Autonomous noisy oscillations in biochemical and mesoscopic systems require nonequilibrium driving and therefore dissipation. A striking conjecture by Oberreiter, Barato, and Seifert (OBS) proposes a universal lower bound on the entropy produced per oscillation period in terms of the coherence number of the slowest oscillatory mode L. Oberreiter et al. [Phys. Rev. E 106, 014106 (2022)]. Here we derive a weaker but rigorous lower bound that preserves the OBS structure while introducing a mode-uniformity factor that quantifies how evenly the oscillatory eigenmode is distributed across states in the steady-state inner product. The result makes explicit that an eigenvalue-only prefactor can fail when the dominant oscillatory mode is localized. We also outline a proof-of-principle route for estimating this factor from low-dimensional data under single-mode dominance and sufficiently informative measurements, and derive an eigenvector-free corollary using only the smallest stationary probability. Translation-invariant Markov jump processes on a ring provide a symmetry-protected class with $\eta =1$, so the refinement reduces to the OBS form; the drift-diffusion limit on a circle saturates the bound.

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• Nonequilibrium & irreversible thermodynamics
• Nonequilibrium statistical mechanics
• Stochastic processes
• Data analysis
• Markovian processes
• Master equation
• Time series analysis