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For \\[ P=\\conv\\{v_0,\\ldots,v_4\\},\\qquad Q=\\conv\\{v_0,\\ldots,v_5\\}, \\] where \\[ \\begin{aligned} v_0&=(-1,-3,-1), & v_1&=(3,2,-2), & v_2&=(0,2,1),\\\\ v_3&=(-1,-3,3), & v_4&=(-2,0,1), & v_5&=(-1,0,-2), \\end{aligned} \\] all five vertices of $P$ remain vertices of $Q$, but \\[ \\PWidth(P)^2=\\frac{48}{353} \\quad\\text{and}\\quad \\PWidth(Q)^2=\\frac{36}{133}. \\] Thus vert"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.29555","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-31T15:48:01Z","cross_cats_sorted":[],"title_canon_sha256":"abe4a0b83d09608175dbc6fdf0e9f673a3d0d63464d20342100e6ba0ae316b5c","abstract_canon_sha256":"1906759ba00324c41638c4823bb177dc6ee4c47f8ea36390c5bd51d3a1e7c78b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-03T01:38:24.706420Z","signature_b64":"bAYFSF64YhmVNzKJZ6wCUJl4TVUnG/AA0aUtR3cIhh100313Z4Kl/6wYkcED8HVHgQZ93NyDX6h9sC2tfwobDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"334e844a0f45d5f8c24252c20594ddfdec51198cfe2ff3acabace22b916e190c","last_reissued_at":"2026-08-03T01:38:24.704821Z","signature_status":"signed_v1","first_computed_at":"2026-08-03T01:38:24.704821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pyramidal Width Can Increase Under Vertex Insertion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Jinze Zhao","submitted_at":"2026-07-31T15:48:01Z","abstract_excerpt":"Lacoste-Julien and Jaggi conjectured in 2015 that the pyramidal width of a polytope cannot increase when a vertex is added, provided that every old point remains a vertex. We give an exact counterexample with six integer points in $\\R^3$. For \\[ P=\\conv\\{v_0,\\ldots,v_4\\},\\qquad Q=\\conv\\{v_0,\\ldots,v_5\\}, \\] where \\[ \\begin{aligned} v_0&=(-1,-3,-1), & v_1&=(3,2,-2), & v_2&=(0,2,1),\\\\ v_3&=(-1,-3,3), & v_4&=(-2,0,1), & v_5&=(-1,0,-2), \\end{aligned} \\] all five vertices of $P$ remain vertices of $Q$, but \\[ \\PWidth(P)^2=\\frac{48}{353} \\quad\\text{and}\\quad \\PWidth(Q)^2=\\frac{36}{133}. \\] Thus vert"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29555","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.29555/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.29555","created_at":"2026-08-03T01:38:24.705708+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.29555v1","created_at":"2026-08-03T01:38:24.705708+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.29555","created_at":"2026-08-03T01:38:24.705708+00:00"},{"alias_kind":"pith_short_12","alias_value":"GNHIISQPIXK7","created_at":"2026-08-03T01:38:24.705708+00:00"},{"alias_kind":"pith_short_16","alias_value":"GNHIISQPIXK7RQSC","created_at":"2026-08-03T01:38:24.705708+00:00"},{"alias_kind":"pith_short_8","alias_value":"GNHIISQP","created_at":"2026-08-03T01:38:24.705708+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X","json":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X.json","graph_json":"https://pith.science/api/pith-number/GNHIISQPIXK7RQSCKLBALFG57X/graph.json","events_json":"https://pith.science/api/pith-number/GNHIISQPIXK7RQSCKLBALFG57X/events.json","paper":"https://pith.science/paper/GNHIISQP"},"agent_actions":{"view_html":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X","download_json":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X.json","view_paper":"https://pith.science/paper/GNHIISQP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.29555&json=true","fetch_graph":"https://pith.science/api/pith-number/GNHIISQPIXK7RQSCKLBALFG57X/graph.json","fetch_events":"https://pith.science/api/pith-number/GNHIISQPIXK7RQSCKLBALFG57X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X/action/storage_attestation","attest_author":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X/action/author_attestation","sign_citation":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X/action/citation_signature","submit_replication":"https://pith.science/pith/GNHIISQPIXK7RQSCKLBALFG57X/action/replication_record"}},"created_at":"2026-08-03T01:38:24.705708+00:00","updated_at":"2026-08-03T01:38:24.705708+00:00"}