Black-hole interiors admitting an inner (Cauchy) horizon are dynamically unstable through the Poisson–Israel–Ori mass-inflation mechanism, an instability governed by the Dray–’t Hooft–Redmount relation and widely regarded as a dynamical realization of strong cosmic censorship. We show that a specific class of anisotropic interiors – static, spherically symmetric sources supported by a marginally divergent but integrable energy density $\rho \sim A/r^2$, with $A>1/(8\pi )$ and mild screening at finite radius – is kinematically incompatible with this mechanism. For such sources the compactness $q(r)=m(r)/r$ decreases monotonically from a finite, super-critical value at the center to $1/2$ at the (unique) event horizon, producing a strictly positive throughout the trapped region and excluding any second zero of the lapse. We then show that this static gap has dynamical consequences in which it enters quantitatively. (i) Under the null energy condition and the dynamical form of the screening condition, $2m/r$ is non-decreasing along both future null directions of a trapped double-null domain, so the gap propagates under nonlinear spherically symmetric evolution and the lapse cannot approach zero there. (ii) For ingoing radiation, described exactly by a generalized Vaidya geometry, the gap bounds the blueshift: the advanced time spent in the gapped region is at most $2(h-\epsilon )/{\Delta }_{\epsilon }$, whereas it diverges at a Cauchy horizon. (iii) For counter-streaming null dust on the fluid background, the radiative part $\mu$ of the mass obeys exactly ${\partial }_u{\partial }_v\mu =2{\partial }_u\mu {\partial }_v\mu /(r|f|)$, of which the Dray–’t Hooft–Redmount relation is the thin-shell form; the coupling between ingoing and outgoing fluxes that drives mass inflation is therefore $2/(r|f|)$, and on the interior annulus it is uniformly bounded precisely when the gap is positive. This yields an explicit a priori bound on the mass for boundary data below a threshold set by the gap and, in the Ori thin-shell model, an e-folding rate bounded by the influx luminosity instead of tending to the inner-horizon surface gravity ${\kappa }_{-}$. Marginally screened integrable-singularity interiors thus realize a class of sources for which the Poisson–Israel–Ori amplification channel is structurally inoperative in the null-dust and thin-shell models in which mass inflation was originally established.