An elementary factorization criterion

This criterion constructs no distributional cross-line product. Its use requires an already specified candidate $\tilde{M}$.

The simplest degree-mixing example is

Indeed,

Under the supplied reconstruction convention,

The two separate terms reconstruct to opposite nonzero distributions. Thus highest presentation degree is not an intrinsic reconstructed symbol. This does not invalidate estimates on a specified presentation module; it requires descent before promoting a presentation-dependent splitting.