Recently, we announced the construction of a new nonpolynomial closed-string field theory which successfully reproduced all $N$-point tree amplitudes, thus solving a long-standing problem. This action generalized the four-string tetrahedron interaction that we introduced earlier, which was required to derive the Shapiro-Virasoro model. However, we gave no derivation of the nonpolynomial action. In this paper we present the detailed analysis of its derivation. We calculate all nonpolynomial graphs up to eighth order and their coefficients, and even the coefficients of several infinite series of polyhedra. We use gauge invariance to calculate all coefficients. Because of an enormous redundancy created by a web of overconstrained equations, we have multiple checks that our coefficients are correctly calculated.