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Given an integer $R\\ge1$, consider a function $g:\\Delta_{nR}^{(n)}\\to\\mathbb R$ satisfying the complete oriented-simplex relations \\[ \\sum_{i=1}^n g(\\beta+e_i)=0, \\qquad \\beta\\in\\Delta_{nR-1}^{(n)}, \\] where $e_i$ is the $i$th standard basis vector. We prove that a nonzero value at the balanced point forces the support-cardinality estimate with optimal growth exponent: if $g(R,\\ldots,R)\\"},"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":"2608.02707","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-08-03T17:27:46Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"45a8df1cd14d132dbc1fd73d9fb66482819a28315d71fabe130bea30bbdb785e","abstract_canon_sha256":"364735e3c9d85b44ace5ff90de253d6f9a0eb5907ae1c420f15a1995b9d3cf5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-05T00:39:28.996853Z","signature_b64":"zMHKu8b5Pt5WKt136If27zk517/PrAMFWnA5yh+oKwSEp6k80FX1uQCJU8+XmEMFRtw2plmas/H9NkDTwYIZDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d49b8267cf15d1f0ab7e271dfda2bbd7ac076b8e7ab2e587e14666ea722feb4","last_reissued_at":"2026-08-05T00:39:28.993797Z","signature_status":"signed_v1","first_computed_at":"2026-08-05T00:39:28.993797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrete Unique Continuation on Simplex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Linjun Li","submitted_at":"2026-08-03T17:27:46Z","abstract_excerpt":"For integers $N\\ge0$ and $n\\ge2$, let \\[ \\Delta_N^{(n)} =\\left\\{\\alpha\\in\\mathbb Z_{\\ge 0}^n: \\alpha_1+\\cdots+\\alpha_n=N\\right\\}. \\] We formulate a discrete unique-continuation problem on this lattice simplex. Given an integer $R\\ge1$, consider a function $g:\\Delta_{nR}^{(n)}\\to\\mathbb R$ satisfying the complete oriented-simplex relations \\[ \\sum_{i=1}^n g(\\beta+e_i)=0, \\qquad \\beta\\in\\Delta_{nR-1}^{(n)}, \\] where $e_i$ is the $i$th standard basis vector. We prove that a nonzero value at the balanced point forces the support-cardinality estimate with optimal growth exponent: if $g(R,\\ldots,R)\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.02707","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/2608.02707/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":"2608.02707","created_at":"2026-08-05T00:39:28.994995+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.02707v1","created_at":"2026-08-05T00:39:28.994995+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.02707","created_at":"2026-08-05T00:39:28.994995+00:00"},{"alias_kind":"pith_short_12","alias_value":"TVE3QJT46FOR","created_at":"2026-08-05T00:39:28.994995+00:00"},{"alias_kind":"pith_short_16","alias_value":"TVE3QJT46FOR6CVX","created_at":"2026-08-05T00:39:28.994995+00:00"},{"alias_kind":"pith_short_8","alias_value":"TVE3QJT4","created_at":"2026-08-05T00:39:28.994995+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.11602","citing_title":"Quantitative Unique Continuation on Simplex and $\\mathbb{Z}^n$","ref_index":12,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV","json":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV.json","graph_json":"https://pith.science/api/pith-number/TVE3QJT46FOR6CVX4JY57WRLXV/graph.json","events_json":"https://pith.science/api/pith-number/TVE3QJT46FOR6CVX4JY57WRLXV/events.json","paper":"https://pith.science/paper/TVE3QJT4"},"agent_actions":{"view_html":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV","download_json":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV.json","view_paper":"https://pith.science/paper/TVE3QJT4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.02707&json=true","fetch_graph":"https://pith.science/api/pith-number/TVE3QJT46FOR6CVX4JY57WRLXV/graph.json","fetch_events":"https://pith.science/api/pith-number/TVE3QJT46FOR6CVX4JY57WRLXV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV/action/storage_attestation","attest_author":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV/action/author_attestation","sign_citation":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV/action/citation_signature","submit_replication":"https://pith.science/pith/TVE3QJT46FOR6CVX4JY57WRLXV/action/replication_record"}},"created_at":"2026-08-05T00:39:28.994995+00:00","updated_at":"2026-08-05T00:39:28.994995+00:00"}