Francesco Sacco

Authored papers (3)

• Topological constraints on self-organization in locally interacting systems2026

  Topological constraints on interaction graphs determine whether a locally interacting system can sustain long-range ordered phases, and therefore whether it can self-organize toward a system-level goal. By analyzing free-energy scaling under domain-wall formation across three model systems—the Potts model, autoregressive models, and hierarchical networks—the paper establishes necessary conditions on graph topology for an ordered phase to exist. The core method introduced is a domain-wall free-energy scaling analysis applied comparatively across these three substrate classes, tracking how the combinatorics of graph connectivity either suppress or permit spontaneous ordering. On planar graphs with the interaction topology characteristic of flat autoregressive language models, domain-wall formation is combinatorially cheap relative to system size, so long-range coherence degrades as sequence length grows; this is a structural, not a training, limitation. Hierarchical and multiscale networks—the topology prevalent in biological systems such as morphogenetic and neural substrates—make domain-wall formation sufficiently costly that ordered phases persist across scales. Published in the Phil. Trans. R. Soc. A theme issue on world models (384, issue 2320, 2026), the paper operationalizes the claim that all intelligence is collective intelligence via free-energy geometry, arguing that architectural hierarchy is not merely a performance heuristic but a topological prerequisite for sustained coherent self-organization, with direct implications for why biological morphogenesis succeeds at pattern maintenance where flat LLMs structurally cannot.

• Topological constraints on self-organisation in locally interacting systems2025

  Topology of local interactions is the decisive factor determining whether a system can sustain long-range order, and decoder-only transformer architectures are provably unable to maintain such order for arbitrarily long output sequences. By generalizing the Landau–Lifshitz scaling argument and Peierls' domain-wall counting to a broad universality class via a Topological Equivalence Theorem (Theorem 1), the paper shows that any local Hamiltonian on a graph shares asymptotically equivalent free energy with a nearest-neighbour Ising model on the same combinatorial structure—meaning the existence or non-existence of a phase transition reduces entirely to graph topology. Three model systems are analyzed: the one-dimensional windowed Potts model, AR(ω) autoregressive models (Corollary 2), and hierarchical clique networks. For the Potts chain, domain-wall entropy scales as log(L−1) while energy is bounded by ωE^max, forcing ΔF negative for sufficiently large sequence length L at any nonzero temperature. Transformer attention with a finite context window ω maps directly onto the AR(ω) framework (Proposition 2 via Theorem 3), inheriting the same no-go result. Conversely, biological systems organized as nested cliques—cells forming tissues, tissues forming organs—admit a non-empty critical temperature range of hierarchical order (Proposition 3), achievable when the clique count ℓ and size nmax satisfy ℓ/r > nmax^nmax/e. The paper argues this constitutes a principled thermodynamic explanation for why autoregressive LLMs exhibit coherence failures on long tasks while multicellular organisms maintain large-scale morphogenetic order, and proposes that stigmergy and embodiment function as evolutionary responses to this topological no-go constraint.

• Topological constraints on self-organisation in locally interacting systems (Sacco, Sakthivadivel, Levin 2025)

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  sacco-2025-topological.md
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  sacco-2025-topological.md