Detecting reorganization onset via an operator commutator: Kuramoto, Floquet, and discrete scale invariance

The work introduces a two-dimensional operator-based diagnostic framework built from a participation operator (P) and a rigidity operator (M). Two dimensionless observables are defined:

• χ: an effective-dimension ratio tracking dimensional selection and redistribution.

• η: a normalized Frobenius commutator measuring misalignment between participation and rigidity.

The manuscript develops the mathematical properties of the framework, including variational stationary states, commutator bounds, and invariance under Lindblad dynamics, and evaluates the diagnostic across three benchmark systems:

1. Kuramoto synchronization networks

   • η peaks before the conventional synchronization threshold (K_c) in 107 of 109 ensemble realizations across multiple network topologies and system sizes.

   • Finite-size scaling is consistent with a positive asymptotic precursor gap.

   • Direct comparison against pairwise transfer entropy shows earlier detection and substantially lower seed-to-seed variance.

2. Floquet-driven quantum systems

   • The same operator construction distinguishes sustained-coherence and selection-relaxation regimes in a periodically driven transverse-field Ising model.

3. Discrete-scale-invariant model spectra

   • The framework recovers imposed log-periodic scaling ratios through collapse-based analysis with sub-percent accuracy.

   • This benchmark serves as a methodological validation of sensitivity to hidden scale structure.