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 $\left(\genfrac{}{}{0ex}{}{N-1}{k-1}\right)|J_N|\Vert h\Vert$; 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 ${\Delta }_{N,L}=E_{\ast }(1-\cos {\theta }_L)$, where $\tan (L{\theta }_L/2)=1/\sin {\theta }_L$ and $0<{\theta }_L<\pi /L$. In particular, ${\Delta }_{N,L}/E_{\ast }={\pi }^2/(2L^2)-2{\pi }^2/L^3+O(L^{-4})$, 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 $E_{\ast }(1-1/\sqrt{2})$. A fixed-weight control penalty selects $\left(\genfrac{}{}{0ex}{}{N}{q}\right)$ ground states without reducing the gap. These exact finite results expose clock and locality resource tradeoffs; they do not establish an emergent geometric phase.

arXiv comment10 pages, 1 figure. Ancillary finite-matrix verification scripts included