Theorem: The complete three-parameter contact family

Every permitted response differs from ${{\Pi }^{\mathrm{R}}}_{\mathrm{ref}}$ by exactly a tensor of the form

The three tensors are linearly independent. Here $\alpha$ has mass dimension two and $\beta ,\gamma$ are dimensionless in the stated four-dimensional convention.

Conversely, every contact (eq:contacts) gives a permitted extension.

Proof. Two extensions agreeing off zero differ by a distribution supported at zero. A point-supported distribution is a finite sum of delta derivatives. The scaling-degree bound restricts its derivative order to at most four, so its Fourier transform is a tensor polynomial of degree at most four.

Parity-even Lorentz covariance and the pair symmetries give the general polynomial form

where $A,B$ have degree at most two in $s$, $C,D$ at most one, and $E$ is constant. Contracting with $p^a$ gives the independent equations

Writing $C=c_0+c_{1}s$, $D=d_0+d_{1}s$, these require

There are therefore exactly three free real coefficients.

The degree-two solution is proportional to $\mathsf{E}$. At degree four the independent spin-two and spin-zero solutions are ${\mathsf{Q}}_2,{\mathsf{Q}}_0$. Their independence follows on a timelike rest frame: ${\mathsf{Q}}_2,{\mathsf{Q}}_0$ are independent projectors multiplied by $s^2$, while $\mathsf{E}$ has degree two. This proves (eq:contacts).

Each displayed contact is real, local, transverse, pair-symmetric, and of scaling degree at most eight. Its support at zero is compatible with retarded support, proving the converse.