Nucleosynthesis Fences the Early Universe: Boost and Shear Budgets for Any Bounced History

Any modification of the early expansion rate strong enough to shrink the sound horizon by the amount the Hubble tension requires must answer to an older and less negotiable witness: the light-element abundances, which fix the expansion rate during nucleosynthesis through the effective number of relativistic species, N_eff = 2.99 ± 0.17 in the Planck analysis. This paper converts the required expansion boosts, computed in the companion sound-horizon analysis, into that currency, and derives the corresponding budget for the residual shear a bounce leaves behind. A boost of 9.2 percent — the smallest the sound horizon accepts, in its most favourable window — if present at the nucleosynthesis epoch would register as ΔN_eff = 1.44, more than four times the two-sigma allowance of 0.34; the boosts for the higher switch windows register as 2.20 and 5.90. Inverting, the expansion boost at the nucleosynthesis epoch is capped at 2.26 percent (at ΔN_eff = 0.34) or 1.13 percent (at 0.17): any successful modification must therefore switch on *below* redshift ~10^9, after nucleosynthesis closes. For shear the arithmetic is sharper. A free shear density grows against radiation by the square of the redshift ratio, a factor of 8.25 × 10^11 between recombination and nucleosynthesis, so the BBN cap forces the shear fraction at recombination below 5.5 × 10^−14 of the radiation density: a bounce may be born anisotropic, but by the time the microwave background forms, its shear must have been erased to fourteen orders of magnitude below the radiation content. These are hard fences, computed from published bounds; any bounced or boosted history that does not state how it lives inside them has not finished its specification. This study is computed from stated assumptions; it is not a measurement.