  We present a mechanism for producing oscillations along the lift of the
Teichm\"uller geodesic flow to the (real) Hodge bundle, as the basepoint
surface is deformed by a unipotent element of $\text{SL}_2(\mathbb{R})$.
Invoking Chen-M\"oller, we apply our methods to all but finitely many strata in
genus $4$, those exhibiting a varying Lyapunov-exponents phenomenon.
