Alex Shvets

For the mixed CM point (a,b,c) = (1/6, 1/3, 1), define Aₙᵐⁱˣ := 108ⁿ[zⁿ] ₂F₁(1/6, 1/3; 1; z)³. For every split prime p ≥ 7, p ≡ 1 (mod 3), and every m ≥ 1, we prove unconditionally Aₘₚᵐⁱˣ ≡ Aₘᵐⁱˣ (mod p⁴). The exponent 4 exceeds the generic weight-3 Hodge-gap prediction of 3; the extra factor of p is a CM enhancement attached to j = 0. We also establish the matching unconditional inert-prime obstruction (p ≡ 2 mod 3), both as a formal-parameter congruence on the q-side and as a coefficient-level Cartier parity law modulo p. The proof uses the modular realization on Γ₀(3) with parameter t = u/(1+27u)², a Lagrange–Bürmann reduction to three Cartier identities Λₚ(Cₘᵢₓ·Uₚ^ℓ) ≡ 0 (mod p⁴) for ℓ = 1, 2, 3, a saturated weak q-expansion lattice on the rigidified stack X₀(3) handling vertical integrality, and a length-three Witt–Cartier pole estimate at the elliptic point P₋ driven by μ₃-equivariance of the canonical Frobenius lift.

We give an independent eta-product derivation of the level-8 Apéry limit lim Bₙ⁽⁸⁾/sₙ = (7/32)ζ(3), where sₙ = Σ_{k=0}^n C(n,k)²·C(2k,n)² and Bₙ⁽⁸⁾ is the rational companion sequence satisfying the same cubic recurrence with initial values B₀⁽⁸⁾ = 0, B₁⁽⁸⁾ = 1. This value was identified numerically by Almkvist–van Straten–Zudilin and proved by Golyshev via Beukers's Atkin–Lehner modular method; it was later recomputed by Golyshev–Kerr–Sasaki in the motivic / normal-function framework. The continued fraction PCF((2n+1)(3n²+3n+1), −n⁶) = 8/(7ζ(3)) already appears in Batut–Olivier and was later rediscovered by the Ramanujan Machine as conjecture Z1. The contribution of the present paper is an explicit rederivation, in the eta-product normalization, of the already-known level-8 Apéry limit: the eta-product verification of the Wronskian identity, the normalization of the Eichler integral, the residue computation of the Fricke period polynomial, and the elementary continuant conversion.

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) = (1/2, 1, 3/2). We place these kernels in a unified Sym² 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) = 108x²/(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 Sym². We further prove an inverse classification: for a fixed Sym²-type Riemann scheme, the one-parameter family of Fuchsian operators contains a unique Sym²(Gauss) point, cut out by the closed-form condition λ₀ = 2γ₁γ₂(1−2α) on the accessory parameter. Finally, a Belyi-pullback scan over 5040 configurations produces 11 additional integer sequences of the form [xⁿ]λⁿ·₂F₁(a,b;c;φ(x))²; we prove their integrality and place them in the same Sym²-pullback framework.

We prove that the ratio Bₙ/Dₙ of the Apéry-like sequence Bₙ to the Domb numbers Dₙ converges to (7/24)ζ(3), and that Σ_{n=1}^∞ 64ⁿ/(n³·Dₙ·Dₙ₋₁) = (56/3)ζ(3). As a corollary we establish the value Z₂ = 12/(7ζ(3)) conjectured by the Ramanujan Machine project. The proof uses level-6 eta products, Atkin–Lehner involutions, and Eichler integrals of weight-4 modular forms.

We prove that the symmetric-cube coefficients Aₙ = (−27)ⁿ[zⁿ] ₂F₁(1/3, 1/3; 1; z)³ satisfy the supercongruence A(mp) ≡ A(m) (mod p⁴) for every prime p ≥ 5 and every m ≥ 1. The proof rests on three ingredients: (i) the modular identification F(t(τ)) = η(τ)⁹/η(3τ)³ with t(τ) = η(3τ)¹²/η(τ)¹², whose logarithmic derivative is the weight-5 Eisenstein series C(q) = 3E₅(χ₀, χ₃) on Γ₀(3); (ii) exact congruences cₘₚʳ ≡ cₘₚʳ⁻¹ (mod p⁴ʳ) for the coefficients of C, combined with a Lagrange–Bürmann extraction; and (iii) a Hecke descent on weakly holomorphic forms, where the defect is expanded in the two-dimensional space of weight-5 forms on Γ₀(3) with character χ₃, spanned by C and tC, via a cusp-adapted basis, with the second cusp handled by the Fricke involution W₃. As an independent result, we show that the Mao–Tian cubic recurrence drops from order 3 to order 2 at the specialization (1/3, 1/3, 1).