Why odd-space and odd-time dimensions in even-dimensional spaces?

We are answering the question why 4-dimensional space has the metric 1+3 by making a general argument from a certain type of equations of motion linear in momentum for any spin (except spin zero) in any even dimension d. All known free equations of motion for non-zero spin for massless fields belong to this type of equations. Requiring Hermiticity of the equations of motion operator as well as irreducibility with respect to the Lorentz group representation, we prove that only metrics with the signature corresponding to q time + (d−q) space dimensions with q being odd exist. Correspondingly, in four dimensional space, Nature could only make the realization of 1+3-dimensional space.