Small numerators canceling small denominators: Is Dyson’s hierarchical model solvable?

We present an analytical method to solve Dyson’s hierarchical model, involving the scaling variables near the high-temperature fixed point. The procedure seems plagued by small denominators as in perturbative expansions near integrable systems in Hamiltonian mechanics. However, in 36 cases considered, a zero denominator always comes with a zero numerator. We conjecture that these cancellations occur in general, allowing the application of the analytical method and suggesting that the model has remarkable features reminiscent of the integrable systems.