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Precisely, if $(X,\\omega_X)$ is a $4$-dimensional star-shaped domain and $\\omega_Y$ is a symplectic form Poincar\\'e dual to $[A]$ then \\[(X,\\omega_X)\\text{ embeds into }(Y,\\omega_Y)\\text{ symplectically } \\implies c^{\\text{ECH}}_k(X,\\omega_X) \\le c^{\\text{alg}}_k(Y,A)\\] We give three applications to toric embedding problems: (1) these obstru"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2008.10125","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2020-08-23T22:52:26Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"b2be816f2bc3cf9833135ffa79234344ae808e24bfc6081f56a2addbc1a3639c","abstract_canon_sha256":"393dd84211d257084970a598937c9a29eeef4bfa585c092deefeb922dc6cc1ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:22:05.687503Z","signature_b64":"nQbCmG72zz75ak+VoZ0G/d5IvFpt+9+ywpFuLZmrShN/4Ow4ERDu3ekyC7wOh24BR/1yiA3NwNTxSq4XFPMMCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4eaf2f8c688069641a0cbbf6c7ae2f2ed4bc736a41755b0a1d0cf222204c9527","last_reissued_at":"2026-07-05T02:22:05.687004Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:22:05.687004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"ECH embedding obstructions for rational surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.SG","authors_text":"Ben Wormleighton, Julian Chaidez","submitted_at":"2020-08-23T22:52:26Z","abstract_excerpt":"Let $(Y,A)$ be a smooth rational surface or a possibly singular toric surface with ample divisor $A$. We show that a family of ECH-based, algebro-geometric invariants $c^{\\text{alg}}_k(Y,A)$ proposed by Wormleighton obstruct symplectic embeddings into $Y$. Precisely, if $(X,\\omega_X)$ is a $4$-dimensional star-shaped domain and $\\omega_Y$ is a symplectic form Poincar\\'e dual to $[A]$ then \\[(X,\\omega_X)\\text{ embeds into }(Y,\\omega_Y)\\text{ symplectically } \\implies c^{\\text{ECH}}_k(X,\\omega_X) \\le c^{\\text{alg}}_k(Y,A)\\] We give three applications to toric embedding problems: (1) these obstru"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.10125","kind":"arxiv","version":3},"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/2008.10125/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2008.10125","created_at":"2026-07-05T02:22:05.687057+00:00"},{"alias_kind":"arxiv_version","alias_value":"2008.10125v3","created_at":"2026-07-05T02:22:05.687057+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2008.10125","created_at":"2026-07-05T02:22:05.687057+00:00"},{"alias_kind":"pith_short_12","alias_value":"J2XS7DDIQBUW","created_at":"2026-07-05T02:22:05.687057+00:00"},{"alias_kind":"pith_short_16","alias_value":"J2XS7DDIQBUWIGQM","created_at":"2026-07-05T02:22:05.687057+00:00"},{"alias_kind":"pith_short_8","alias_value":"J2XS7DDI","created_at":"2026-07-05T02:22:05.687057+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3","json":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3.json","graph_json":"https://pith.science/api/pith-number/J2XS7DDIQBUWIGQMXP3MPLRPF3/graph.json","events_json":"https://pith.science/api/pith-number/J2XS7DDIQBUWIGQMXP3MPLRPF3/events.json","paper":"https://pith.science/paper/J2XS7DDI"},"agent_actions":{"view_html":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3","download_json":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3.json","view_paper":"https://pith.science/paper/J2XS7DDI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2008.10125&json=true","fetch_graph":"https://pith.science/api/pith-number/J2XS7DDIQBUWIGQMXP3MPLRPF3/graph.json","fetch_events":"https://pith.science/api/pith-number/J2XS7DDIQBUWIGQMXP3MPLRPF3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3/action/storage_attestation","attest_author":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3/action/author_attestation","sign_citation":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3/action/citation_signature","submit_replication":"https://pith.science/pith/J2XS7DDIQBUWIGQMXP3MPLRPF3/action/replication_record"}},"created_at":"2026-07-05T02:22:05.687057+00:00","updated_at":"2026-07-05T02:22:05.687057+00:00"}