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 non-cancelling transverse and trace asymptotics, this forces |R| →∞ and an asymptotically linear high-curvature theory, f (R)/R → F− > 0, where F− ≡ limv→∞ fR. The curvature coefficient is fixed explicitly by the transverse asymptotics. For regular model classes with a finite high-curvature fRR limit one obtains fRR → 0. More sharply, every eventually viable branch with fRR > 0 is forced to R →−∞ and Λ∞ > −1; the standard scalaron mass parameter then diverges, while an additional differentiable rate condition yields m^2_sc ∼ F/(3fRR) → +∞. In the Einstein frame the scalaron null kinetic contribution is nonnegative, 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.