Both completion legs and the compact collar root

For this interface, fix:

1. a finite-order coordinate operator

   $${X_1}^{\mu }[r]={E_0}^{\mathrm{R}}{D^{\mu }}_{\mathrm{coord}}r,$$

   whose smooth differential coefficients vanish for $t\le 0$;

2. a bilinear scalar differential expression

   $$V_{1,g}[r]\varphi =\sum _{I,\alpha ,\beta }{v}_{\alpha \beta ,I}(x)({\partial }^{\alpha }{r}_I)({\partial }^{\beta }\varphi ),$$

   with fixed smooth coefficients supported in a compact switched region.

All coefficients and signs are input data. They are not inferred from similarly named operators in another manuscript.

Define the two-leg completion

Its first term is the coordinate leg; its second is the switched scalar leg. Let ${\alpha }_{f,g}[\varphi ]$ be the exact formal-transpose root:

The star here denotes formal differential transpose, not a quantum product.