On the Spectral Structure of Deviations from the Temporal Attractor

Recent work has argued that physical time should be treated not as a coordinate or external parameter, but as an operationally defined quantity governed by a temporal rate β. Within this framework, a stable infrared temporal attractor ⟨β⟩ emerges, providing a universallarge-scale flow of physical time while excluding any meaningful temporal microstructure in horizon regimes. The present work addresses a remaining open question: whether any residual temporal structure may persist once such an infrared attractor has formed. We formulate a minimal spectral hypothesis, according to which deviations from the attractor, δβ = β − ⟨β⟩, may exhibit scale-dependent organization under further coarse-graining. The hypothesis does not introduce new microscopic dynamics or reinterpret existing experimental data, butconstrains the possible asymptotic behavior of operational time measurements. We show that the hypothesis is sharply falsifiable. If all residual deviations vanish under sufficient coarse-graining, the attractor description is complete. If, however, correlated deviations persist in the infrared, their behavior must obey nontrivial scaling constraints. In this sense, the spectral hypothesis functions as a structural consistency condition on any theory that treats physical time as an emergent, operational quantity.