{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VW3YZFDLK7X4ERNGSTEGH4XQEC","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":"12195487a1097d283f71ee3b9704194664d83b8990e6e391f1ca84847515ff40","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T16:15:18Z","title_canon_sha256":"b79d984584b66569cded81ff30250a7790407438cca12bff75bb0039d85f01fb"},"schema_version":"1.0","source":{"id":"2608.06226","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.06226","created_at":"2026-08-07T01:40:59Z"},{"alias_kind":"arxiv_version","alias_value":"2608.06226v1","created_at":"2026-08-07T01:40:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.06226","created_at":"2026-08-07T01:40:59Z"},{"alias_kind":"pith_short_12","alias_value":"VW3YZFDLK7X4","created_at":"2026-08-07T01:40:59Z"},{"alias_kind":"pith_short_16","alias_value":"VW3YZFDLK7X4ERNG","created_at":"2026-08-07T01:40:59Z"},{"alias_kind":"pith_short_8","alias_value":"VW3YZFDL","created_at":"2026-08-07T01:40:59Z"}],"graph_snapshots":[{"event_id":"sha256:f6a4b0911f24d1ab9bbae2c93f3cfb391f481af43ce44ceae86411cd2ce76664","target":"graph","created_at":"2026-08-07T01:40:59Z","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/2608.06226/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the energy $\\mathcal{E}(s) := E_{s}[\\Phi_s]$ of the one-dimensional fractional Allen--Cahn layer solution $\\Phi_s$, defined as the unique odd, increasing solution of $(-\\Delta)^{s} \\Phi_s = \\Phi_s - \\Phi_s^{3}$ with $\\Phi_s(\\pm\\infty)=\\pm 1$ and $\\Phi_s(0)=0$, for $s\\in(1/2,1]$. Our main results are a sharp qualitative and quantitative description of the energy $\\mathcal{E}$ on this interval. We show that the energy is continuous and strictly decreasing, and we obtain explicit asymptotic expansions at both endpoints. At the upper endpoint we prove $\\mathcal{E}(s) = \\frac{2\\sqrt{2}}{3}","authors_text":"Alvaro Carballeira, Damien Galant, Javier G\\'omez-Serrano","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T16:15:18Z","title":"The energy of fractional Allen--Cahn layers in dimension one"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06226","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:b14063a3143d05ace817c4a5854780b722e34c57d297b83944f247b0bca49f44","target":"record","created_at":"2026-08-07T01:40:59Z","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":"12195487a1097d283f71ee3b9704194664d83b8990e6e391f1ca84847515ff40","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T16:15:18Z","title_canon_sha256":"b79d984584b66569cded81ff30250a7790407438cca12bff75bb0039d85f01fb"},"schema_version":"1.0","source":{"id":"2608.06226","kind":"arxiv","version":1}},"canonical_sha256":"adb78c946b57efc245a694c863f2f020ac0812592ff787c4f5f0ce848b71ef87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"adb78c946b57efc245a694c863f2f020ac0812592ff787c4f5f0ce848b71ef87","first_computed_at":"2026-08-07T01:40:59.327145Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-07T01:40:59.327145Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oRsTIybWtGabwKBC4wrfVH2ch+T3AbGvqH3cd+ODjv7NbkteyMpDQEnclMVOOPyjavwIdQUT1XHMTL8w5bGLAg==","signature_status":"signed_v1","signed_at":"2026-08-07T01:40:59.328889Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.06226","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b14063a3143d05ace817c4a5854780b722e34c57d297b83944f247b0bca49f44","sha256:f6a4b0911f24d1ab9bbae2c93f3cfb391f481af43ce44ceae86411cd2ce76664"],"state_sha256":"c25c7b7522939e7c1b761cfc0816d86d6131c710d14bf5249504961ef4afc426"}