Definition: The joint-map class

Let ${\mathcal{K}}_{\mathrm{int}}$ be the space of actual reconstructed internal kernels obtained from the finite presentations in the copied precomposition package, with their tensor indices retained.

The class considered here consists of linear operator maps

that:

1. on the named reconstructed sum give the specified contraction with the grouped cross-lines and mixed vertex Hessian;

2. retain both $K_5$ placements and the residual variable $z$;

3. agree with the canonical composition on its proved domain;

4. are defined on arbitrary compact tests meeting $N$;

5. have a stated and proved complete-graph wavefront bound appropriate to their composition.

The product data and ${\mathfrak{C}}_{\mathrm{part}}$ are unchanged.

The domain consists of reconstructed kernels, not arbitrary amplitude arrays. No representative-independent operation on all free-shell quotient classes is asserted.

A generalized composition belongs to this class only after its relation, amplitude, and remainder hypotheses are supplied for the present target. No common product phase is introduced.