Consider learning what "Helly sets" are, here in the guise of "RS-covers": an edge-covering set of complete sets of vertices; you will be rewarded with the many puzzles of designing "widgets" via old-school math graphs: sets of vertices and edges.

A "widget" for 3-column "OR" would combine with a "tree" to give an effective reduction of "Clique Graph" to "3SAT", that is, an alternate form of the proof that Clique Graph is equivalent to 3SAT.

A "widget" for "4-OR" would combine with the "variable" widgets to give a direct equivalence of "Clique Graph" to "4SAT", the latter being substantially more intuitive and useful in direct form than "3SAT". And likewise for "nSAT", n>4.

Algorithmic advances have sped up each iteration of the code, first the Python code (trusted as correct for what it does) and now the C++ code (in prototype form, trusted as correct thus far). The far goal is to continue to operate near to P versus NP, while upgrading the algorithm also to learn how and why P is not equal (we would assume) to NP. It is to let the machine learn to "think" in graphs and triangles, tantalizing the neural network engineers or the working mathematician-meets-computer scientist of today.