Notes on the bootstrap of four-point conformal integrals

We set up a bootstrap workflow to study four-point conformal integrals in position space, using leading singularities, single-valued multiple polylogarithmic ans\"atze and boundary data from expansion by regions. These four-point conformal integrals are general in the sense that they are generated by the four-point projections of all possible $f$-graphs, including all non-planar $f$-graph sectors. For three-loop cases, fourteen of the fifteen inequivalent integrand basis can be directly calculated by \texttt{HyperlogProcedures} and the last one is fixed by Gram identity. Then we concentrate on how far the bootstrap workflow can go for four-loop cases, though it works for three-loop cases as well. We show that integrals with several leading singularities can be made tractable by decomposing them into pieces with simpler cut structure. Some four-loop integrals which can not be calculated or very hard to be calculated by other methods for now are obtained in this way. We also provide a package with skill files which is suitable to be read and used by current AI models.