Regulated coordinate coefficient equations and data

{Term ledger for (eq:X1eq)–(eq:X2eq).}

• Ordinary metric terms: ${\sigma }_{\epsilon }{C}^{\mu }[h]$; ${\sigma }_{\epsilon }{C}^{\mu }[k]$; $-{\sigma }_{\epsilon }{\partial }_{\alpha }{a_2}^{\alpha \mu }[h]$; $-{\sigma }_{\epsilon }{\partial }_{\alpha }({a_1}^{\alpha \beta }[h]{\partial }_{\beta }{{\mathsf{X}}_1}^{\mu })$.

• Regulator-derivative terms actually present: those with one factor ${\sigma }_{\epsilon }^{\prime }$. No ${\sigma }_{\epsilon }^{\prime \prime }$ term occurs.

• Switch terms: none; $g$ does not enter the coordinate prescription.

• Data: zero Cauchy data for ${\mathsf{X}}_1,{\mathsf{X}}_2$; the data of ${\mathsf{X}}_q$ are $(x^{\mu },{{\delta }^{\mu }}_0)$.

The second coefficient is derived from the frozen regulated problem, not obtained by attaching ${\sigma }_{\epsilon }$ to the unswitched equation.