{
  "abstract": "Exact permutation covariance does not itself supply a sparse metric geometry. For permutation-covariant, full-support $k$-body interaction orbits, the incident interaction norm scales as $\\binom{N-1}{k-1}|J_N|\\|h\\|$; such orbits cannot have uniformly finite range in metrics of diverging diameter. A permutation-invariant Hamiltonian diagonal in the graph basis has ground-state multiplicity at least $N!/|\\mathrm{Aut}(G)|$ whenever $G$ is a ground graph. We give a quantitative version for approximately symmetric energies, together with fixed-radius connectivity and unitary-isometry obstructions. The trace-distance asymmetry of an arbitrary state also bounds its probability of selecting one labeled graph. We then construct two exactly solvable finite relational history Hamiltonians. A cyclic clock with a reversible incidence update has a unique ground state; gauge reduction yields a cycle Laplacian with a rank-one pin and the exact gap $Δ_{N,L}=E_* (1-\\cosθ_L)$, where $\\tan(Lθ_L/2)=1/\\sinθ_L$ and $0<θ_L<π/L$. In particular, $Δ_{N,L}/E_*=π^2/(2L^2)-2π^2/L^3+O(L^{-4})$, independently of the number of vertices. In an alternative order-free architecture, one clock qubit per unordered pair gives commuting four-local terms, $2^N$ ground states and a size-independent gap $E_*(1-1/\\sqrt2)$. A fixed-weight control penalty selects $\\binom Nq$ ground states without reducing the gap. These exact finite results expose clock and locality resource tradeoffs; they do not establish an emergent geometric phase.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14081",
  "author_total": 1,
  "authors": [
    {
      "name": "Maxim V. Churilov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bknz1j8d3ws5d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14081",
  "fulltext_url": null,
  "id": "bknz1j8d3ws5d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14081",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum many-body systems"
    ]
  ],
  "openalex": "W7221771214",
  "pdf_url": "https://arxiv.org/pdf/2610.14081",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bknz1j8d3ws5d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.14081",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14081"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians",
  "title_text": "Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/bknz1j8d3ws5d",
  "venue": "arXiv",
  "versions": []
}