The Metrological Non-Constructibility of Time in Canonical Quantum Gravity

The Wheeler–DeWitt (WDW) equation, ˆHΨ = 0, plays a central role in canonical quantum gravity but lacks an explicit time parameter. This feature is traditionally diagnosed as the “prob lem of time.” In this paper, developed within the Sequential Time Theory (STT) program, we argue that the absence of a time parameter is not an anomaly but an inevitable structural con sequence of time metrology. We define physical closure independently of time-theoretic concepts and show that for a physically closed universe, the global external time-derivative operator ∂t is non-constructible under STT metrological premises. The proof demonstrates that linear global evo lution requires a calibration map common across the physical state space, which a closed universe fundamentally lacks. Furthermore, we position relational time programs (e.g., Page–Wootters) and semiclassical approaches as non-competing cases; these frameworks successfully reconstruct inter nal conditional dynamics without defining a global external time. The timelessness of the WDW equation thus emerges as a structural metrological consequence of enforcing physical closure upon a state-independent calibration.