  We further develop the gravitational model, Thomas-Whitehead Gravity (TW
Gravity), that arises when projective connections become dynamical fields. TW
Gravity has its origins in geometric actions from string theory where the TW
projective connection appears as a rank two tensor, $\mathcal{D}_{ab}$, on the
spacetime manifold. Using a Gauss-Bonnet (GB) action built from the
$(\mathrm{d}+1)$-dimensional TW connection, and applying the tensor
decomposition $\mathcal{D}_{ab} = D_{ab} + 4\Lambda /(\mathrm{d}(\mathrm{d}-1))
g_{ab}$, we arrive at a gravitational model made up of a
$\mathrm{d}$-dimensional Einstein-Hilbert + GB action sourced by $D_{ab}$ and
with cosmological constant $\Lambda$. The $\mathrm{d}=4$ action is studied and
we find that $\Lambda \propto 1/J_0$, with $J_0$ the coupling constant for
$D_{ab}$. For $\Lambda$ equal to the current measured value, $J_0$ is on the
order of the measured angular momentum of the observable Universe. We view this
as $\Lambda$ controlling the scale of patches of the Universe that acquire
angular momentum, with the net angular momentum of multiple patches vanishing,
as required by the cosmological principle. We further find a universal axial
scalar coupling to all fermions where the trace, $\mathcal{D} =
\mathcal{D}_{ab}g^{ab}$ acts as the scalar. This suggests that $\mathcal{D}$ is
also a dark matter portal for non-standard model fermions.
