The paper establishes a universal factorial bound for the growth of correlation functions produced by topological recursion. For any regular spectral curve that has only simple ramification points, the authors prove that the n‑point, genus‑g correlator grows at most like ((2g-2+n)!) as the genus (g) tends to infinity. This rate matches the expected factorial growth that appears in many enumerative problems, such as counting maps, computing Weil–Petersson volumes, and evaluating intersection numbers on moduli spaces. The result supplies a rigorous upper bound for a wide class of curve‑counting problems in the large‑genus regime and provides a necessary step toward a full resurgence analysis of the associated asymptotic series.

The proof combines several modern tools. The authors use the language of quantum Airy structures to encode the recursion, and they analyse local spectral curves and their associated Frobenius manifolds. Boundedness conditions on the recursion kernels allow them to control the combinatorial sums over stable graphs that appear in the explicit formulae for the free energies (F_g). By comparing the Airy‑structure amplitudes with the topological‑recursion amplitudes, they derive both upper and lower bounds. While the lower bounds are less uniform and involve an exponential factor that does not match the upper bound, the upper bounds are sharp up to a constant prefactor. The constant (A) appearing in the bound is shown to give a lower bound on the distance of the nearest singularity of the Borel transform of the n‑point function from the origin.

A key application is to the Painlevé I equation. The authors recover an upper bound on the Painlevé I amplitudes that is close to the exact large‑genus asymptotics obtained by Kapaev. They also derive large‑genus asymptotics for the Painlevé I free energies and show that the bound is not far from the exact values. This illustrates how the general theorem can be specialised to concrete integrable systems and intersection‑theory problems. The paper further discusses how the method could be extended to irregular spectral curves, provided a uniform upper bound for intersection numbers involving the Θ‑class is available. The authors note that new techniques would be required to handle non‑simple ramification points, such as those arising in Witten r‑spin theory.

The work was carried out by G. Borot, B. Eynard, and A. Giacchetto. It was initiated while Borot and Giacchetto were affiliated with the Max‑Planck‑Institut für Mathematik in Bonn, with support from the Max‑Planck‑Gesellschaft. The project was completed during a stay of Borot at IHES, where he benefited from excellent working conditions. Giacchetto received an ETH Fellowship (22‑2 FEL‑003) and a Hermann‑Weyl‑Instructorship from the Forschungsinstitut für Mathematik at ETH Zürich. The research was partially funded by the ERC Synergy project “Recursive and Exact New Quantum Theory” (ReNewQuantum). The paper was submitted to arXiv on 26 September 2024 and is classified under primary 14H10, 14H70 and secondary 37K20, 05A16.