A unique local Hamiltonian with window length ω can be associated to any AR(ω) model (Theorem 3)

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• All local Hamiltonians on lattices with the same combinatorial structure have asymptotically equivalent free energies (Theorem 1)finding0.762
  Topological equivalence theorem for local Hamiltonians
• 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)finding0.748
  Establishes that the boundary is a modelling choice not determined by the underlying physics
• For one-dimensional local Hamiltonian with m>1 stored patterns at non-zero temperature, domain wall formation is thermodynamically favourable (Theorem 2)finding0.737
  No ordered phase in 1D with multiple stored patterns
• 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.finding0.733
  Core result demonstrating topological constraints on self-organization
• Windowed Hamiltonianframework0.720
  Core technical concept: a Hamiltonian where spin-spin interactions are defined only within finite windows ω, enabling generic analysis across diverse lattices
• Local Hamiltonianconcept0.714
  Sum of windowed Hamiltonians with same window length; a key construct introduced in the paper to model local interactions on graphs
• Purely local-interaction systems on 1D topologies cannot maintain long-range order at finite temperature.claim0.707
• Bell's theorem: experiments by Aspect, Clauser, and Freedman show instantaneous correlations between distant quantum particles, violating local realism.finding0.706
  A foundational empirical result undermining mechanistic separability, cited as evidence that the whole influences local events.