The Lawrence-Sullivan construction is the right model of $I^{+}$

We prove that the universal enveloping algebra of the Lawrence-Sullivan construction is a particular perturbation of the complete Baues-Lemaire cylinder of $S^0$. Together with other evidences we present, this exhibits the Lawrence-Sullivan construction as the right model of $I^{+}$. From this, we also deduce a generalized Euler formula on Bernoulli numbers.