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Furthermore, a formula for $G(d,k)$ is given, showing that e.g. $G(d,k)=1$ if $k\\ge \\left\\lfloor\\frac{d+1}{2}\\right\\rfloor$ or if both $d$ and $k$ are even, and also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the number of $k$-faces of some simple $d$-polytope).\n  This question has previously been studied"},"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":"math/9612218","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"1996-12-11T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"eb0cf081185b3a02dadf75c615dbcab3a94d8f604d7d155e0f61cc63d1ad11e4","abstract_canon_sha256":"68354e20753c7c88c876d284007bfa6a4d92353e887d22fb8c38f4a0e1768bea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:37.348380Z","signature_b64":"CSDo8ekoG2uBTB14XP15JKNAgi4IhXpQvWrKfURrv/TMV9h7EBxnJaJyFtfIJmx0L07+1jAjf3xOJXjydTs3Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f980cce0526c86902b9bb7c9e14e43fef55fd8e33d3a622d79b9659eac658436","last_reissued_at":"2026-05-18T01:05:37.347836Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:37.347836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of faces of a simple polytope","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anders Bj\\\"orner, Svante Linusson","submitted_at":"1996-12-11T00:00:00Z","abstract_excerpt":"Consider the question: Given integers $k<d<n$, does there exist a simple $d$-polytope with $n$ faces of dimension $k$? We show that there exist numbers $G(d,k)$ and $N(d,k)$ such that for $n> N(d,k)$ the answer is yes if and only if $n\\equiv 0\\quad \\pmod {G(d,k)}$. Furthermore, a formula for $G(d,k)$ is given, showing that e.g. $G(d,k)=1$ if $k\\ge \\left\\lfloor\\frac{d+1}{2}\\right\\rfloor$ or if both $d$ and $k$ are even, and also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the number of $k$-faces of some simple $d$-polytope).\n  This question has previously been studied"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9612218","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":""},"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":"math/9612218","created_at":"2026-05-18T01:05:37.347940+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9612218v1","created_at":"2026-05-18T01:05:37.347940+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9612218","created_at":"2026-05-18T01:05:37.347940+00:00"},{"alias_kind":"pith_short_12","alias_value":"7GAMZYCSNSDJ","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_16","alias_value":"7GAMZYCSNSDJAK43","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_8","alias_value":"7GAMZYCS","created_at":"2026-05-18T12:25:47.700082+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/7GAMZYCSNSDJAK43W7E6CTSD73","json":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73.json","graph_json":"https://pith.science/api/pith-number/7GAMZYCSNSDJAK43W7E6CTSD73/graph.json","events_json":"https://pith.science/api/pith-number/7GAMZYCSNSDJAK43W7E6CTSD73/events.json","paper":"https://pith.science/paper/7GAMZYCS"},"agent_actions":{"view_html":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73","download_json":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73.json","view_paper":"https://pith.science/paper/7GAMZYCS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9612218&json=true","fetch_graph":"https://pith.science/api/pith-number/7GAMZYCSNSDJAK43W7E6CTSD73/graph.json","fetch_events":"https://pith.science/api/pith-number/7GAMZYCSNSDJAK43W7E6CTSD73/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73/action/storage_attestation","attest_author":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73/action/author_attestation","sign_citation":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73/action/citation_signature","submit_replication":"https://pith.science/pith/7GAMZYCSNSDJAK43W7E6CTSD73/action/replication_record"}},"created_at":"2026-05-18T01:05:37.347940+00:00","updated_at":"2026-05-18T01:05:37.347940+00:00"}