{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:SXCITAEMRXDBSCME67CNNG6V47","short_pith_number":"pith:SXCITAEM","canonical_record":{"source":{"id":"2607.15213","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:17:12Z","cross_cats_sorted":["math.AG","math.MG","math.NT"],"title_canon_sha256":"26c983062e722bd397a0c50443413270c42986ff88c960c3ce9eccf605a87a10","abstract_canon_sha256":"c4b34cf107dfcdd8b6cbbc5237d0e3d0e9fb8da45ed97b45d2a1142f5c54850f"},"schema_version":"1.0"},"canonical_sha256":"95c489808c8dc6190984f7c4d69bd5e7f6f579ca297acb9db930749a12db64f5","source":{"kind":"arxiv","id":"2607.15213","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.15213","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"arxiv_version","alias_value":"2607.15213v1","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.15213","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"pith_short_12","alias_value":"SXCITAEMRXDB","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"pith_short_16","alias_value":"SXCITAEMRXDBSCME","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"pith_short_8","alias_value":"SXCITAEM","created_at":"2026-07-17T01:22:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:SXCITAEMRXDBSCME67CNNG6V47","target":"record","payload":{"canonical_record":{"source":{"id":"2607.15213","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:17:12Z","cross_cats_sorted":["math.AG","math.MG","math.NT"],"title_canon_sha256":"26c983062e722bd397a0c50443413270c42986ff88c960c3ce9eccf605a87a10","abstract_canon_sha256":"c4b34cf107dfcdd8b6cbbc5237d0e3d0e9fb8da45ed97b45d2a1142f5c54850f"},"schema_version":"1.0"},"canonical_sha256":"95c489808c8dc6190984f7c4d69bd5e7f6f579ca297acb9db930749a12db64f5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-17T01:22:14.851764Z","signature_b64":"afrN2cx7IX1fawftYP3Q2blQiY84f6xysxMTVeID0DaU0bxBU540iwq4ogpdQgHBrCqGchffiVXur5oX28rvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95c489808c8dc6190984f7c4d69bd5e7f6f579ca297acb9db930749a12db64f5","last_reissued_at":"2026-07-17T01:22:14.850969Z","signature_status":"signed_v1","first_computed_at":"2026-07-17T01:22:14.850969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.15213","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":"2026-07-17T01:22:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TVN2VjBZoA7LeoMlulEGyoUGi4eVICe4+mTWYUR6lt2c6ckaFb6uXbdTBSzgXbkzBZUJpCEAprgpJHweZ4iMAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T18:28:33.142070Z"},"content_sha256":"05861e6b3382812d8280e5e97f7abbe2d63bc3cb56cb1a2efb42c07f3468f06e","schema_version":"1.0","event_id":"sha256:05861e6b3382812d8280e5e97f7abbe2d63bc3cb56cb1a2efb42c07f3468f06e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:SXCITAEMRXDBSCME67CNNG6V47","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic Brill-Noether Existence at the Half-Canonical Degree: Energy Pairing, Cheeger Inequality and Covering Radii","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.MG","math.NT"],"primary_cat":"math.CO","authors_text":"Madhusudan Manjunath","submitted_at":"2026-07-16T17:17:12Z","abstract_excerpt":"We study asymptotic versions of the Brill-Noether existence conjecture on graphs via techniques inspired by the geometry of numbers. We confirm an asymptotic version of the conjecture at (and near) the half-canonical degree in several well-connected families of graphs. They include expander graphs of even valence, almost-Ramanujan graphs of a fixed valence at least five and certain random graphs. In particular, for any fixed $k \\geq 5$, almost all simple, connected, $k$-regular graphs satisfy the Brill-Noether existence conjecture at the half-canonical degree up to a constant factor. The key t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15213","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/2607.15213/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":"2026-07-17T01:22:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"066NW8Hd1iRnzf5WhGnkLN5lx9N6BC02XV2J+9Q+VjD4a3E3U+Tj1xJ20+GvHXS2lw/9dVb7VLA3mGqWY0xeBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T18:28:33.142663Z"},"content_sha256":"c97c787bad9c28a88845a76fbe4844d638f63ea011e9c60e9b680c59a6a02bae","schema_version":"1.0","event_id":"sha256:c97c787bad9c28a88845a76fbe4844d638f63ea011e9c60e9b680c59a6a02bae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SXCITAEMRXDBSCME67CNNG6V47/bundle.json","state_url":"https://pith.science