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

TL;DR

This paper explores order-3 pi-formulas, identifies related kernels, and unifies them within a symmetric square framework, revealing new integer sequences and their algebraic properties.

Contribution

It explicitly links order-3 pi-formulas to order-2 kernels, recasts kernels in a unified framework, and classifies associated Fuchsian operators using CMF techniques.

Findings

Identified all three kernels as rescaled Apéry-like sequences and hypergeometric twists.

Recast kernels within a symmetric square framework and analyzed their algebraic structure.

Discovered 11 new integer sequences related to Belyi pullbacks and proved their integrality.

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.