Smooth-curvature high-force extension law ($⟨\mathcal{L}-L⟩/L\sim B^2⟨({\kappa }_0^{\prime }{)}^2⟩/(2F^2)$)

For a stationary quenched intrinsic curvature with power spectrum $S_{\kappa }(q)$ and finite second spectral moment, the bulk deficit satisfiesThe moment is $⟨({\kappa }_0^{\prime }{)}^2⟩$ for mean-square differentiable curvature. Infinite-moment disorder and finite endpoint boundary layers can change the exponent, so this is a smooth bulk law rather than a universal law for all random polymers.