Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians
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 binomN−1k−1|JN||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!/|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 ΔN,L=E* (1−cosθL), where tan(LθL/2)=1/sinθL and 0<θL<π/L. In particular, ΔN,L/E*=π2/(2L2)−2π2/L3+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, 2N ground states and a size-independent gap E*(1−1/√2). 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.
Related
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BibTeX
@misc{churilov2026prelocality,
title = {{Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians}},
author = {Maxim V. Churilov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14081},
eprint = {2610.14081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14081},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians
AU - Maxim V. Churilov
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.14081
UR - https://arxiv.org/abs/2610.14081
AB - 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 binomN−1k−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!/|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/√2). 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.
ER -