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Conjectured universal Painlevé I double-scaling limit along the critical surface

Demonstrate the conjectured scaling limit for the isomonodromic τ-function of the (3,4) string equation along any non-tangential direction beneath the tangent plane of the real critical surface: identify a constant C=C(P0, u)>0 and a polynomial Q(t1,t2,t5) such that, with the specified rescalings X5(x|T), X2(x|T), X1(x|T), the normalized differential converges to the Painlevé I Hamiltonian differential, namely lim_{T→∞} d log[ τ(X5, X2, X1) / exp(Q(X5, X2, X1)) ] = −𝓗(x) dx.

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Background

Beyond the two specific double-scaling limits rigorously analyzed in the paper (along the curves γ+ and γ−), the authors propose a broader universality statement covering arbitrary directions from points on the critical surface within the admissible region.

This conjecture predicts that, after an explicit renormalization by a polynomial in the parameters, the τ-function degenerates to the Painlevé I τ-function in a universal manner governed by the Hamiltonian 𝓗(x).

References

Conjecture There exists a choice of constant C = C(P_0,{f u })>0 and a polynomial Q(t_1,t_2,t_5) such that, if we define \hat{\tau}0 := e{Q}, considered as a differential in the variable x, \begin{equation} \lim{T\to \infty} {\bf d} \log \frac{\tau(X_5(x|T),X_2(x|T),X_1(x|T))}{\hat{\tau}_0(X_5(x|T),X_2(x|T),X_1(x|T))} = -\mathcal{H}(x)dx, \end{equation} where \mathcal{H}(x) is the Painlev{e} I Hamiltonian.

— Asymptotic Properties of a Special Solution to the (3,4) String Equation (2507.22646 - Hayford, 30 Jul 2025) in Concluding Remarks, item 2 (Conjecture)