What zero-past uniqueness does and does not say

The identities

use fields vanishing near the initial surface. They are not identities for arbitrary Cauchy data. For example, for a general smooth scalar $u$,

where

Both sides solve the same equation with zero data. Dropping the homogeneous subtraction changes the prescription.

Nor is homogeneous uniqueness of a wave equation a claim that every zero-past solution of an unrestricted gauge-degenerate Einstein equation vanishes. Compact pure gauges already refute that stronger assertion.