{"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"}