General designs and an irrationality exponent for Catalan's constant

The preprint "Catalan's constant is irrational" (OpenAI, 24 September 2026) proves the irrationality of Catalan's constant by comparing two bounds for a sequence of determinants built from five integer parameters; the certified gap between the two bounds is about 0.00013. We state four conditions on the parameters under which the argument of the preprint goes through unchanged, verify each step of the preprint for general parameters, and exhibit a second choice of parameters that satisfies the four conditions with a certified gap of 0.0026, by exact computations of the same kind as in the preprint. For this choice we bound all minors of the determinants, by an energy estimate with fewer nodes and by bounds for Schur polynomials, and obtain an upper bound below two thousand seven hundred and fifteen for the irrationality exponent of Catalan's constant; measurements of the determinants near the constant indicate that the method gives about two thousand one hundred, and show which bounds account for the difference. We also show that the best gap the real-place estimate of the preprint can certify for given parameters is the value of an equilibrium problem for a logarithmic energy of two measures, and compute it for both choices. Attempts to extend the method to the Dirichlet beta value at four are reported in a companion note. Scripts, run logs and coefficient files: https://github.com/MarcosCarreira/catalan-designs. This note was prepared with the assistance of Claude (Anthropic). All computations were run by the author on his own computer. It has not been refereed. https://github.com/MarcosCarreira/catalan-designs