Purely local-interaction systems on 1D topologies cannot maintain long-range order at finite temperature.

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• Purely local autoregressive systems cannot maintain long-range coherence at finite temperature.claim0.832
• 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.finding0.777
  Core result demonstrating topological constraints on self-organization
• In hierarchical systems with independent cliques, there exist parameter regimes where individual cliques maintain uniform magnetisation while others flip.finding0.769
  Shows how hierarchical topology enables local order within global flexibility; explains biological multiscale organization
• For one-dimensional local Hamiltonian with m>1 stored patterns at non-zero temperature, domain wall formation is thermodynamically favourable (Theorem 2)finding0.768
  No ordered phase in 1D with multiple stored patterns
• There exists a non-empty critical temperature range of hierarchical behaviour (Proposition 3)finding0.767
  Proof that the conditions of Theorem 4 are realisable in a range of temperatures
• Topology is the critical factor differentiating systems capable of long-range order from those that are not.claim0.765
  Key interpretive position: topological properties of interaction graphs determine whether systems can self-organize, independent of substrate
• The results generalise readily to non-equilibrium systems where scaling relationships remain similar (e.g., dynamic or localised scaling).claim0.760
  Claim about broader applicability of the scaling argument
• We conclude that certain systems cannot remain ordered over long ranges for all time, and suggest this no-go theorem as a source of fitness pressure for biological phenomena like stigmergy and embodiment.quote0.752
  Final concluding statement linking the mathematical result to biological evolution