Relational (conditional) time, environment-induced decoherence, and the decoherent-histories formalism are often encountered as three disconnected and mathematically demanding topics in the curriculum. We present a compact bridge module that unifies them in a single solvable clock–system–environment model. Starting from a Page–Wootters-type conditional-state construction on C+S+E, the reduced dynamics of a qubit system S undergoing pure dephasing is governed by a single environment overlap factor g(t). Working within the same Stinespring dilation, we show that the same g(t) multiplies the off-diagonal elements of the decoherence functional for a two-time history family in which the first event is a pointer (which-path) alternative. The result is not new physics but a transparent, checkable “consistency check” that helps students see, in one place, that the emergence of an effective time arrow in open systems and the consistency of classical histories are enforced by the same information-leakage mechanism.
Soyoung Kim (Tue,) studied this question.