{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:QRJUN4YUTIRELZUO7WRCWJILBR","short_pith_number":"pith:QRJUN4YU","canonical_record":{"source":{"id":"2412.21040","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-12-30T16:03:56Z","cross_cats_sorted":[],"title_canon_sha256":"57684308a5773ac0a1f775116ac8540f379a8ff473f4c72e6e507b6358cfca9b","abstract_canon_sha256":"6237578b3dab24810aa73344685eb83417f6e654747096392de68eaf4c4e1344"},"schema_version":"1.0"},"canonical_sha256":"845346f3149a2245e68efda22b250b0c7b8cc62dd52dc131ef3d45b53bde66ea","source":{"kind":"arxiv","id":"2412.21040","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.21040","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"arxiv_version","alias_value":"2412.21040v2","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.21040","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"pith_short_12","alias_value":"QRJUN4YUTIRE","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"pith_short_16","alias_value":"QRJUN4YUTIRELZUO","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"pith_short_8","alias_value":"QRJUN4YU","created_at":"2026-07-05T09:56:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:QRJUN4YUTIRELZUO7WRCWJILBR","target":"record","payload":{"canonical_record":{"source":{"id":"2412.21040","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-12-30T16:03:56Z","cross_cats_sorted":[],"title_canon_sha256":"57684308a5773ac0a1f775116ac8540f379a8ff473f4c72e6e507b6358cfca9b","abstract_canon_sha256":"6237578b3dab24810aa73344685eb83417f6e654747096392de68eaf4c4e1344"},"schema_version":"1.0"},"canonical_sha256":"845346f3149a2245e68efda22b250b0c7b8cc62dd52dc131ef3d45b53bde66ea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:56:05.072310Z","signature_b64":"N1Fn6mfRalxS9JfDoF9yonkOAY37+jZnpGEuuBSnwPE3SELBoMOk9R7bOQoCXcPZIzj1fEzI1OW4FugL6a3PDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"845346f3149a2245e68efda22b250b0c7b8cc62dd52dc131ef3d45b53bde66ea","last_reissued_at":"2026-07-05T09:56:05.071835Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:56:05.071835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.21040","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:56:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g8yuy+eC+XL2b/AX3vyJMIcUAo+KFQ7u1yEfdKJHKXSi8BdGTVH9ambDNpRKll0AdKmRInqP51qPt6Cln+l9CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T07:40:26.452902Z"},"content_sha256":"710d126725dceddcf44fa3037ffda95025af60ee374bff65648b6944ec1292f7","schema_version":"1.0","event_id":"sha256:710d126725dceddcf44fa3037ffda95025af60ee374bff65648b6944ec1292f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:QRJUN4YUTIRELZUO7WRCWJILBR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gradient catastrophes and an infinite hierarchy of H\\\"older cusp-singularities for 1D Euler","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Isaac Neal, Steve Shkoller, Vlad Vicol","submitted_at":"2024-12-30T16:03:56Z","abstract_excerpt":"We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with non-constant entropy. Specifically, for all integers $n\\geq 1$, we prove that there exist classical solutions, emanating from smooth, compressive, and non-vacuous initial data, which form a cusp-type gradient singularity in finite time, in which the gradient of the solution has precisely $C^{0,\\frac{1}{2n+1}}$ H\\\"older-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\\infty}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.21040","kind":"arxiv","version":2},"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/2412.21040/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":"2026-07-05T09:56:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jxGqIowOPqGmGfFfNjrIpUXx9e0Wsqbet/aUrWb8OWC35wiyuKvL+AATTw/lOFzWYP330HXP5uhdl8mKujGZAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T07:40:26.453787Z"},"content_sha256":"2cf9aea10c6a2fa6225e87cb320443acc9e0a9e6ae282558b34bb3f34169bdf1","schema_version":"1.0","event_id":"sha256:2cf9aea10c6a2fa6225e87cb320443acc9e0a9e6ae282558b34bb3f34169bdf1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QRJUN4YUTIRELZUO7WRCWJILBR/bundle.json","state_url