A determinantal structure for Wolstenholme-type congruences of multiple harmonic sums

For the elementary (height-one) multiple harmonic sums H(s,k) we study the 3 ×3 JacobiTrudi-type determinant D(s) = det H(s,2i−j) . We prove that at the shifted arguments s= p−2, p−3 one has D(s) ≡0 (mod p2) for every prime p≥11, independently of the regularity of p a Wolstenholme-type vanishing that persists away from the classical point p−1 and is carried by cancellation at the level of the determinant rather than by the entries. The proof rests on a structural theorem: for every g ≥1 the g×g determinant equals the staircase Schur function s(g,g−1,...,1) evaluated at 1,1/2,...,1/s, and is therefore a polynomial in the odd power sums alone the even ones cancel identically, a consequence of the fact that staircases are 2-cores, via the MurnaghanNakayama rule. We recover the exact second-order expansions in Bernoulli numbers (independently derived by R. Tauraso via a theorem of Zhao) and use them to characterize the sporadic primes of deeper vanishing: 17, 41 in one family and 47, 733, 853 in the other the last two predicted by the formula before, and conrmed by, direct computation. The determinantal viewpoint extends to a g ×g family whose rank and cancellation depth are determined in a companion note deposited simultaneously; the formulas above are the two linear instances of its general sporadic criterion.