finding

active

finding:all-local-hamiltonians-on-lattices-with-the-same-combinatorial-structure-have-asymptotically-equivalent-free-energies-theorem-1

All local Hamiltonians on lattices with the same combinatorial structure have asymptotically equivalent free energies (Theorem 1)

Topological equivalence theorem for local Hamiltonians

Source paper

extracted_from

Topological constraints on self-organisation in locally interacting systems

(2025) · Francesco Sacco · Dalton A R Sakthivadivel · Michael Levin

Neighborhood — ranked by edge-count

Claims (1)

claim

Topology is the critical factor differentiating systems capable of long-range order from those that are not.
supports
Key interpretive position: topological properties of interaction graphs determine whether systems can self-organize, independent of substrate

Communities (4)

community

Causal emergence in biological systems
members_of
Examines how macro-scale causal power exceeds micro-scale in living and learning systems.
Hierarchical structure and multiscale coherence in physical systems
members_of
How graph topology and hierarchical interaction patterns enable or prevent phase transitions and ordered states, from statistical mechanics to biological organization.
Lattice topology and thermodynamic phase transitions
members_of
How combinatorial structure of local Hamiltonians determines free energy equivalence classes and long-range order feasibility across temperature regimes.
Lattice Hamiltonian free energy universality
members_of
Theorem showing combinatorial lattice structure determines asymptotic free energy equivalence across local Hamiltonians.

Concepts (1)

concept

Local Hamiltonian
about
Sum of windowed Hamiltonians with same window length; a key construct introduced in the paper to model local interactions on graphs

Questions (1)

question

What is the functional distinction between simple language models and multicellular organisms, and can generative AI harness that property to achieve long-range order?
answered_by
Core motivating question; drives investigation of topological differences between biological and artificial systems

Related by similarity (8)

cosine ≥ 0.65 · no typed edge

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

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.finding0.769
Core result demonstrating topological constraints on self-organization
A unique local Hamiltonian with window length ω can be associated to any AR(ω) model (Theorem 3)finding0.762
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)finding0.744
Establishes that the boundary is a modelling choice not determined by the underlying physics
Bell's theorem: experiments by Aspect, Clauser, and Freedman show instantaneous correlations between distant quantum particles, violating local realism.finding0.736
A foundational empirical result undermining mechanistic separability, cited as evidence that the whole influences local events.
Local symmetries are pervasive in nature (sun, trees, crystals, bodies) and are associated with minimum energy and least-action principles.claim0.734
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.claim0.734
For one-dimensional local Hamiltonian with m>1 stored patterns at non-zero temperature, domain wall formation is thermodynamically favourable (Theorem 2)finding0.723
No ordered phase in 1D with multiple stored patterns
Purely local-interaction systems on 1D topologies cannot maintain long-range order at finite temperature.claim0.721