A Frobenius structure in two summation variables and integral p-adic interpolation of the Almkvist–Zudilin numbers (ζ)

The Almkvist–Zudilin numbers (ζ), u(n) = Σ_{k,l} C(n,k)² C(n,l) C(k,l) C(k+l,n) = 1, 3, 27, 309, 4059, …, form one of the fifteen sporadic Apéry-like sequences. Gorodetsky proved in 2019, through q-congruences, that u(mp^r) ≡ u(mp^(r−1)) (mod p^(3r)) for every prime p ≥ 5. In earlier papers the author showed that, for seven of the fifteen sporadic sequences, such congruences are shadows of a stronger fact, a p-adic analytic interpolation of r ↦ u(mp^r) with p-integral coefficients, obtained from a Frobenius structure in the summation variable of a single binomial sum. The sequence (ζ) is one of the eight for which no such single-sum structure is known; its natural representation is a double sum. This paper shows that the method works in two summation variables for (ζ). Main results. Theorem A. For every prime p ≥ 5 and every m ≥ 1, u(mp^r) = Φ_m(p^r) + (−1)^r Ψ_m(p^r) for all r ≥ 0, where Φ_m and Ψ_m are power series that converge on a disc of radius larger than 1, with Ψ_m = 0 when p ≡ 1 (mod 6). Among series converging on the unit disc the pair is unique. Their coefficients are p-adic integers, and Φ_m(z) − Φ_m(0) and Ψ_m(z) lie in p³ z³ Z_p[[z]]. Corollary B. u(mp^r) ≡ u(mp^(r−1)) (mod p^(3r)) for all primes p ≥ 5 and all m, r ≥ 1. This is Gorodetsky's theorem; the proof here is new and independent of his. Corollary C. The normalized differences (u(mp^r) − u(mp^(r−1)))/p^(3r) are congruent to L_0(m) + (−1)^r L_1(m) modulo p^(r−1). In particular their limits exist along each parity class of r, and from r = 2 on, for each class, the congruence modulo p^(3r+1) holds either for all r or for none. Theorem D (structure). There is a 6×6 matrix Θ(y) of p-adic power series with V(py) = Θ(y)V(y), where V(x) is a vector of six double moments of the summand. The module of moments has dimension exactly 6 for every level y ≠ −1, and the reduction to its basis is polynomial in the level. Θ(y) ≡ Θ(0) + y·Θ'(0) (mod p³) with an explicit Hodge form. Θ(0) is diagonalizable with eigenvalues 1, p, p, p², p³ and χ(p)p³, where χ(p) = (−3/p). Θ commutes with an explicit shift matrix R(x) of determinant 27 that realizes the recurrence in the upper index, and det Θ(y) = χ(p)·p^10·27^((p−1)y). Θ has analytic eigen-sections for its six eigenvalues, and they can be chosen to form a matrix with p-adic integral Taylor coefficients. Method. (1) Rhombus digits. Splitting k = pk' + t and l = pl' + u with the usual digits produces a pole along k' + l' = y − 1. With a system of balanced digits in the plane, for which t + u stays in [0, p−1], the Frobenius weights have no poles at all, and their zeros are explicit. (2) The module of moments. Its dimension is exactly 6. The lower bound is proved with a filtration and an explicit family of syzygies between the two difference operators; a second explicit family of relations makes the reduction polynomial in the level, although the module jumps at y = −1. (3) Frobenius and the image. That the Frobenius operator maps the image of the difference operators at level py into the image at level y is a telescoping over the digits. In two variables the rhombus has three kinds of exits, and the third produces a diagonal difference operator, which lies in the image by an explicit identity. (4) The spectrum. The eigenvalues 1, p, p, p² come from jets at the origin. The other two are detected by functionals built from sums of (−1)^l C(k,l)·g(k,l) over triangles in the summation lattice; these sums are quasi-polynomials of period 6, because Σ_l (−1)^l C(σ−l, l) is periodic of period 6 in σ, and Frobenius acts on the period by multiplication by p. This is where χ(p) = (−3/p) comes from, without any congruence for truncated hypergeometric sums. (5) Eigen-sections. At the diagonal point k = l = y all the weights except one vanish to second order. Evaluating the reduction identities and their jets there gives eigen-sections for the eigenvalues 1, p, p, p², and the obstructions at the resonances are killed by two Volkenborn functionals on the coordinate axes, which are exact eigenfunctionals of Frobenius. Since the first coordinates of these jets are 1, 0, 0, 0, the coefficients of degree 1 and 2 of Φ_m vanish; this is why the supercongruence is not needed as an input. (6) The determinant and integrality. The determinant formula comes from the commutation with the shift, proved with twisted modules of moments. With the Hodge form, an integral basis of eigenvectors and the integrality criterion of the author's previous paper, adapted to a 6×6 matrix with repeated eigenvalues, the eigen-sections are integral and Theorem A follows. Observations (not proved). The sign: for p ≡ 5 (mod 6) the alternating part L_1(m) of Corollary C vanishes numerically, although the series Ψ_m itself does not (for p = 5 this is shown with exact values of u(5^r m)). The paper reduces the vanishing of L_1(m) to an explicit identity for the first row of Θ(0), involving the Volkenborn integral I_2 = ∫ x^(−2) over Z_p^×, which holds to the full precision computed (mod 5^10 and 7^8). One more digit: numerically the normalized differences stabilize modulo p^r, for all parities. This is the analogue of the conjecture of the previous paper for the Apéry, Domb, (δ) and (ε) sequences. Verification. Apart from one step, the proofs do not depend on computer calculations. That step is a symbolic computation: two 6×6 determinants of multiplication maps between modules of moments, and the shift matrix obtained from them (Lemma 5.2 of the paper). It is reproduced by the scripts in this record. Independently of the proofs, the following were checked numerically: the weights and their zeros, the reduction and its degree, the identities between difference operators, the shift matrix against exact moments, the Frobenius matrix for p = 5 and 7 (Hodge form, eigenvalues, commutation and determinant), the diagonal eigen-sections, and, with exact integers, the corollaries and the observations for p = 5, 7, 11 and 13. Limits and open questions. Among the eight sporadic sequences without a single-sum Frobenius structure, Zagier's sequence F is also a double sum. In our computations its module of moments changes basis at every integer level y ≥ 3 that we tested; if this persists, this representation gives no analytic Frobenius matrix. The sequences (η) and s18 are single sums of a different shape. We do not know whether the method extends to them. The two observations above remain open. Authorship. The author of this work is Roberto Blanco Gómez. The structural tests behind this work were carried out with SyntheticMind, the mathematical engine that the author created, designed and programmed himself in Rust. The AI system Claude (Anthropic) collaborated as an assistant, under the author's direction, in developing and checking the arguments, in independent review passes over the proofs, in the verification scripts and in drafting, translating and editing the text. The author has checked every statement and every proof and takes full responsibility for the content. Files: Blanco_Gomez_2026_zeta_two_variable_Frobenius_EN.pdf: the paper. Blanco_Gomez_2026_zeta_two_variable_Frobenius_ES.pdf: the Spanish version, with the same content. verification.zip: the verification scripts (Python 3 with SymPy), README.txt, run_all.sh and the complete output. sources.zip: LaTeX sources of both versions.