Balanced digits and the p-adic analyticity of Zagier's sporadic sequence E along the towers mp^r

Zagier's sporadic Apéry-like sequence E(n) = Σ_k C(n,k) C(2k,k) C(2n−2k,n−k) (OEIS A081085: 1, 4, 20, 112, 676, 4304, …) is one of the fifteen sporadic sequences of Zagier, Almkvist–Zudilin and Cooper. In previous work (doi:10.5281/zenodo.23928953 and doi:10.5281/zenodo.23935071) we proved, for six of them, that for every prime p ≥ 5 and every m ≥ 1 the whole sequence r ↦ u(mp^r) is the restriction to z = p^r of one or two p-adic power series, convergent on a disc of radius larger than 1, with bounds uniform in m; and we showed that for the other nine every known representation violates a specific hypothesis of the method. For E the obstruction was a pole of the Frobenius weights of its usual binomial representation, while the only other known representation, of Clausen type, is degenerate. This paper removes the obstruction for E, so that seven of the fifteen sporadic sequences are now covered. Main result. Theorem 1. Let p ≥ 5 be a prime, χ(p) = (−1/p) and 0 < σ < 1/3. There is a constant K = K(p,σ) such that for every m ≥ 1 E(mp^r) = Φ_m(p^r) + χ(p)^r Ψ_m(p^r) (r ≥ 0), with power series Φ_m(z) = Σ_a c_a(m) z^a and Ψ_m(z) = Σ_a d_a(m) z^a satisfying |c_a(m)|, |d_a(m)| ≤ K·p^(−σa), c_1(m) = 0 and d_0(m) = d_1(m) = 0. If p ≡ 1 (mod 4), then Ψ_m = 0 and E(mp^r) is given by a single power series in p^r. The pair (Φ_m, Ψ_m) is uniquely determined by the sequence (E(mp^r))r≥0. The statement has the same shape as the result for Zagier's sequence C, with (−1/p) in place of (−3/p). The character (−1/p) is the one that the divided-difference test of the previous paper had detected numerically for E. How the obstruction is removed. Two extensions of the framework of the previous paper: the recurrence in the summation index may have a polynomial on both sides, M(x,k)T(x,k+1) = N(x,k)T(x,k), with difference operator D_yφ(k) = φ(k−1)M(y,k−1) − φ(k)N(y,k); with it the usual representation of E is admissible, of type I, with constant leading coefficient −4 and a shift matrix of constant determinant 128; the Frobenius splitting k = pk′ + t uses balanced digits, |t| ≤ (p−1)/2; then no factorial with a coefficient ±2 in k crosses a multiple of p, and the pole disappears. The paper checks, result by result, which parts of the general theorem change and how. The sign comes out elementarily. At y = 0 the Frobenius weights reduce to ω_t(0,k′) = (−1)^t (pk′)²/(pk′+t)², and the third eigenvalue of the constant term of the Frobenius matrix is p² Σ|t|≤(p−1)/2 (−1)^t = (−1/p) p². For Zagier's sequence C the corresponding eigenvalue (−3/p) p² needed a Dwork-type congruence, proved from the theorem of Mellit and Vlasenko; here no congruence is needed. The same alternating sum, restricted to t < 0, shows that the resonant obstruction of order two vanishes when p ≡ 1 (mod 4). The remaining sequences. Balanced digits also remove the pole for the usual representations of the Domb numbers and of (ε), which gives nothing new. A search over representations that are admissible in the extended sense finds, for the eight remaining sporadic sequences, only one representation beyond the known ones, for s10 = Σ_k C(n,k)⁴, which we have not found in the literature: s10(n) = Σ_k (−1)^(n−k) C(n,k) C(2k,k) C(2n−2k,n−k) C(n+k,k). Its weights cannot be made analytic: the factorial (n+k)! requires t ≥ 0, the factorial (2n−2k)! requires |2t| ≤ p−1, and no complete system of residues satisfies both. Zagier's sequence F is a binomial transform of the Franel numbers, but this representation is degenerate, because the transform sends the singular point 1/8 of the Franel equation to infinity. The covered sequences are now A, (γ), (α), (δ), (ε), C and E; the other eight are degenerate (B, D, F, s10) or have no single-sum representation of the required shape ((ζ), (η), s7, s18). Compatibility with congruences. Given Theorem 1, a congruence E(mp^r) ≡ E(mp^(r−1)) (mod p^(2r)) for all large r is equivalent to v_p(c_2(m)) ≥ 2 and, if p ≡ 3 (mod 4), v_p(d_2(m)) ≥ 2. In the computed data, v_p(E(mp^r) − E(mp^(r−1))) is 2r or 2r+1 for 5 ≤ p ≤ 19, 1 ≤ m ≤ 4 and all r ≥ 1 with mp^r ≤ 16000 (83 values). When p ≡ 3 (mod 4) the data also point to c_2(m) = 0; this is not proved. Verification. The proofs do not depend on computer calculations. All the identities used in them were nevertheless checked by computer, in exact rational arithmetic or in truncated p-adic arithmetic, first with the author's scripts and then by a second, independently written check that did not reuse the first (with its own coefficients of Morita's p-adic gamma function, its own weights and its own reduction): the shift matrix and its determinant; the formula for the weights, against exact values for p ≤ 23, and the absence of poles, for all primes p < 600; the weights at y = 0; the Frobenius matrices for p = 5, …, 23, with V(py) = Θ(y)V(y) to full precision; the constant term and its eigenvalues; the identity behind the new Frobenius map on the modules; the eigen-sections and the resonant coefficient; negative controls (removing the exponential factor, or changing the sign of the weights with t < 0, breaks V(py) = Θ(y)V(y)); and the exact values E(mp^r) along the towers. The record contains the scripts (Python 3 with SymPy), run_all.sh, which runs all of them (about twenty minutes), and the complete output. Limits and open questions. Eight of the fifteen sporadic sequences remain open. The coefficients of Φ_m and Ψ_m are not known to be p-integral, so the theorem does not by itself give the supercongruences. The primes 2 and 3 are excluded. The vanishing of c_2(m) for p ≡ 3 (mod 4) is only supported numerically. Authorship. The author of this work is Roberto Blanco Gómez. The structural tests that led to this paper (module of moments, poles of the Frobenius weights, shift certificates and the catalogue of the fifteen sporadic sequences) are implemented in 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 the verification scripts and the search for representations, in an independent review of the proofs, and in drafting and translating the text. AI tools are not authors of this work. The author has checked the arguments and computations and takes full responsibility for the content. Files. Blanco_Gomez_2026_Zagier_E_towers_EN.pdf: the paper. Blanco_Gomez_2026_Zagier_E_towers_ES.pdf: the Spanish version, with the same content. verification.zip: the verification scripts, README.txt, run_all.sh and the complete output. sources.zip: LaTeX sources of both versions.