For one-dimensional local Hamiltonian with m>1 stored patterns at non-zero temperature, domain wall formation is thermodynamically favourable (Theorem 2)

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• Free-energy scaling under domain-wall formation in Potts, autoregressive, and hierarchical networks shows that combinatorics of interactions on a graph prevent or allow spontaneous ordering.finding0.795
  Core result demonstrating topological constraints on self-organization
• Purely local-interaction systems on 1D topologies cannot maintain long-range order at finite temperature.claim0.768
• Free-energy scaling under domain-wall formationmethod0.761
  Key analytical technique used across three model systems to determine constraints on long-range order.
• A unique local Hamiltonian with window length ω can be associated to any AR(ω) model (Theorem 3)finding0.737
  Mapping autoregressive models to spin systems
• The total Hamiltonian H_U is equally consistent with any factorisation of the Hilbert space, so nothing about the physics privileges any one boundary placement (Zanardi 2002)finding0.736
  Establishes that the boundary is a modelling choice not determined by the underlying physics
• The deepest invariant of all living structures is that local symmetries are so densely packed that the highest possible density of local symmetries occurs, without having an overall symmetry.claim0.729
• Purely local autoregressive systems cannot maintain long-range coherence at finite temperature.claim0.724
• All local Hamiltonians on lattices with the same combinatorial structure have asymptotically equivalent free energies (Theorem 1)finding0.723
  Topological equivalence theorem for local Hamiltonians