Battery Seeds Through a Bounce: The No-Net-Gain Theorem for Adiabatic Histories

The Biermann battery, in which misaligned electron pressure and density gradients generate current, is the standard source of seed magnetic fields that need no pre-existing field. This paper computes battery seeds generated during the hot contracting phase of a bounced cosmology and follows them to the present. The central result is a small theorem of adiabatic histories: flux freezing from the generation epoch to today gives a present field equal to the seed times the square of the generation scale factor, with the bounce cancelling exactly, because the amplification of contraction is repaid in full by the dilution of expansion. For battery generation at 1e10 kelvin on one per cent of the horizon, the seed is 2.08e-04 gauss and the present value 1.55e-23 gauss, leaving a factor 6.47e+13 still to be supplied by non-adiabatic amplification if a nanogauss field is the target. The bounce does not rescue battery seeds; it merely gives them an earlier, longer contracting epoch in which to be made. Computed from a stated order-of-magnitude source model, not measured.