The Scale-Invariance Condition: Which Contraction Histories Can Source a Flat Magnetic Spectrum

A primordial magnetic field that speaks to observations on many scales should have a nearly flat energy spectrum, and flatness is a condition on the history that produced it, not an automatic outcome. Within a stated model in which the contraction is a power law in time and the electromagnetic coupling is a power law in the scale factor, this paper derives the flatness condition and tabulates the coupling exponent it demands across contraction histories. For a matter-like contraction the required exponent is 2, the value found in the single-history coupling analysis. As the contraction stiffens, the conformal-time exponent shrinks and the requirement grows steeply: a stiff equation of state of nine demands an exponent of 56, and the stiffest case tabulated demands nearly six hundred, which no reasonable coupling supplies. Soft contractions are therefore the natural home of coupling-driven magnetogenesis with a flat spectrum, and stiff, ekpyrotic contractions must either accept a tilted spectrum or use a different mechanism. The result is a map from bounce constructions to the magnetogenesis they can support. Computed within the stated model, not measured.