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

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 Apéry-like sequences $A036917$ and $A002895$ (Domb numbers, case~$(\alpha )$), while the Catalan kernel is a hypergeometric twist of the Gauss-square coefficient sequence at $(a,b,c)=(\frac{1}{2},1,\frac{3}{2})$. We place these kernels in a unified ${\mathrm{Sym}}^2$ 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 $\phi (x)=108x^2\mathrm{/}(1-4x{)}^3$ 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 ${\mathrm{Sym}}^2$. We further prove an inverse classification: for a fixed ${\mathrm{Sym}}^2$-type Riemann scheme, the one-parameter family of Fuchsian operators contains a unique ${\mathrm{Sym}}^2(\mathrm{G}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s})$ 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 $[x^n]{\lambda }^n{}_2{F}_1(a,b;c;\phi (x){)}^2$; we prove their integrality and place them in the same ${\mathrm{Sym}}^2$-pullback framework.