Prepared-source input

The supplied XII-C source theorem gives the Euler source of the selected weighted cubic density as

For a time-only switch, its reported general layer formula is

The coefficient labels $C_2$ and $M_2$ belong to XII-C. Their general component formulas are not reconstructed here XIIC.

For the calculations below, the necessary explicit input is the reported pure-trace restriction:

In that sector, XII-C reports $C_2=0$, so the coefficient of ${\sigma }^{\prime \prime }$ in (eq:time-layer) vanishes.

These are imported formulas, not a new derivation of the full cubic Euler source. For fixed smooth input fields, the source evaluations considered below are smooth tensor fields.

The quadratic polarization convention is

for each homogeneous quadratic map $B$. Complex inputs are evaluated through complex bilinear extension, not through a Hermitian polarization.