{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:SWAI3QODBJC3Y62NNZ37V6GJST","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":"ae161f52b37e94a382e1bb8f31db06e3dcd2a52ead1fd53659e0269e8806cb05","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-12-23T09:40:59Z","title_canon_sha256":"ae49f81fdedde0cf5cd97f5c9d3e9aec7d1f0626908307cde2a820d56ff05ab9"},"schema_version":"1.0","source":{"id":"1612.07915","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.07915","created_at":"2026-05-18T00:23:40Z"},{"alias_kind":"arxiv_version","alias_value":"1612.07915v2","created_at":"2026-05-18T00:23:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.07915","created_at":"2026-05-18T00:23:40Z"},{"alias_kind":"pith_short_12","alias_value":"SWAI3QODBJC3","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"SWAI3QODBJC3Y62N","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"SWAI3QOD","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:d11dc981dff3162b43df8db03c94b7cd41a290f682d1508a48ab6a41dc55bdb2","target":"graph","created_at":"2026-05-18T00:23:40Z","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":"For a nonempty polyhedral set $P\\subset \\mathbb R^d$, let $\\mathcal F(P)$ denote the set of faces of $P$, and let $N(P,F)$ be the normal cone of $P$ at the nonempty face $F\\in\\mathcal F(P)$. We prove that the function $\\sum_{F\\in\\mathcal F(P)}(-1)^{\\text{dim} F} 1_{F-N(P,F)}$ equals $1$ if $P$ is bounded, or $0$ if $P$ is unbounded and line-free. Previously, this formula was known to hold everywhere outside some exceptional set of Lebesgue measure $0$ or for polyhedral cones. The case of a not necessarily line-free polyhedral set is also covered by our general theorem.","authors_text":"Daniel Hug, Zakhar Kabluchko","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-12-23T09:40:59Z","title":"An inclusion-exclusion identity for normal cones of polyhedral sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07915","kind":"arxiv","version":2},"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":"sha256:4c5a5e4f5be26949863f1d7f3171ac904a61bb3f42f0201f7d635e8c07b82047","target":"record","created_at":"2026-05-18T00:23:40Z","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":"ae161f52b37e94a382e1bb8f31db06e3dcd2a52ead1fd53659e0269e8806cb05","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-12-23T09:40:59Z","title_canon_sha256":"ae49f81fdedde0cf5cd97f5c9d3e9aec7d1f0626908307cde2a820d56ff05ab9"},"schema_version":"1.0","source":{"id":"1612.07915","kind":"arxiv","version":2}},"canonical_sha256":"95808dc1c30a45bc7b4d6e77faf8c994c99397ee9143f176647c6ccee802807b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95808dc1c30a45bc7b4d6e77faf8c994c99397ee9143f176647c6ccee802807b","first_computed_at":"2026-05-18T00:23:40.361116Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:23:40.361116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ngIEN+Ab2iBCPuDOdZIHcnc1YjsM7c0f02DrG8+INeG46ATvYOqA8PZv4R30El2MSND+RDFPIFzp7qu15CdiBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:23:40.361888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.07915","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c5a5e4f5be26949863f1d7f3171ac904a61bb3f42f0201f7d635e8c07b82047","sha256:d11dc981dff3162b43df8db03c94b7cd41a290f682d1508a48ab6a41dc55bdb2"],"state_sha256":"d74710894d09fe6ccbd272b9e1ff9a58881fc3d5be95c5ecae2f85b1b6cc101b"}