Order-3 π-formulas, Apéry-like kernels, and Clausen functoriality for Conservative Matrix Fields

We show that the printed order-3 recurrences for π and Catalan’s constant in Raz et al. are shifted summation lifts of explicit order-2 kernels. The kernels are identified as rescalings of A036917 and A002895 (Domb), and a Gauss-square specialization. All three arise from a unified mechanism: the symmetric square of a Gauss hypergeometric CMF, combined with summation, and in the Domb case with a Belyi pullback and twist. We further give an explicit square-gauge for Gauss CMFs, recover the kernels via the differential component, and prove a pullback–twist functoriality result. This establishes a direct CMF-level link between the Raz et al. framework and Apéry-like sequences.