued_at":"2026-06-24T01:15:33.179522Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:15:33.179522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.24532","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":"2026-06-24T01:15:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YxOSJZ9uuayPsKkqesSG5kSrgiBgwKijlSu2DyAhmUqlU5nkBhs4DX8Yit2SEM+p/Vwidj9FeON/yOqjGcChDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T09:10:43.833570Z"},"content_sha256":"d211ffddd4645a6aef7407a685bceadaa9e2341148d3cd167998d4693112938e","schema_version":"1.0","event_id":"sha256:d211ffddd4645a6aef7407a685bceadaa9e2341148d3cd167998d4693112938e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GNMTWPERPUGVN5LIVXWRL2QVH3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Elements in $K_4$ and regulator maps of Fermat curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.KT"],"primary_cat":"math.NT","authors_text":"David T.-B. G. Lilienfeldt, Fran\\c{c}ois Brunault, Yusuke Nemoto","submitted_at":"2026-06-23T13:01:58Z","abstract_excerpt":"We construct explicit elements in the group $K_4^{(3)}$ of the Fermat curves $x^N+y^N=1$ for all $N\\geq 3$. The construction, which is uniform in $N$, uses polylogarithmic complexes and a map of de Jeu to $K$-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to $\\frac{3}{2}\\zeta(3)N^2$ as $N\\to +\\infty$. Moreover, we derive f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24532","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/2606.24532/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":"2026-06-24T01:15:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T8mh5SbQENjyHwnOp2XqQeyM3RpkeF5o61Z9U43B5GQ9GPnFKSUoal3E2CM9fZpe/7QuHeNbOsEC92AlBr2kCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T09:10:43.833951Z"},"content_sha256":"9b2d57730b5dd8a8169d503d7485ae808a3f7df2e014ee612c3fadb74e3ac6de","schema_version":"1.0","event_id":"sha256:9b2d57730b5dd8a8169d503d7485ae808a3f7df2e014ee612c3fadb74e3ac6de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GNMTWPERPUGVN5LIVXWRL2QVH3/bundle.json","state_url":"https://pith.science/pith/GNMTWPERPUGVN5LIVXWRL2QVH3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GNMTWPERPUGVN5LIVXWRL2QVH3/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":"2026-06-26T09:10:43Z","links":{"resolver":"https://pith.science/pith/GNMTWPERPUGVN5LIVXWRL2QVH3","bundle":"https://pith.science/pith/GNMTWPERPUGVN5LIVXWRL2QVH3/bundle.json","state":"https://pith.science/pith/GNMTWPERPUGVN5LIVXWRL2QVH3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GNMTWPERPUGVN5LIVXWRL2QVH3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GNMTWPERPUGVN5LIVXWRL2QVH3","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":"82852a75cd1b675ed9c58ff9bfd66dbbee61e6e8abd52ef32f9e741c7b61dfbd","cross_cats_sorted":["math.AG","math.KT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T13:01:58Z","title_canon_sha256":"6e2e94daf431a9d2a7d1c2685338abb3fecbab40f7f287b630d6dd654ca1d46c"},"schema_version":"1.0","source":{"id":"2606.24532","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24532","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24532v1","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24532","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_12","alias_value":"GNMTWPERPUGV","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_16","alias_value":"GNMTWPERPUGVN5LI","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_8","alias_value":"GNMTWPER","created_at":"2026-06-24T01:15:33Z"}],"graph_snapshots":[{"event_id":"sha256:9b2d57730b5dd8a8169d503d7485ae808a3f7df2e014ee612c3fadb74e3ac6de","target":"graph","created_at":"2026-06-24T01:15:33Z","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/2606.24532/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We construct explicit elements in the group $K_4^{(3)}$ of the Fermat curves $x^N+y^N=1$ for all $N\\geq 3$. The construction, which is uniform in $N$, uses polylogarithmic complexes and a map of de Jeu to $K$-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to $\\frac{3}{2}\\zeta(3)N^2$ as $N\\to +\\infty$. Moreover, we derive f","authors_text":"David T.-B. G. Lilienfeldt, Fran\\c{c}ois Brunault, Yusuke Nemoto","cross_cats":["math.AG","math.KT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T13:01:58Z","title":"Elements in $K_4$ and regulator maps of Fermat curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24532","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:d211ffddd4645a6aef7407a685bceadaa9e2341148d3cd167998d4693112938e","target":"record","created_at":"2026-06-24T01:15:33Z","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":"82852a75cd1b675ed9c58ff9bfd66dbbee61e6e8abd52ef32f9e741c7b61dfbd","cross_cats_sorted":["math.AG","math.KT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-23T13:01:58Z","title_canon_sha256":"6e2e94daf431a9d2a7d1c2685338abb3fecbab40f7f287b630d6dd654ca1d46c"},"schema_version":"1.0","source":{"id":"2606.24532","kind":"arxiv","version":1}},"canonical_sha256":"33593b3c917d0d56f568aded15ea153ee9da3116817650e81156ae19b0560bd2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33593b3c917d0d56f568aded15ea153ee9da3116817650e81156ae19b0560bd2","first_computed_at":"2026-06-24T01:15:33.179522Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-24T01:15:33.179522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uja99k/5vVk/lrrsa6Lqk5dcc5OjFdFT0w9FSWVZV3ATwgAhK6hGJgOGGP5nZPz/1VQLMMlRnwTkFqmjhGOmDg==","signature_status":"signed_v1","signed_at":"2026-06-24T01:15:33.179865Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.24532","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d211ffddd4645a6aef7407a685bceadaa9e2341148d3cd167998d4693112938e","sha256:9b2d57730b5dd8a8169d503d7485ae808a3f7df2e014ee612c3fadb74e3ac6de"],"state_sha256":"1d1f68dcc8f5375fa1b5b30ee728a790e4ef3e109ef8d0b74c390e0558173341"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BWXrzxov00IQywkEsLoS4cP+hMjb6OHGocCGcwMsqbnulI/kWwdZcTgPM934UWOPSWld7BVBdDnaXoImZzskAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T09:10:43.836045Z","bundle_sha256":"d1ef6ce07fd0eebf858cd9d0b79f4c0b294f2351dc2cf9c7baf3d57e13874d87"}}