p-adic analyticity of sporadic Apéry-like sequences along the towers mp^r

The fifteen sporadic Apéry-like sequences of Zagier, Almkvist–Zudilin and Cooper (among them the Apéry numbers for ζ(2) and ζ(3), the Franel numbers and the Domb numbers) satisfy supercongruences u(mp^r) ≡ u(mp^(r−1)) (mod p^(2r) or p^(3r)), established case by case. In previous work (doi:10.5281/zenodo.23928953) we proved a statement of a different kind for the Apéry, Franel and Domb numbers: for every prime p ≥ 5 and every m ≥ 1, the whole sequence r ↦ u(mp^r) is the restriction to z = p^r of a single p-adic power series, convergent on a disc of radius larger than 1, with bounds uniform in m. All the p-adic digits of u(mp^r), for all r, are then encoded in one sequence of coefficients. This paper isolates the properties of a binomial summand that make the method work, proves a general theorem under them, applies it to three further sporadic sequences, so that six of the fifteen are covered, and explains exactly why each of the remaining nine resists. Main results. Theorem 1 (general theorem). Let p ≥ 5 and let T(n,k) be an admissible summand with shift data: two first-order recurrences in k and in n, a regular three-dimensional module of moments Q_p[k]/D_y(Q_p[k]) whose leading coefficient does not depend on the parameter y (types I and II), Frobenius weights T(py, pk′+t)/T(y,k′) without poles, and a map realizing the recurrence in n with constant determinant δ_R. Let κ = 1/3 (type I) or κ_p = (1/3 − 1/(p−1))/2 (type II), and 0 < σ < κ. Then the vector V(n) of the first three moments satisfies V(py) = Θ(y)V(y) with Θ analytic on |y| ≤ p^σ, Θ commutes with the recurrence in the upper index, Θ(y+1)R(y) = R(py+p−1)⋯R(py)Θ(y), and det Θ(y) = det Θ(0)·δ_R^((p−1)y). If Θ(0) has eigenvalues 1, ±p^(e1), ±p^(e2) and Θ has a normal form, then for every m ≥ 1 u(mp^r) = Φ_m(p^r) + (−1)^r Ψ_m(p^r) (r ≥ 0), with power series whose coefficients are bounded by K·p^(−σa), K independent of m. Φ_m and Ψ_m are unique, do not depend on σ and converge on |z| < p^κ. Theorem 2 (six sporadic sequences). For every prime p ≥ 5, Theorem 1 applies to the Franel numbers A, the Apéry numbers (γ), the Domb numbers (α), the Almkvist–Zudilin numbers (δ), the sequence (ε) and Zagier's sequence C: for A, (γ), (α), (δ) and (ε), u(mp^r) = Φ_m(p^r) for a single power series; for (α), (δ), (ε) its coefficients of z and z² vanish; for C, u(mp^r) = Φ_m(p^r) + (−3/p)^r Ψ_m(p^r), with Φ_m without term in z and Ψ_m = O(z²); numerically the signed term is really present (p = 5, 11), and the sign comes from a Dwork-type congruence for the central binomial sums Σ_{k<p^n} C(2k,k), which we prove from the Dwork congruences of Mellit and Vlasenko. The results for (δ), (ε) and C are new; those for A, (γ), (α) recover the previous paper, now with a uniform treatment of the shift in the upper index. For (ε) we use a new representation of Clausen type, u(n) = Σ_k 4^(n−2k) C(n+2k,4k) (4k)!/k!⁴, which we prove with a Zeilberger certificate. Compatibility with the known supercongruences. An elementary proposition shows that, for expansions Φ_m(p^r) ± Ψ_m(p^r), a congruence modulo p^(er) for all large r is equivalent to the vanishing of the coefficients of order 1, …, e−1 together with v_p(c_e) ≥ e. With the theorems of Osburn–Sahu (2011, 2013) and Osburn–Sahu–Straub (2016) this gives v_p(c_2(m)) ≥ 2 for C and v_p(c_3(m)) ≥ 3 for (ε), Domb and Apéry. For (δ) the congruence modulo p^(3r) was proved for r = 1 by Amdeberhan and Tauraso and is open for r ≥ 2; we show that, for large r, it is equivalent to v_p(c_3(m)) ≥ 3. Theorem 3 (obstructions for the other nine). For B, D, E, F, (ζ), (η), s7, s10, s18 every known representation violates a specific hypothesis, for a proved reason: either the module is degenerate (the leading coefficient of the difference operator in k is (Σa_i)y + Σb_i − j, which vanishes on a set of parameters dense in Z_p, because the y-coefficients of the linear factors of N(y,k) do not balance), or a factorial with a coefficient ±2 in k on the "wrong side" produces a pole of the Frobenius weights along k′ = y − 1/2 or k′ = (y − 1)/2, or no single hypergeometric sum is known. The Apéry numbers for ζ(2) are discussed in detail: both classical representations are degenerate, with poles of the reduction at the positive (resp. negative) integers; searches among the summands of the admissible shape, over large grids of parameters, found no other representation. Conjecture 4. For each of the fifteen sporadic sequences and all but finitely many primes p there is a character χ with u(mp^r) = Φ_m(p^r) + χ(p)^r Ψ_m(p^r), two analytic functions with bounds uniform in m. Numerical evidence (divided differences on the nodes p^r, p = 5, 7, 11, 13) supports it and suggests the character in each case. Why this matters. 1. It extends a statement that controls all the p-adic digits along the towers at once, rather than a fixed number of them, from three to six sporadic sequences, and it does so through one general theorem with checkable hypotheses. 2. For Zagier's sequence C it shows that the towers are given by two analytic functions, one of them multiplied by (−3/p)^r; numerically the second one does not vanish, consistently with the fact that C satisfies congruences only modulo p^(2r) along the towers. 3. It identifies precisely where the method stops, with proved obstructions and a simple mechanism behind each of them. 4. Unlike the Dwork crystals of Beukers and Vlasenko, which work with the generating function in its variable t and give congruences of a fixed depth, the Frobenius structure here lives in the summation variable k with the upper index as an analytic parameter, and gives analytic interpolation in p^r; the two points of view are complementary and meet in the compatibility statements. Verification. The proofs do not depend on computer calculations. As a safeguard every object was recomputed independently with the Python scripts in this record (exact rational arithmetic, truncated p-adic arithmetic and SymPy; run_all.sh runs them, and the logs are included): the identities and certificates, the Frobenius matrices for p = 5, 7, 11, 13 with V(py) = Θ(y)V(y), the eigenvalues, det Θ and the commutation, the sections and the closed formula against exact moments, the Dwork-type lemma for C, the tower valuations, the structural tests and pole computations for the other nine sequences, the searches for representations, and the divided-difference test for all fifteen sequences. Limits and open questions. Nine of the fifteen sequences remain open; each needs a new representation or a new idea (degenerate modules, double sums). The coefficients of Φ_m and Ψ_m are not known to be p-integral, so the theorem does not by itself reprove the supercongruences. The primes 2 and 3 are excluded. Authorship. The author of this work is Roberto Blanco Gómez. The structural tests that singled out the new cases were first implemented in SyntheticMind, the mathematical reasoning engine that the author created, designed and programmed himself in Rust. The AI system Claude (Anthropic; models Claude Fable 5.1 and Claude Opus 5.5) collaborated as an assistant, under the author's direction, in developing and checking the arguments, in the verification scripts and searches, in review passes, in the literature search 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_sporadic_analyticity_EN.pdf: the paper. Blanco_Gomez_2026_sporadic_analyticity_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 logs, including the searches. sources.zip: LaTeX sources of both versions.