An alternative extension of the punctured numerical kernel

This is a numerical extension, not automatically a compatible quantum insertion prescription. Exchange symmetry, reality, field-equation relations, and any gauge conditions must be imposed on it separately.

For the insertion (eq:IR), the chosen route is the Hadamard-subtracted smooth remainder, not an unannounced identification with every raw Taylor extension.

There is no additional off-diagonal light-cone extension in this internal rod contraction: equal-time distinct points are spacelike. Other graphs may have light-cone products and require their own analysis.