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 $\binom{N-1}{k-1}|J_N|\|h\|$; 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 $\Delta_{N,L}=E_* (1-\cos\theta_L)$, where $\tan(L\theta_L/2)=1/\sin\theta_L$ and $0<\theta_L<\pi/L$. In particular, $\Delta_{N,L}/E_*=\pi^2/(2L^2)-2\pi^2/L^3+O(L^{-4})$, 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 $E_*(1-1/\sqrt2)$. 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.

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

arXiv:2610.14081 [quant-ph]

(or arXiv:2610.14081v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.14081 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

generate_cyclic_gap.py

requirements.txt

verification_results.json

verify_relational_histories.py

verify_state_concentration.py

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