Definition: Joint-map class; transcribed

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

The class consists of linear operator maps

which:

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 the class only after its relation, amplitude, and remainder hypotheses are supplied for the present target. No common product phase is introduced.