{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:J7ZLWM2M5KWOMUWE3TLNOVZU72","short_pith_number":"pith:J7ZLWM2M","canonical_record":{"source":{"id":"2606.11159","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T17:41:34Z","cross_cats_sorted":[],"title_canon_sha256":"ee4a7243c8ad95282ed819a6a3859ccb0ab0d3af8ddbc3570e62f27c311bfa36","abstract_canon_sha256":"a1ee786403f1413f7f2614f4a38ce07685dc7529403ec53f9cd9f996c524b8b7"},"schema_version":"1.0"},"canonical_sha256":"4ff2bb334ceaace652c4dcd6d75734fe95cc93cd5aec597e2666d16b4c87846f","source":{"kind":"arxiv","id":"2606.11159","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11159","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11159v1","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11159","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"pith_short_12","alias_value":"J7ZLWM2M5KWO","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"pith_short_16","alias_value":"J7ZLWM2M5KWOMUWE","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"pith_short_8","alias_value":"J7ZLWM2M","created_at":"2026-06-10T01:11:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:J7ZLWM2M5KWOMUWE3TLNOVZU72","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11159","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T17:41:34Z","cross_cats_sorted":[],"title_canon_sha256":"ee4a7243c8ad95282ed819a6a3859ccb0ab0d3af8ddbc3570e62f27c311bfa36","abstract_canon_sha256":"a1ee786403f1413f7f2614f4a38ce07685dc7529403ec53f9cd9f996c524b8b7"},"schema_version":"1.0"},"canonical_sha256":"4ff2bb334ceaace652c4dcd6d75734fe95cc93cd5aec597e2666d16b4c87846f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:11:14.522304Z","signature_b64":"jVoeeEvVO7kHh7ABa4J2vL6JYkyDTcrNHSCAYQUgqXoHxAw/AXenpaUI2suqg+IChbZu1BD2EuoylZJexcZuCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ff2bb334ceaace652c4dcd6d75734fe95cc93cd5aec597e2666d16b4c87846f","last_reissued_at":"2026-06-10T01:11:14.521442Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:11:14.521442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11159","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":"2026-06-10T01:11:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Orf6hucv/fhzLeaI19IxUkMxx4xzQ/bsXzlH36ubbMZ3DDybdajLSes5dYNuJ+lQZMP2oYcU9v3bKnewCtVRCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T13:12:09.736367Z"},"content_sha256":"80062f06252574589ab12fb6f755bb79148c898358735cb0b855a181d764f346","schema_version":"1.0","event_id":"sha256:80062f06252574589ab12fb6f755bb79148c898358735cb0b855a181d764f346"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:J7ZLWM2M5KWOMUWE3TLNOVZU72","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Unidirectional Entropic Solutions of the Pressureless Euler Alignment System","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Joshua O. Adeleke, Trevor M. Leslie","submitted_at":"2026-06-09T17:41:34Z","abstract_excerpt":"We study the pressureless Euler Alignment system with unidirectional velocity (u,0,...,0). By re-casting the system as a family of coupled scalar balance laws, one for each horizontal slice of R^d, we are able to prove existence, uniqueness, and stability within the class of unidirectional solutions, under the assumption of a bounded Lipschitz communication protocol. The nonlocal coupling between horizontal slices provides the system with a rich structure that is absent from the 1D setting and also constitutes the main technical difficulty of the present work. We construct our solutions as lim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11159","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/2606.11159/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":"2026-06-10T01:11:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"36h2Bz/OVvLHIcDECN32Qq7iDSi3w4MOtAUtlo91tJ/zSH86b2Ju2PXs4vO0zyI1QZL5aXbZhOGjroBFnibLCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T13:12:09.737236Z"},"content_sha256":"7758f2325e5043cc3ba937e7dcbd9b18b07a9081ef1ba1dbde4577410a673704","schema_version":"1.0","event_id":"sha256:7758f2325e5043cc3ba937e7dcbd9b18b07a9081ef1ba1dbde4577410a673704"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J7ZLWM2M5KWOMUWE3TLNOVZU72/bundle.json","state_url":"https://pith.science