Order-3 pi-formulas, Apery-like kernels, and Clausen functoriality for Conservative Matrix Fields
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911584332283904 |
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| author | Shvets, Alex |
| author_facet | Shvets, Alex |
| contents | Raz, Shalyt, Leibtag, Kalisch, Weinbaum, Hadad, and Kaminer recently showed that formulas for $π$ 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 $π$-kernels are explicit rescalings of the sporadic Apéry-like sequences $A036917$ and $A002895$ (Domb numbers, case~$(α)$), 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 $π$-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 $ϕ(x)=108x^2/(1-4x)^3$ 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 $λ_0=2γ_1γ_2(1-2α)$ on the accessory parameter. Finally, a Belyi-pullback scan over $5040$ configurations produces $11$ additional integer sequences of the form $[x^n]λ^n\,{}_2F_1(a,b;c;ϕ(x))^2$; we prove their integrality and place them in the same $\operatorname{Sym}^2$-pullback framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09723 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Order-3 pi-formulas, Apery-like kernels, and Clausen functoriality for Conservative Matrix Fields Shvets, Alex Number Theory 33C20, 11B83, 33C45 Raz, Shalyt, Leibtag, Kalisch, Weinbaum, Hadad, and Kaminer recently showed that formulas for $π$ 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 $π$-kernels are explicit rescalings of the sporadic Apéry-like sequences $A036917$ and $A002895$ (Domb numbers, case~$(α)$), 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 $π$-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 $ϕ(x)=108x^2/(1-4x)^3$ 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 $λ_0=2γ_1γ_2(1-2α)$ on the accessory parameter. Finally, a Belyi-pullback scan over $5040$ configurations produces $11$ additional integer sequences of the form $[x^n]λ^n\,{}_2F_1(a,b;c;ϕ(x))^2$; we prove their integrality and place them in the same $\operatorname{Sym}^2$-pullback framework. |
| title | Order-3 pi-formulas, Apery-like kernels, and Clausen functoriality for Conservative Matrix Fields |
| topic | Number Theory 33C20, 11B83, 33C45 |
| url | https://arxiv.org/abs/2604.09723 |