Three sourced successor problems

sourced successor problems{ Use a named gauge group and support class, for example the initial-surface-fixing specification ${\mathcal{G}}_{+,t_0}$ of Section sec:nontransfer, rather than silently importing the C1 residual group.

S1. Invariant coupled jet. Define and solve the full Einstein–scalar coefficient equations, including stress compatibility and sourced Cauchy constraints. Specify permitted metric asymptotics. The coupled first-order scalar result alone does not solve S1.

S2. Gauge-fixed response. Choose the reduced operator, gauge-fixing, and initial-data or retarded prescription. Construct its forced responses and check which satisfy the invariant constraints. Existence of a reduced inverse alone does not solve S1.

S3. Quotient functional. On a named quotient of S1, or a separately justified quotient associated with S2, construct the complete reduced functional or prove a new descent obstruction. A coordinate dressing or one forced response alone does not establish S3.}