Definition: Three successor objects

For a selected sourced Einstein–scalar model, distinguish:

1. Invariant coupled jet. The coefficients solve the expanded invariant equations, including the stress compatibility and Cauchy constraints implied by the chosen action and normalization. The allowed gauge transformations must preserve that equation space.

2. Gauge-fixed forced response. A named reduced operator is inverted with a declared gauge-fixing and support or initial-data prescription. For example, one may prescribe de Donder-reduced wave equations and zero response data at $t_0$. Such a reduced response must be checked separately against the invariant constraints.

3. Reduced functional. A complete coefficient functional is shown to be constant on a named quotient of (i), or on a separately justified quotient of (ii). Its correction includes both the forced scalar coefficient and the coordinate compensator.

The full sourced successor comprising these objects is not constructed in this paper. Existence of (ii) alone does not imply existence of (i), and it does not establish descent in (iii).