Existence of solutions to the 3×3 Painlevé I parametrix Riemann–Hilbert problem

Prove the existence (and, as appropriate, uniqueness) of a solution to the 3×3 Painlevé I model Riemann–Hilbert problem introduced in the appendix, with the specified jump matrices on the rays γ_k and the asymptotic normalization involving the Hamiltonian 𝓗(x) and Painlevé I transcendent q(x), at least for x away from poles and for the stated Stokes data (e.g., κ=1).

Background

In the η<0 critical regime, the analysis requires a 3×3 Painlevé I parametrix. The paper formulates the corresponding Riemann–Hilbert problem, including the jump configuration and asymptotic expansion tied to the Painlevé I Hamiltonian and solution q(x).

While heuristic construction via a generalized Laplace transform from the classical 2×2 Painlevé I problem is suggested, a rigorous existence proof for the 3×3 parametrix is not provided, and establishing it would solidify the foundation of the critical analysis.