/pith/SXCITAEMRXDBSCME67CNNG6V47/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SXCITAEMRXDBSCME67CNNG6V47/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":"2026-08-05T18:28:33Z","links":{"resolver":"https://pith.science/pith/SXCITAEMRXDBSCME67CNNG6V47","bundle":"https://pith.science/pith/SXCITAEMRXDBSCME67CNNG6V47/bundle.json","state":"https://pith.science/pith/SXCITAEMRXDBSCME67CNNG6V47/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SXCITAEMRXDBSCME67CNNG6V47/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:SXCITAEMRXDBSCME67CNNG6V47","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":"c4b34cf107dfcdd8b6cbbc5237d0e3d0e9fb8da45ed97b45d2a1142f5c54850f","cross_cats_sorted":["math.AG","math.MG","math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:17:12Z","title_canon_sha256":"26c983062e722bd397a0c50443413270c42986ff88c960c3ce9eccf605a87a10"},"schema_version":"1.0","source":{"id":"2607.15213","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.15213","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"arxiv_version","alias_value":"2607.15213v1","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.15213","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"pith_short_12","alias_value":"SXCITAEMRXDB","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"pith_short_16","alias_value":"SXCITAEMRXDBSCME","created_at":"2026-07-17T01:22:14Z"},{"alias_kind":"pith_short_8","alias_value":"SXCITAEM","created_at":"2026-07-17T01:22:14Z"}],"graph_snapshots":[{"event_id":"sha256:c97c787bad9c28a88845a76fbe4844d638f63ea011e9c60e9b680c59a6a02bae","target":"graph","created_at":"2026-07-17T01:22:14Z","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/2607.15213/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study asymptotic versions of the Brill-Noether existence conjecture on graphs via techniques inspired by the geometry of numbers. We confirm an asymptotic version of the conjecture at (and near) the half-canonical degree in several well-connected families of graphs. They include expander graphs of even valence, almost-Ramanujan graphs of a fixed valence at least five and certain random graphs. In particular, for any fixed $k \\geq 5$, almost all simple, connected, $k$-regular graphs satisfy the Brill-Noether existence conjecture at the half-canonical degree up to a constant factor. The key t","authors_text":"Madhusudan Manjunath","cross_cats":["math.AG","math.MG","math.NT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:17:12Z","title":"Asymptotic Brill-Noether Existence at the Half-Canonical Degree: Energy Pairing, Cheeger Inequality and Covering Radii"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15213","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":"sha256:05861e6b3382812d8280e5e97f7abbe2d63bc3cb56cb1a2efb42c07f3468f06e","target":"record","created_at":"2026-07-17T01:22:14Z","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":"c4b34cf107dfcdd8b6cbbc5237d0e3d0e9fb8da45ed97b45d2a1142f5c54850f","cross_cats_sorted":["math.AG","math.MG","math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:17:12Z","title_canon_sha256":"26c983062e722bd397a0c50443413270c42986ff88c960c3ce9eccf605a87a10"},"schema_version":"1.0","source":{"id":"2607.15213","kind":"arxiv","version":1}},"canonical_sha256":"95c489808c8dc6190984f7c4d69bd5e7f6f579ca297acb9db930749a12db64f5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95c489808c8dc6190984f7c4d69bd5e7f6f579ca297acb9db930749a12db64f5","first_computed_at":"2026-07-17T01:22:14.850969Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-17T01:22:14.850969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"afrN2cx7IX1fawftYP3Q2blQiY84f6xysxMTVeID0DaU0bxBU540iwq4ogpdQgHBrCqGchffiVXur5oX28rvAA==","signature_status":"signed_v1","signed_at":"2026-07-17T01:22:14.851764Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.15213","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:05861e6b3382812d8280e5e97f7abbe2d63bc3cb56cb1a2efb42c07f3468f06e","sha256:c97c787bad9c28a88845a76fbe4844d638f63ea011e9c60e9b680c59a6a02bae"],"state_sha256":"1eae4288767cc46bdff83b6b063427f4217ed072b47cf1071c97f52fe270f453"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N7ow/igIdQng87OFbb7QlmvD7BSANAW3HH34JuaM+19yWjvZAVn8pa+n1oKm8uOpAH98dQRXJLR3bB9N73Z5Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T18:28:33.149812Z","bundle_sha256":"d04aa017f0a0c278198cf2e25fab663691a3d20a9cab57eaa9170adaa452603d"}}