Theorem: Exact seed action

For compact smooth scalar $W$ supported in $U$,

Proof. In the $47$-chart,

Expand the derivative of $fL$ before transposing. The outer minus derivative supplies a plus test derivative; three seed derivatives supply a minus sign, and two supply a plus sign. The three resulting terms are (eq:P47).

For $K_{5,+}$,

These line signs cancel. Transposition of the outer derivative and the three seed derivatives gives another cancellation, yielding (eq:P5p). The second placement gives (eq:P5m).

For $K_6$, the line-only derivative is $-{\partial }_{s^{\alpha }}R(s)$. Transposing the outer derivative and this line derivative gives (eq:P6).

For response rows, (eq:response) becomes

Its sign cancels on transposition. In $K_8$, the inner derivative gives ${\partial }_{s^{\rho }}B=(s_{\rho }/2)R$ on the separated future support. The outer minus derivative then gives (eq:P8). The root terms give (eq:P9d) and (eq:P9m), with the original mass sign retained.

Finally apply the measure (eq:RB) once or twice. The charts have unit absolute determinant, giving the ordinary-integral formulas.