{"paper":{"title":"Mayer Path Homology","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Mayer path homology equips directed path complexes with an N-nilpotent differential to produce a finer invariant than standard path homology.","cross_cats":[],"primary_cat":"math.AT","authors_text":"Dilan Karaguler, Guo-Wei Wei","submitted_at":"2026-05-15T18:21:42Z","abstract_excerpt":"We introduce Mayer path homology, a new homology theory for directed path complexes obtained by equipping path complexes with an $N$-nilpotent differential. The main novelty of this work is the introduction of an $N$-differential on path complexes, giving rise to $N$-chain complexes of $\\partial$-invariant paths and Mayer path homology groups $H_n^{N,q}(P)$. We prove that this construction defines a canonical invariant of directed graphs and is more sensitive than standard path homology, distinguishing directed network motifs that ordinary path homology cannot separate. We further establish a "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This construction defines a canonical invariant of directed graphs and is more sensitive than standard path homology, distinguishing directed network motifs that ordinary path homology cannot separate.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The N-nilpotent differential on path complexes produces well-defined homology groups that remain invariant under the directed graph structure and are strictly finer than classical path homology.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Mayer path homology equips path complexes with an N-differential to yield homology groups that distinguish directed graph motifs more finely than standard path homology.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Mayer path homology equips directed path complexes with an N-nilpotent differential to produce a finer invariant than standard path homology.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7dd77192ca706b8d6523fdbfcd5e294fa770a1f0495f51a54928c8dccf386e07"},"source":{"id":"2605.16525","kind":"arxiv","version":1},"verdict":{"id":"fb3abfd2-750c-4656-9387-c84d5452041d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:24:39.481057Z","strongest_claim":"This construction defines a canonical invariant of directed graphs and is more sensitive than standard path homology, distinguishing directed network motifs that ordinary path homology cannot separate.","one_line_summary":"Mayer path homology equips path complexes with an N-differential to yield homology groups that distinguish directed graph motifs more finely than standard path homology.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The N-nilpotent differential on path complexes produces well-defined homology groups that remain invariant under the directed graph structure and are strictly finer than classical path homology.","pith_extraction_headline":"Mayer path homology equips directed path complexes with an N-nilpotent differential to produce a finer invariant than standard path homology."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16525/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T21:31:20.266601Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T21:31:19.485187Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:33:23.077475Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:21:56.941554Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5d0a2cd2908ab40b53b8bd667bc30bdccb354949788e462653b35265bd915ca8"},"references":{"count":20,"sample":[{"doi":"","year":2023,"title":"Path topology in molecular and materials sciences.The journal of physical chemistry letters, 14(4):954–964, 2023","work_id":"fc5be529-5a9e-4d87-817b-8480b9220f45","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Samir Chowdhury and Facundo M´ emoli.Persistent Path Homology of Directed Networks, pages 1152–1169","work_id":"1412e272-93b2-4ab0-9b84-b334800fabc0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Dey, Tianqi Li, and Yusu Wang","work_id":"3a592b28-a1af-40c5-a576-782aa0a1656c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Mayer-homology learning predic- tion of protein-ligand binding affinities.Journal of computational biophysics and chemistry, 24(02):253–266, 2025","work_id":"b259cbce-062b-4e8f-95d7-d00089de0475","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Network modeling and topology of aging.Physics Reports, 1101:1–65, 2025","work_id":"3a5bdc33-846f-4b7f-b793-8f006030160a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":20,"snapshot_sha256":"979c3f61102a52e7548f987f64d311b40333e5ab0a883cda93506aaab2990e45","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"fb2d75a562cb0ac8b6c577c8bc678f808ac702b5a24bce674b3ba0677b259447"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}