Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds

We continue to develop Bootstrability -- a method merging Integrability and Conformal Bootstrap to extract CFT data in integrable conformal gauge theories such as $\mathcal{N}$=4 SYM. In this paper, we consider the 1D defect CFT defined on a $\frac{1}{2}$-BPS Wilson line in the theory, whose non-perturbative spectrum is governed by the Quantum Spectral Curve (QSC). In addition, we use that the deformed setup of a cusped Wilson line is also controlled by the QSC. In terms of the defect CFT, this translates into two nontrivial relations connecting integrated 4-point correlators to cusp spectral data, such as the Bremsstrahlung and Curvature functions -- known analytically from the QSC. Combining these new constraints and the spectrum of the $10$ lowest-lying states with the Numerical Conformal Bootstrap, we obtain very sharp rigorous numerical bounds for the structure constant of the first non-protected state, giving this observable with seven digits precision for the 't Hooft coupling in the intermediate coupling region $\frac{\sqrt{\lambda}}{4\pi}\sim 1$, with the error decreasing quickly at large 't Hooft coupling. Furthermore, for the same structure constant we obtain a $4$-loop analytic result at weak coupling. We also present results for excited states.