A proof of a conjecture of P. Bala on the squares of the Apéry numbers for ζ(2)

We prove a conjecture of P. Bala (OEIS A005258, July 8, 2024): for every n >= 0, a(n)^2 = Sum_{k=0..n} (-1)^(n+k) C(n,k) C(n+k,k) T(n,k), where a(n) = Sum_k C(n,k)^2 C(n+k,k) are the Apéry numbers associated with zeta(2) (A005258) and T(n,k) = Sum_j C(n,j) C(n+j,j) C(k,j) C(k+j,j) is the symmetric array A143007, whose diagonal consists of the Apéry numbers associated with zeta(3) (A005259). The right-hand side is a double sum. By creative telescoping we show that it satisfies the same recurrence of order three as a(n)^2, namely the symmetric square of Apéry's recurrence, whose leading coefficient does not vanish for n >= 0, and we compare three initial values. Two telescoping certificates give a pair of relations for the inner sum T(n,k), one in each variable; a third certificate, made of two polynomials of total degree 15, gives the recurrence of the outer sum. All certificates are printed in the paper, together with the boundary analysis. The certificates were found by a goal-directed search of the SyntheticMind engine, which splits the identity into subgoals (an annihilating recurrence for each side), guesses relations from exact values and then proves them symbolically. The proof does not depend on the software: every step is either a short argument given in the paper or an identity between explicit polynomials, and all of them were checked independently with SymPy. The use of AI is described in the paper. As of October 3, 2026 the statement was still marked as a conjecture in the OEIS, and we have not found a proof in the literature. Files: Blanco_Gomez_2026_Bala_conjecture_EN.pdf is the paper; Blanco_Gomez_2026_conjetura_Bala_ES.pdf is the Spanish version (same content); bala_certificates.txt contains the certificates in plain text, readable by any computer algebra system; verify_bala.py is the independent verification script (Python with SymPy; it runs in about a minute and a half and prints ALL CHECKS PASSED).