{
  "abstract": "We give a deterministic polynomial-resource black-box identity test for noncommutative algebraic branching programs (ABPs), using ordinary matrix evaluations of the output polynomial. Given an alphabet of size $m\\ge1$ and an ABP vertex bound $S\\ge1$, we construct one fixed tuple of integer matrices of order $R=(m+2)d(d+1)/2$, where $d=(m+2)S$, with entry bit length $O(d\\log(d+1))$. Every nonzero polynomial computed by an ABP with at most $S$ vertices has a nonzero evaluation at this tuple. The construction accesses neither the hidden ABP nor its intermediate gates. A rectangular variant improves the order to $O(m^{3/2}S^2)$. Matrix order, description length, and all returned entries are counted; the total resources are polynomial. The result holds in characteristic zero and in characteristic $p>\\binom d2$. The main ingredient is a fixed polynomial frame spanning the joint cyclic module of every matrix tuple from every prescribed vector. A compensating block makes a tagged encoding nilpotent of index at most three, while preserving the original cyclic module. A differential exterior-vector argument then gives a single triangular recurrence with polynomially many outputs. Its shared left-linear circuit is converted into actual integer query matrices by coefficient duality. Two-sided pairing further gives deterministic nonadaptive reconstruction of an equivalent ABP from polynomially many output-only matrix queries. The same frame gives polynomial-cardinality explicit trace separators for simultaneous-conjugation orbit closures, and a fixed nilpotent substitution for bounded-dimensional recognizable series. The latter also tests rational formulas whose given syntax is defined at the scalar origin. We provide complete proofs, construction costs, and exact implementation checks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37677",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhipeng Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26hxhrkez3rhs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37677",
  "fulltext_url": null,
  "id": "26hxhrkez3rhs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37677",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37677",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26hxhrkez3rhs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37677",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37677"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Explicit matrix hitting points for noncommutative algebraic branching programs",
  "title_text": "Explicit matrix hitting points for noncommutative algebraic branching programs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/26hxhrkez3rhs",
  "venue": "arXiv",
  "versions": []
}