Both internal placements of K 5

For the first placement use

For the second use

The copied maps

are diffeomorphisms onto their images.

On either patch $d_{\pm }$ is nonzero and future directed. Near its null cone choose smooth coordinates

for $d_{\pm }$. The derivative of ${d_{\pm }}^2$ is nonzero there, so these are ordinary regular coordinates.

Let $u_{+}=a$ and $u_{-}=z$. An adapted chart is

Its dimension is sixteen, and $r_{\pm }$ retains the residual metric incidence rather than testing it away.

The construction uses separate normal Fourier transformations of already defined tensor factors. It assigns no common product phase to the six-branch quantum coefficient.

amplitude module

Let $\theta$ be the outer normal frequency dual to $q$. For $j\ge 0$ and a four-dimensional multiindex $\beta$, set

For an adapted chart $V$ and an integer $m\ge -2$, let ${{\mathfrak{A}}_{\mathrm{R}}}^m(V)$ consist of finite expressions

with the finite inherited tensor components understood.

The coefficients carry their usual compact-set smooth seminorms. The frequency factors are distributions, not ordinary smooth symbols at $p(\kappa )=0$.

The module is filtered by the polynomial degree $j+|\beta |$. Its reconstruction is defined by

Normal inverse Fourier transformation is taken on each tensor factor, and coefficient multiplication is performed afterward.

This is a specific finite module with fixed generators. It is not the unrestricted space of tempered amplitudes. No ordinary principal-symbol bundle over a single Lagrangian is asserted.

The polynomial coefficient presentation is unambiguous as a frequency amplitude: multiplication by $M_{\mathrm{R}}$ annihilates no nonzero polynomial, since $M_{\mathrm{R}}=-1/p$ on a nonempty open set. The reconstruction need not be injective, because physical coefficient identities such as $q\delta (q)=0$ may occur. Injectivity of reconstruction is not required.

estimates through the pole

For a distribution $A$ in five frequency variables, $A(L\cdot )$ denotes pullback by the dilation $(\theta ,\kappa )\mapsto (L\theta ,L\kappa )$.