Low energy gauge couplings in grand unified theories and high precision physics

I generalize the leading log relations between low energy SU(3){sub QCD}, SU(2){sub {rvec I}} and U(l){sub Y} effective gauge couplings to include all one-loop threshold effects of matter fields in oblique vector self energy quantum corrections for both supersymmetric and non-supersymmetric SU(5) grand unified theories. These always involve an exactly conserved current from the unbroken SU(3){sub QCD} {times} U(L){sub QED} subgroup; this fact strongly constrains any non-decoupling of heavy states as well as the generic character of threshold effects. Relations between low energy gauge couplings depend on the details of the spectra of both the superheavy and low mass sectors; I display the common origin of the logs appropriate to superheavy matter states, which can be found with well known renormalization group techniques, and the combination of logs and polynomials appropriate for light matter states, which cannot. Relations between any two or all three low energy effective gauge couplings do not depend on the top quark or standard model Higgs` masses. Neither do they depend on neutral color singlet states such as other neutral color singlet Higgs` or higgsinos, neutrinos, zinos or photinos. Further, they do not depend on degenerate SU(5) matter representations, of either spin 0 or spin 1/2 of any mass; matter representations of SU(5) can affect such relations only if there is mass splitting within them. The b quark splitting from the {tau} and {nu}{sub {tau}} can affect the relation between gauge couplings for {vert_bar}q{sub 2}{vert_bar} {yields} m{sub b}{sup 2} as can hadronic resonances and multi-hadron states for lower {vert_bar}q{sub 2}{vert_bar}. New mass-split representations of light states, such as occur in supersymmetric theories, can also affect such relations