Integral Frobenius structures and supercongruences for seven sporadic Apéry-like sequences

The fifteen sporadic Apéry-like sequences of Zagier, Almkvist–Zudilin and Cooper are conjectured to satisfy the supercongruences u(mp^r) ≡ u(mp^(r−1)) (mod p^(λr)) for every prime p ≥ 5, with λ = 3 or λ = 2 depending on the sequence. These congruences were proved one sequence at a time, by different methods, the last case being settled by Alinquant and Osburn in 2024, who remarked that there was no general framework explaining them. This paper gives a uniform argument for the seven sporadic sequences that admit a Frobenius structure in the summation variable: the Franel numbers, the Apéry numbers, the Domb numbers, the Almkvist–Zudilin sequences (δ) and (ε), and Zagier's sequences C and E. Main results. In earlier papers the author showed that, for these seven sequences and every prime p ≥ 5, u(mp^r) = Φ_m(p^r) + (−1)^r Ψ_m(p^r) for power series Φ_m and Ψ_m that converge on a disc of radius larger than 1. Their coefficients were not known to be p-integral. Theorem A. The coefficients of Φ_m and Ψ_m are p-adic integers, and Φ_m(z) − Φ_m(0) and Ψ_m(z) lie in p^λ z^λ Z_p[[z]], with λ = 3 for the Franel, Apéry, Domb, (δ) and (ε) sequences and λ = 2 for C and E. Corollary B. The supercongruences u(mp^r) ≡ u(mp^(r−1)) (mod p^(λr)) hold for the seven sequences, all primes p ≥ 5 and all m, r ≥ 1. These are known; the point is the uniform proof. Corollary C. The normalized differences (u(mp^r) − u(mp^(r−1)))/p^(λr) stabilize modulo p^(r−1), along each parity class of r when a sign is present. In particular their limits exist, and from r = 2 on, for each class, the congruence modulo p^(λr+1) holds either for all r or for none. Theorem D. For the Franel numbers, C and E there is one more digit: the coefficients of degree at least λ+1 lie in p^(λ+1) Z_p, and the normalized differences stabilize modulo p^r, from r = 1 on. The modulus p^r cannot be improved in general. Conjecture. The same holds for the Apéry, Domb, (δ) and (ε) sequences. It would follow from a bound for the Frobenius matrix modulo p^4 that has been checked numerically for p = 5 and 7, and the resulting congruence has been checked with exact integers. Method. (1) An integrality criterion. If the Frobenius matrix Θ(y), written in an integral eigenbasis of Θ(0), has Taylor coefficients divisible by the denominators that appear in the recursion for its eigen-sections, then the eigen-sections are integral and the coefficients of the interpolation are divisible by p^λ. Resonances are harmless, because an analytic eigen-section exists. A refinement of the criterion gives one more power of p.(2) A Hodge form. For the seven summands the Frobenius matrix satisfies Θ(y) ≡ B(py)·diag(1, p, p²) (mod p³): its columns are divisible by 1, p, p², and it depends on y only through py. This comes from the shape of the summand, through bounds for the Morita coefficients of the p-adic gamma function and for the denominators of the reduction.(3) The first row. The first row of Θ must be congruent to (1, 0, 0) modulo p^λ. For the Franel numbers, C and E this is immediate; for the Apéry numbers it reduces to the classical congruence Σ_{t<p} t^(−2) ≡ 0 (mod p). For the Domb, (δ) and (ε) sequences it is a genuine arithmetic statement, equivalent to congruences for truncated hypergeometric sums modulo p², such as Σ_{t=1}^{p−1} C(2t,t)C(3t,t)/(t·27^t) ≡ 3·log(3^(p−1))/p (mod p²) for (δ). These are obtained without computing any sum, from an exact eigen-section with vanishing first coordinate.(4) Linear algebra. The Hodge form, the first row and the valuations 0, 1, κ of the eigenvalues of Θ(0) give the hypotheses of the criterion. For C and E, where λ = 2, the Hodge form modulo p³ also gives the refined criterion. For the Franel numbers the Hodge form holds modulo p^4: the reduction respects the total degree, and the one weight coefficient that could fall short, a Morita term at p = 5, is saved by a divisibility by p. This gives Theorem D. Verification. The proofs do not depend on any computation. Every statement was nevertheless checked numerically with the Python scripts in this record (standard library only, a little over one minute in total; run_all.sh runs them all and the complete output is included). For p = 5 and 7 they compute the Frobenius matrices of the seven sequences from their definitions and check the Hodge form, the first-row congruence, the integrality condition and its refinement, and the bounds for the eigen-sections. With exact integers they check the supercongruences, the stabilization of the next digit, Theorem D and the conjecture for several primes. The checks of the new statements were written by the author and again, independently, in separate review passes. Limits and open questions. The other eight sporadic sequences have no known Frobenius structure in the summation variable, so the method does not reach them yet. The extra digit for the Apéry, Domb, (δ) and (ε) sequences is conjectural; a proof would need the Frobenius matrix modulo p^4, beyond what the method of the Franel case gives. The limits of the normalized differences are identified only for the Apéry numbers, through Liu's congruence and the author's earlier work. Authorship. The author of this work is Roberto Blanco Gómez. The structural tests behind the series of papers that this one continues were carried out with SyntheticMind, the mathematical engine that the author created, designed and programmed himself in Rust. The AI system Claude (Anthropic) collaborated extensively 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_integral_Frobenius_supercongruences_EN.pdf: the paper. Blanco_Gomez_2026_integral_Frobenius_supercongruences_ES.pdf: the Spanish version, with the same content. verification.zip: the verification scripts (Python 3, standard library only), README.txt, run_all.sh and the complete output. sources.zip: LaTeX sources of both versions.