  We constrain X-ray spectral shapes for the ensemble of AGN based on the shape
of the Cosmic X-ray Background (CXB). Specifically, we rule out regions of
X-ray spectral parameter space that do not reproduce the CXB in the energy
range 1-100 keV. The key X-ray spectral parameters are the photon index,
{\Gamma}; the cutoff energy, Ecutoff; and the reflection scaling factor, R.
Assuming each parameter follows a Gaussian distribution, we first explore the
parameter space using a Bayesian approach and a fixed X-ray luminosity function
(XLF). For {\sigma}_E = 36 keV and {\sigma}_R = 0.14, fixed at the observed
values from the Swift-BAT 70-month sample, we allow <R>, <Ecutoff > and
<{\Gamma}> to vary subject to reproducing the CXB. We report results for
{\sigma}_{\Gamma} = 0.1-0.5. In an alternative approach, we define the
parameter distributions, then forward model to fit the CXB by perturbing the
XLF using a neural network. This approach allows us to rule out parameter
combinations that cannot reproduce the CXB for any XLF. The marginalized
conditional probabilities for the four free parameters are: <R> =
0.99^{+0.11}_{-0.26}, <Ecutoff> = 118^{+24}_{-23}, {\sigma}_{\Gamma} =
0.101^{+0.097}_{-0.001} and <{\Gamma}> = 1.9^{+0.08}_{-0.09}. We provide an
interactive online tool for users to explore any combination of <Ecutoff>,
{\sigma}_E, <{\Gamma}>, {\sigma}_{\Gamma}, <R> and {\sigma}_R including
different distributions for each absorption bin, subject to the integral CXB
constraint. The distributions observed in many AGN samples can be ruled out by
our analysis, meaning these samples can not be representative of the full AGN
population. The few samples that fall within the acceptable parameter space are
hard X-ray-selected, commensurate with their having fewer selection biases.