/pith/J7ZLWM2M5KWOMUWE3TLNOVZU72/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J7ZLWM2M5KWOMUWE3TLNOVZU72/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":"2026-08-13T13:12:09Z","links":{"resolver":"https://pith.science/pith/J7ZLWM2M5KWOMUWE3TLNOVZU72","bundle":"https://pith.science/pith/J7ZLWM2M5KWOMUWE3TLNOVZU72/bundle.json","state":"https://pith.science/pith/J7ZLWM2M5KWOMUWE3TLNOVZU72/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J7ZLWM2M5KWOMUWE3TLNOVZU72/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:J7ZLWM2M5KWOMUWE3TLNOVZU72","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":"a1ee786403f1413f7f2614f4a38ce07685dc7529403ec53f9cd9f996c524b8b7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T17:41:34Z","title_canon_sha256":"ee4a7243c8ad95282ed819a6a3859ccb0ab0d3af8ddbc3570e62f27c311bfa36"},"schema_version":"1.0","source":{"id":"2606.11159","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11159","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11159v1","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11159","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"pith_short_12","alias_value":"J7ZLWM2M5KWO","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"pith_short_16","alias_value":"J7ZLWM2M5KWOMUWE","created_at":"2026-06-10T01:11:14Z"},{"alias_kind":"pith_short_8","alias_value":"J7ZLWM2M","created_at":"2026-06-10T01:11:14Z"}],"graph_snapshots":[{"event_id":"sha256:7758f2325e5043cc3ba937e7dcbd9b18b07a9081ef1ba1dbde4577410a673704","target":"graph","created_at":"2026-06-10T01:11:14Z","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/2606.11159/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the pressureless Euler Alignment system with unidirectional velocity (u,0,...,0). By re-casting the system as a family of coupled scalar balance laws, one for each horizontal slice of R^d, we are able to prove existence, uniqueness, and stability within the class of unidirectional solutions, under the assumption of a bounded Lipschitz communication protocol. The nonlocal coupling between horizontal slices provides the system with a rich structure that is absent from the 1D setting and also constitutes the main technical difficulty of the present work. We construct our solutions as lim","authors_text":"Joshua O. Adeleke, Trevor M. Leslie","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T17:41:34Z","title":"Unidirectional Entropic Solutions of the Pressureless Euler Alignment System"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11159","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":"sha256:80062f06252574589ab12fb6f755bb79148c898358735cb0b855a181d764f346","target":"record","created_at":"2026-06-10T01:11:14Z","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":"a1ee786403f1413f7f2614f4a38ce07685dc7529403ec53f9cd9f996c524b8b7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-09T17:41:34Z","title_canon_sha256":"ee4a7243c8ad95282ed819a6a3859ccb0ab0d3af8ddbc3570e62f27c311bfa36"},"schema_version":"1.0","source":{"id":"2606.11159","kind":"arxiv","version":1}},"canonical_sha256":"4ff2bb334ceaace652c4dcd6d75734fe95cc93cd5aec597e2666d16b4c87846f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ff2bb334ceaace652c4dcd6d75734fe95cc93cd5aec597e2666d16b4c87846f","first_computed_at":"2026-06-10T01:11:14.521442Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:11:14.521442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jVoeeEvVO7kHh7ABa4J2vL6JYkyDTcrNHSCAYQUgqXoHxAw/AXenpaUI2suqg+IChbZu1BD2EuoylZJexcZuCg==","signature_status":"signed_v1","signed_at":"2026-06-10T01:11:14.522304Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11159","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:80062f06252574589ab12fb6f755bb79148c898358735cb0b855a181d764f346","sha256:7758f2325e5043cc3ba937e7dcbd9b18b07a9081ef1ba1dbde4577410a673704"],"state_sha256":"8de6fe7b2c0d6d420eb5a8f27f4bb8b2541816d7de41d9f51970e47373ac8582"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N5bNo9W2zWe5rITWGQ8zM0z1NHKOaZfXBXiO181Rl5Xmj/h7Kpkah7GmLCYy21BdoQRJ2wBDIIYK19iWdtXLCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T13:12:09.795274Z","bundle_sha256":"85f6a2a65215249f764db37d1c67df52b96c6a3668da70bf6d57434e687b1591"}}