This paper gives an explicit formula for the n‑gon graph contribution to the integrand of the cosmological wavefunction in de Sitter space for a conformally coupled phi^3 theory. In plain terms, the authors find a closed‑form special function that reproduces the mathematical piece of the wavefunction coming from a simple loop graph with n sides. The new functions are a B‑type version of previously studied “quadrangular polylogarithms,” and the result is an equality at the level of full functions, not only at the level of a symbol or derivative data.

The authors build their answer from combinatorics of a 4n‑gon. They sum over centrally symmetric quadrangulations of that polygon and attach a specific cross‑ratio to each quadrangle. The functions they define have weight n, a standard measure of complexity for polylogarithms, and they show this weight‑n function equals the n‑gon wavefunction coefficient. The proof links a recursive differential equation obeyed by the physics quantity to a recursive algebraic relation (a coproduct formula) satisfied by the quadrangular polylogarithms. An overall constant is fixed by taking a soft kinematic limit where one momentum goes to zero.

A key mathematical idea is the symbol, which is a way to record the algebraic structure of a polylogarithm without keeping all analytic detail. The symbol’s pieces, called letters, come from a set tied to a finite‑type cluster algebra of type B. The authors find that every letter that appears together in a symbol word sits in a single cluster — a strong compatibility property called total cluster compatibility. This is what lets them use the combinatorics of quadrangulations to build the exact function. Technically, passing from the better‑known A‑type case to B‑type requires folding a polygon by a symmetry. That folding produces special “self‑conjugate” quadrangles whose associated cross‑ratios appear as square roots, and the choice of sign for those roots is important in the construction.