A proof of P. Bala's congruence T(np^k) ≡ T(np^(k-1)) (mod p^k) for the tangent numbers, via the Kummer congruences for the Genocchi numbers without the exception (p-1) | m

Let T(n) = 2^(2n) (2^(2n) - 1) |B_(2n)| / (2n) be the tangent numbers (OEIS A000182): 1, 2, 16, 272, 7936, ... In the OEIS entry, P. Bala conjectured (December 24, 2021) that T(n p^k) == T(n p^(k-1)) (mod p^k) for every prime p and all n, k >= 1, except for (p, n, k) = (2, 1, 1) and (2, 1, 2). We prove it, and the exponent is sharp. For odd p the statement follows from the Kummer congruences for the Genocchi numbers G_m = 2 (1 - 2^m) B_m in the form G_m/m == G_m’/m’ (mod p^k) whenever m == m’ (mod (p-1) p^(k-1)) and m, m’ >= k+1, with no hypothesis (p-1) does not divide m. This is the classical Kummer congruence in its regularized form (the numbers (1 - c^m)(1 - p^(m-1)) B_m/m are the moments of Mazur’s regularized Bernoulli measure); we give an elementary, self-contained proof through the alternating power sums Sumj < p^M (-1)^j j^(m-1), which the Genocchi numbers approximate p-adically by the Euler polynomials. The case (p-1) | 2n, where B_m has p in its denominator and the classical Kummer congruence does not apply, is exactly the case covered by this form. For p = 2 the statement follows from the exact 2-adic valuation v_2(T(j)) = 2j - 2 - v_2(j), which also explains the two exceptions. The same argument proves the analogous congruence E(n p^k) == E(n p^(k-1)) (mod p^k) for the Euler numbers E(n) = A000364(n) and every odd prime p; this statement was produced by the generalization step of the author’s SyntheticMind engine from Bala’s statement, and for p = 2 it is verified numerically and left as a conjecture. Bala’s statement was left as “consistent” (true in 15,197 checked cases, no proof) by the engine in the batch of open statements of the core entries of the OEIS (October 5, 2026). On October 15 the engine was extended with a method for congruences with a free prime that computes exact p-adic valuations, generalizes the statement and proves what it can with a written argument; its first proof covered the case (p-1) does not divide 2n and reported the other case as pending, which the regularized form closed. The proofs use no computation; independent checks (Python 3, exact rational arithmetic with Bernoulli numbers up to B_1200, with negative controls) and the engine’s output are distributed with the note, and the text was reviewed by an independent AI referee with its own programs (zigzag triangle, independent Bernoulli numbers). The use of AI is described in the paper. As of October 15, 2026 the statement was still marked as a conjecture in the OEIS entry.