Order-3 pi-formulas, Apery-like kernels, and Clausen functoriality for Conservative Matrix Fields

Abstract

Raz, Shalyt, Leibtag, Kalisch, Weinbaum, Hadad, and Kaminer recently showed that formulas for $\pi$ can be organized by canonical polynomial recurrences and partially unified by a rank-$2$ Conservative Matrix Field (CMF). We prove that each order-$3$ recurrence explicitly printed in the public Appendix~B.6 of their paper is a shifted summation lift of an explicit order-$2$ kernel, and identify all three kernels: the two $\pi$-kernels are explicit rescalings of the sporadic Ap\'ery-like sequences $A036917$ and $A002895$ (Domb numbers, case~$(\alpha)$), while the Catalan kernel is a hypergeometric twist of the Gauss-square coefficient sequence at $(a,b,c)=(\tfrac12,1,\tfrac32)$. We place these kernels in a unified $\operatorname{Sym}^2$ framework: the first $\pi$-kernel and the Catalan kernel come directly from Gauss-square coefficient sequences, while the Domb kernel is recovered by recasting the classical degree-$3$ Belyi pullback $\phi(x)=108x^2/(1-4x)^3$ and the associated algebraic twist in CMF language. We write an explicit square-gauge matrix for the Gauss CMF, formulate the standard pullback--twist transport in CMF terms, and show that for rank-$2$ objects it is compatible with $\operatorname{Sym}^2$. We further prove an inverse classification: for a fixed $\operatorname{Sym}^2$-type Riemann scheme, the one-parameter family of Fuchsian operators contains a unique $\operatorname{Sym}^2(\mathrm{Gauss})$ point, cut out by the closed-form condition $\lambda_0=2\gamma_1\gamma_2(1-2\alpha)$ on the accessory parameter. Finally, a Belyi-pullback scan over $5040$ configurations produces $11$ additional integer sequences of the form $[x^n]\lambda^n\,{}_2F_1(a,b;c;\phi(x))^2$; we prove their integrality and place them in the same $\operatorname{Sym}^2$-pullback framework.