Finite source implementation

The source specification can be reproduced without selecting a different density:

1. choose the basis and algebraic dual in (eq:duality);

2. expand ${q_s}^{-1}$, $\sqrt{-\det q_s}$, and the declared curvature only through the order needed for $[s^3]$;

3. identify commuting mixed field derivatives by ordinary multiindices;

4. form the coordinate Euler derivatives of $\sigma {\ell }_3$, retaining switch derivatives;

5. apply (eq:sourceconversion);

6. extract (eq:sourcehessian);

7. impose the plateau only after the local source has been formed;

8. contract the full branch–vertex inventory in a common density and test convention.

The mathematical checks are: slot symmetry, total derivative weight, repeated-variable factors, basis covariance, and

No executed component output is claimed.