An explicit presentation model

One local retarded presentation space can be defined independently. For a multi-normal variable $q$, an internal difference $r\in {\mathbb{R}}^4$, and passive variables $s$, take finite sums of formal expressions

with smooth coefficient $a$, polynomial $P$, and the fixed tempered multiplier. Use inverse Fourier transforms in $\theta ,\kappa$ and define

This is well defined by tensor product and smooth multiplication. For a complete finite graph presentation use a direct sum of its declared coordinate charts and reconstructions. Let

with the quotient topology of the chosen presentation space.

This local definition makes the independent descent theorem concrete. It is not claimed to be the entire ${{\mathfrak{A}}_{\mathrm{R}}}^2$ module of XIV-I, whose exact coefficient space, seminorms, uniform annular estimates, and remainder were not supplied in full.