Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians

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 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!/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- θ L), where (Lθ L/2)=1/ θ 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 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.