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LIT-1668: Cycle holonomy captures higher-order compatibility constraints in remote synchronization

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StatusActive — read in full 2026-10-27 (NOTE-1218), from the arXiv preprint v2; worth reading for a clean statement of when edge-wise phase relations can be met around cycles, and for its own caution that the point of maximal cycle frustration is not where remote synchronization loses stability. The spectral criterion is standard mathematics; the synchronization result is one seven-node motif.
Version2
Tagsnetwork-science, complex-systems, mathematics
Cited as “paper” byNOTE-1218 — Torres-Hugas et al. — Cycle holonomy and remote synchronization
Cited as “source” byTHEORY-1082 — On a connected graph whose edges carry phase shifts αᵢⱼ = −αⱼᵢ, the edge relations zⱼ = e^{iαᵢⱼ}zᵢ have a common nonzero solution, equivalently the twisted Laplacian Lα = δα*δα has a zero mode, exactly when the phase accumulated around every cycle is 0 mod 2π; otherwise each relation can be met alone but not all together, and the smallest eigenvalue of Lα, positive, measures the residual incompatibility, which on an isolated ℓ-cycle is largest at holonomy π, THEORY-1087 — In the symmetric Sakaguchi–Kuramoto model on a triangle and a pentagon sharing one vertex, the lag α gives each ℓ-cycle a phase-independent path mismatch (ℓ − 2)α, and the remote-synchronized locked branch passes through an exact two-group state at α = π/3, where the pentagon's mismatch reaches π; but that branch has already lost transverse stability, in the pentagonal symmetry sector, at αc ≃ 1.0164, so the point of maximal cycle frustration marks the geometry of the phase relations, not the end of the synchronized pattern

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Torres-Hugas, L., Duch, J., Gómez, S., & Arenas, A. (2026). Cycle holonomy captures higher-order compatibility constraints in remote synchronization. Communications Physics. doi:10.1038/s42005-026-02914-y. Preprint: arXiv:2604.19682.

Read in full on 2026-10-27 (NOTE-1218).

Why this is here

A21 §4 found it two weeks after publication and gave it “especially high priority”: a constraint that belongs to no edge but appears when relations are composed around a cycle, with a spectral criterion and dynamical consequences. A23 §V cited it for frustration as a third route to organization. It bears on CLAIM-471 and QUESTION-281.

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