Critical cusp cancellation and analytic response for a Bernoulli random interval map

We show that the critical cusp exponent $\beta =-1\mathrm{/}2$ admits $W^{2,1}$ invariant densities in an explicit family of expanding full-branch interval maps. The usual estimates exclude this exponent because two zero-order coefficients in the second derivative of the transfer operator are individually non-integrable. Their signed combination, however, contains the factor $\beta (2\beta +1)$ for a pure power cusp. We exploit its cancellation at $\beta =-1\mathrm{/}2$ to prove a $W^{2,1}$ Lasota–Yorke inequality and a spectral gap for the explicit family. We also identify a separate obstruction caused by an interior cusp value. As an application, a small Bernoulli probability perturbation between two fixed maps has a stationary density that depends real-analytically on the perturbation parameter in $W^{2,1}$, with explicit formulas for all response derivatives.