{"paper":{"title":"Clearing in Liability Networks via Sheaves on Directed Hypergraphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Liability clearing configurations are precisely the global sections of a sheaf on a directed hypergraph.","cross_cats":["math.CT"],"primary_cat":"q-fin.MF","authors_text":"Robert Ghrist","submitted_at":"2026-05-15T09:36:25Z","abstract_excerpt":"We associate to a decorated liability network a liability sheaf on a directed hypergraph whose hyperedges separate the distribution of payments from the collection of receipts. Clearing configurations are precisely the global sections of this sheaf, and the global-section object is canonically the equalizer of the identity and a clearing operator $\\Phi=A\\circ D$ factored into collective distribution $D$ and aggregation $A$; an institution-edge duality identifies it equivalently with the equalizer of the dual operator $D\\circ A$ on the edge side. This identifies liability clearing as a finite-l"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Clearing configurations are precisely the global sections of this sheaf, and the global-section object is canonically the equalizer of the identity and a clearing operator Φ=A∘D factored into collective distribution D and aggregation A; an institution-edge duality identifies it equivalently with the equalizer of the dual operator D∘A on the edge side. This identifies liability clearing as a finite-limit construction in the ambient data category.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a decorated liability network can be represented as a directed hypergraph whose hyperedges separate payment distribution from receipt collection, and that the coefficient category admits finite limits together with constraint subobjects compatible with a finite-limit-preserving functor.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Liability clearing in networks is modeled as global sections of a liability sheaf on directed hypergraphs, identified as a finite-limit construction with existence and uniqueness from lattice and metric theorems on payment objects.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Liability clearing configurations are precisely the global sections of a sheaf on a directed hypergraph.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2540b2d1b0916399a717e10f7d9d11586f2a3a038b0aeb4c85abfab0c2e12ff6"},"source":{"id":"2605.15778","kind":"arxiv","version":1},"verdict":{"id":"ff6cfdc6-7a51-4902-b622-d477f74f57a2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:58:04.227533Z","strongest_claim":"Clearing configurations are precisely the global sections of this sheaf, and the global-section object is canonically the equalizer of the identity and a clearing operator Φ=A∘D factored into collective distribution D and aggregation A; an institution-edge duality identifies it equivalently with the equalizer of the dual operator D∘A on the edge side. This identifies liability clearing as a finite-limit construction in the ambient data category.","one_line_summary":"Liability clearing in networks is modeled as global sections of a liability sheaf on directed hypergraphs, identified as a finite-limit construction with existence and uniqueness from lattice and metric theorems on payment objects.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a decorated liability network can be represented as a directed hypergraph whose hyperedges separate payment distribution from receipt collection, and that the coefficient category admits finite limits together with constraint subobjects compatible with a finite-limit-preserving functor.","pith_extraction_headline":"Liability clearing configurations are precisely the global sections of a sheaf on a directed hypergraph."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15778/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T18:31:18.786961Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:10:15.810803Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:48.752472Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:21:55.929053Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c0588dc99dfe505fc56958c0d7f88bf73d76cfc59926830dc882749a2c974f75"},"references":{"count":33,"sample":[{"doi":"","year":2015,"title":"Daron Acemoglu, Asuman Ozdaglar, and Alireza Tahbaz-Salehi,Systemic risk and stability in financial networks, American Economic Review105(2015), no. 2, 564–608","work_id":"246974a6-f332-4755-94e6-67248da0ac10","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Kartik Anand, Ben Craig, and Goetz von Peter,Filling in the blanks: Network structure and interbank contagion, Quantitative Finance15(2015), no. 4, 625–636","work_id":"a5c70040-4683-4816-8974-3d59eeacf18c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"2025 Sheaf theory: from deep geometry to deep learning","work_id":"46192dc2-30c6-4b4b-b5fa-46af94536e45","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Tathagata Banerjee, Alex Bernstein, and Zachary Feinstein,Dynamic clearing and contagion in finan- cial networks, EuropeanJournalofOperationalResearch321(2025), no.2, 664–675, arXiv:1801.02091","work_id":"b63a98b4-31c1-4ad0-8b71-e8e61285ffe9","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"Tathagata Banerjee and Zachary Feinstein,Impact of contingent payments on systemic risk in financial networks, Mathematics and Financial Economics13(2019), no. 4, 617–636","work_id":"5e1ebaa4-4811-4008-babd-1f7557145bc7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":33,"snapshot_sha256":"222bec2be1b408366b6c2335b79fe77f32d9c2c274b31d59584512c79aa18d36","internal_anchors":1},"formal_canon":{"evidence_count":1,"snapshot_sha256":"962d90a9a5ce58c0e877cefeba8a39cd34fe9a07a04f0ba065059fc66eb554c5"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}