":"https://pith.science/pith/QRJUN4YUTIRELZUO7WRCWJILBR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QRJUN4YUTIRELZUO7WRCWJILBR/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":"2026-08-11T07:40:26Z","links":{"resolver":"https://pith.science/pith/QRJUN4YUTIRELZUO7WRCWJILBR","bundle":"https://pith.science/pith/QRJUN4YUTIRELZUO7WRCWJILBR/bundle.json","state":"https://pith.science/pith/QRJUN4YUTIRELZUO7WRCWJILBR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QRJUN4YUTIRELZUO7WRCWJILBR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:QRJUN4YUTIRELZUO7WRCWJILBR","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":"6237578b3dab24810aa73344685eb83417f6e654747096392de68eaf4c4e1344","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-12-30T16:03:56Z","title_canon_sha256":"57684308a5773ac0a1f775116ac8540f379a8ff473f4c72e6e507b6358cfca9b"},"schema_version":"1.0","source":{"id":"2412.21040","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.21040","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"arxiv_version","alias_value":"2412.21040v2","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.21040","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"pith_short_12","alias_value":"QRJUN4YUTIRE","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"pith_short_16","alias_value":"QRJUN4YUTIRELZUO","created_at":"2026-07-05T09:56:05Z"},{"alias_kind":"pith_short_8","alias_value":"QRJUN4YU","created_at":"2026-07-05T09:56:05Z"}],"graph_snapshots":[{"event_id":"sha256:2cf9aea10c6a2fa6225e87cb320443acc9e0a9e6ae282558b34bb3f34169bdf1","target":"graph","created_at":"2026-07-05T09:56:05Z","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/2412.21040/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with non-constant entropy. Specifically, for all integers $n\\geq 1$, we prove that there exist classical solutions, emanating from smooth, compressive, and non-vacuous initial data, which form a cusp-type gradient singularity in finite time, in which the gradient of the solution has precisely $C^{0,\\frac{1}{2n+1}}$ H\\\"older-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\\infty}$.","authors_text":"Isaac Neal, Steve Shkoller, Vlad Vicol","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-12-30T16:03:56Z","title":"Gradient catastrophes and an infinite hierarchy of H\\\"older cusp-singularities for 1D Euler"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.21040","kind":"arxiv","version":2},"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":"sha256:710d126725dceddcf44fa3037ffda95025af60ee374bff65648b6944ec1292f7","target":"record","created_at":"2026-07-05T09:56:05Z","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":"6237578b3dab24810aa73344685eb83417f6e654747096392de68eaf4c4e1344","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-12-30T16:03:56Z","title_canon_sha256":"57684308a5773ac0a1f775116ac8540f379a8ff473f4c72e6e507b6358cfca9b"},"schema_version":"1.0","source":{"id":"2412.21040","kind":"arxiv","version":2}},"canonical_sha256":"845346f3149a2245e68efda22b250b0c7b8cc62dd52dc131ef3d45b53bde66ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"845346f3149a2245e68efda22b250b0c7b8cc62dd52dc131ef3d45b53bde66ea","first_computed_at":"2026-07-05T09:56:05.071835Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:56:05.071835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N1Fn6mfRalxS9JfDoF9yonkOAY37+jZnpGEuuBSnwPE3SELBoMOk9R7bOQoCXcPZIzj1fEzI1OW4FugL6a3PDg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:56:05.072310Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.21040","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:710d126725dceddcf44fa3037ffda95025af60ee374bff65648b6944ec1292f7","sha256:2cf9aea10c6a2fa6225e87cb320443acc9e0a9e6ae282558b34bb3f34169bdf1"],"state_sha256":"102b70e2d26aab7c208db62cb682f3131437cb2d899726c9f7e1e63a605c8b94"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EA1ft7HbLsWpG2hDKvONDijd41bUAw1KVH/5ubzg7nzYfsGOA0SDSIkUmFxkG4iVI+yTtbW3Ob5SZx7YMICyBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T07:40:26.460145Z","bundle_sha256":"fda04f8b50eaaa4a6b268e12848b74721b7671b7e5d312af04a88029e1cd2e19"}}