DoT: The Duality of Time Postulate

For two alternatives with equal populations, write the accessible density matrix as

The off-diagonal entry retains the coherence available to a later comparison. Testing the superposition $|+_\varphi\rangle=(|0\rangle+e^{\mathrm{i}\varphi}|1\rangle)/\sqrt2$ gives

Scanning $\varphi$ yields a fringe visibility $$\mathcal V=\frac{p_{\max}-p_{\min}}{p_{\max}+p_{\min}}=|\zeta|.$$ A coherent equal superposition has $|\zeta|=1$; an equal incoherent mixture has $\zeta=0$. Both give probabilities $1/2,1/2$ when measured directly in the original channels, yet they have different future-test records.

An environmental interaction can multiply $\zeta$ by an overlap between the two environmental states. If those states are orthogonal, the accessible fringe disappears. Recovering it requires control of the correlations that were omitted from the reduced record. This gives completion a measurable content: a proposed retention rule must predict which values of $\zeta$ remain accessible and under which interventions. Merely calling a configuration completed cannot decide that question.