{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6C5U6OLN47NAU3Y3Y7M6FEIODF","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":"fd44c67f4226295373423be83492e2fab5e02f757581c66adc21d816955325d6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-11-02T03:01:13Z","title_canon_sha256":"4056c80ec4d00bbb24187efbdde81998063278a934af59dd4483215a443cd588"},"schema_version":"1.0","source":{"id":"1811.00719","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.00719","created_at":"2026-05-18T00:01:42Z"},{"alias_kind":"arxiv_version","alias_value":"1811.00719v1","created_at":"2026-05-18T00:01:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.00719","created_at":"2026-05-18T00:01:42Z"},{"alias_kind":"pith_short_12","alias_value":"6C5U6OLN47NA","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6C5U6OLN47NAU3Y3","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6C5U6OLN","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:1cf00322868fd23ab1ee0da18634e6d8261d3f1bc52825923e86d2f7638302d0","target":"graph","created_at":"2026-05-18T00:01:42Z","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"},"paper":{"abstract_excerpt":"In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. In this paper we introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the Mountain Pass Lemma and give an alternate proof of a discrete Lusternik-Schnirelmann Theorem.","authors_text":"Kevin Knudson, Lacey Johnson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-11-02T03:01:13Z","title":"Min-max theory for cell complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00719","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":"sha256:12d29d2be3638c06dcf9544523c251ee3824655a81187f3f916ea79efc9beb9e","target":"record","created_at":"2026-05-18T00:01:42Z","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":"fd44c67f4226295373423be83492e2fab5e02f757581c66adc21d816955325d6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-11-02T03:01:13Z","title_canon_sha256":"4056c80ec4d00bbb24187efbdde81998063278a934af59dd4483215a443cd588"},"schema_version":"1.0","source":{"id":"1811.00719","kind":"arxiv","version":1}},"canonical_sha256":"f0bb4f396de7da0a6f1bc7d9e2910e1962375a7b309291b363f5792e5356620d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f0bb4f396de7da0a6f1bc7d9e2910e1962375a7b309291b363f5792e5356620d","first_computed_at":"2026-05-18T00:01:42.381907Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:42.381907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q74PFMqFw+e454Ns0iQ0bMevNmGWoFaJs7i1KLpHx4wme6WPolVcEVuKRSh9+j0u2prZ7oGF6+MxkDJpwxzoAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:42.382363Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.00719","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:12d29d2be3638c06dcf9544523c251ee3824655a81187f3f916ea79efc9beb9e","sha256:1cf00322868fd23ab1ee0da18634e6d8261d3f1bc52825923e86d2f7638302d0"],"state_sha256":"a4ca6585a5cf2be1d2977b0e260022f1085a7b3a33de153b0e292c894c019f5f"}