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Furthermore, we show that the same result holds for the Milnor fillable structure on Y with the possible exception of Σ(3,4,5), Σ(2,5,7) and Σ(2,3,6k+1) for k≥1. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The canonical orientation on the Brieskorn sphere Y and the precise definition of half-convex Giroux torsion are assumed to hold in the stated cases; the obstruction relies on the correction term vanishing or the absence of torsion without additional verification for all contact structures."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures"},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Brieskorn spheres admit no rational homology ball symplectic fillings for their contact structures except in listed cases."}],"snapshot_sha256":"eb404feacd88815448f9a708562a7dba3552563ac605bf7d8448602010adb709"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"22a4d83cb6b8acb4b60194e2d65437fdd444f46d1f24d46ae0e9b64eb1a3358d"},"paper":{"abstract_excerpt":"Given a canonically oriented Brieskorn sphere $Y=\\Sigma(a_1,...,a_n)$, we confirm some statements conjectured by Gompf. More specifically, we obstruct the existence of rational homology ball symplectic fillings for any contact structure on $-Y$ if $n=3$, and when there is no half convex Giroux torsion for $n>3$. Furthermore, we show that the same result holds for the Milnor fillable structure on $Y$ with the possible exception of $\\Sigma(3,4,5),$ $\\Sigma(2,5,7)$ and $\\Sigma(2,3,6k+1)$ for $k\\geq1$. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correctio","authors_text":"Alberto Cavallo, Antonio Alfieri, Irena Matkovi\\v{c}","cross_cats":["math.SG"],"headline":"Brieskorn spheres admit no rational homology ball symplectic fillings for their contact structures except in listed cases.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-05-13T17:39:50Z","title":"Brieskorn spheres and rational homology ball symplectic fillings"},"references":{"count":28,"internal_anchors":2,"resolved_work":28,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"P. Aceto, D. McCoy and J. H. Park,A survey on embeddings of3-manifolds in definite4-manifolds, arXiv:2407.03692","work_id":"f9fe6a17-943a-4a03-98d4-3b032b79ab93","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"S. Akbulut and K. Larson,Brieskorn spheres bounding rational balls, Proc. Am. Math. Soc.,146(2018), no. 4, pp. 1817–1824","work_id":"c1b24e3c-320c-4edb-933b-9ac8f68884e8","year":2018},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"A. Berman and R. Plemmons,Nonnegative matrices in the mathematical sciences, Volume 9 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994","work_id":"061db4ef-e2c5-4467-bb3b-b848afebaae8","year":1994},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"M. Bhupal and A. Stipsicz,Weighted homogeneous singularities and rational homology disk smoothings, Am. J. Math.,133(2011), no. 5, pp. 1259–1297","work_id":"f5d789f0-a684-4779-93ac-9db488133d98","year":2011},{"cited_arxiv_id":"2604.28174","doi":"","is_internal_anchor":true,"ref_index":5,"title":"Fillable structures on negative-definite Seifert fibred spaces","work_id":"23bc730a-2bd8-4c76-936b-2172efd3700b","year":null}],"snapshot_sha256":"7189781bde4e86c6c449e235dd9898a5614f034c989c48190440e9a6b60fd35f"},"source":{"id":"2605.13812","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T17:31:58.970647Z","id":"4d0e692f-d4dc-4e1f-870e-20e2d340aef7","model_set":{"reader":"grok-4.3"},"one_line_summary":"Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Brieskorn spheres admit no rational homology ball symplectic fillings for their contact structures except in listed cases.","strongest_claim":"we obstruct the existence of rational homology ball symplectic fillings for any contact structure on −Y if n=3, and when there is no half convex Giroux torsion for n>3. Furthermore, we show that the same result holds for the Milnor fillable structure on Y with the possible exception of Σ(3,4,5), Σ(2,5,7) and Σ(2,3,6k+1) for k≥1. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.","weakest_assumption":"The canonical orientation on the Brieskorn sphere Y and the precise definition of half-convex Giroux torsion are assumed to hold in the stated cases; the obstruction relies on the correction term vanishing or the absence of torsion without additional verification for all contact structures."}},"verdict_id":"4d0e692f-d4dc-4e1f-870e-20e2d340aef7"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:99884e6b6f83faf0934aa6282ba135012c092726e07eb17133f04d699e0d0e89","target":"record","created_at":"2026-05-18T02:44:15Z","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":"bc8b2cd1021f69da1e248356e9bfcf07ae7c953b096bccc6e028acc8fe60f16c","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-05-13T17:39:50Z","title_canon_sha256":"6e333c9460e47fa126c1ef0f0de3255f8d218ae24021d4cd1430c9e7e53fc171"},"schema_version":"1.0","source":{"id":"2605.13812","kind":"arxiv","version":1}},"canonical_sha256":"63797788a909fc9fc220ec0a59fa49d4e70a83f75b4f9fd8d22dc0ff2c585848","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63797788a909fc9fc220ec0a59fa49d4e70a83f75b4f9fd8d22dc0ff2c585848","first_computed_at":"2026-05-18T02:44:15.384883Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:15.384883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mQyBW6yl10EjGg0dGXliAeSoXsVXBXAxj55we3A9dIvoYCknEfvbjfD7ROYhyA844Bz2sfKzGhuisls/qWxTDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:15.385485Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13812","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99884e6b6f83faf0934aa6282ba135012c092726e07eb17133f04d699e0d0e89","sha256:25a182f906768e298fb13d3e8c3bc2fc4abfdb021917c4158e4bac0ceb55694d"],"state_sha256":"fda5a1eaed1cf1bd183c48fd5b97ec3d254dbe0b22cc3228e9ff23302b5b2c4f"}