We develop a double-null framework for Cauchy-horizon dynamics in metric $f(R)$ gravity that separates ordinary mass inflation from scalaron cancellation and genuine escape from the blueshift instability. Starting from the exact spherical Hawking-mass transport equation, we show that a regular nondegenerate Cauchy horizon undergoes mass inflation whenever the effective longitudinal source is eventually nonnegative, uniformly dominates the mixed channel, and has a divergent blueshift-weighted integral. Hence bounded mass requires sufficiently strong blueshift-integrable suppression of this source or an independent competing channel. We then analyze the weaker branch in which the scalaron cancels only the leading Price-tail contribution. Under noncanceling transverse and trace asymptotics, this forces $|R|\to \infty$ and an asymptotically linear high-curvature theory, $f(R)/R\to F_{-}>0$, where $F_{-}\equiv {\lim }_{v\to \infty }{f}_R$. The curvature coefficient is fixed explicitly by the transverse asymptotics. For regular model classes with a finite high-curvature $f_{RR}$ limit one obtains $f_{RR}\to 0$. More sharply, every eventually viable branch with $f_{RR}>0$ is forced to $R\to -\infty$ and ${\mathrm{\Lambda }}_{\infty }>-1$; the standard scalaron mass parameter then diverges, while an additional differentiable rate condition yields $m_{\mathrm{sc}}^2\sim F/(3f_{RR})\to +\infty$. In the Einstein frame the scalaron null kinetic contribution is non-negative, so the Jordan-frame cancellation cannot be interpreted as negative scalaron null energy. The result is a local rigidity classification rather than a global theorem of strong cosmic censorship.

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