Partial Row–Column Coupling in Symmetric Arithmetic Circuits for the Determinant: Matching Defect, Alternating Cores, and a Parameterized Complexity Frontier

We study the determinant in the model of symmetric arithmetic circuits, focusing on symmetry groups between simultaneous row–column permutation and independent parity-preserving row and column permutation. For a subgroup H <= D_n, where D_n = {(sigma,tau) in S_n x S_n : sgn(sigma)=sgn(tau)}, we use two complementary parameters. The matching-coupling defect delta_mc(H), introduced here, is the minimum number of row and column indices left unmatched by an H-invariant partial row–column matching. The independent alternating core alpha_ind(H) was introduced in the author's preceding preprint and is the largest r for which H contains an independently acting Alt_r x Alt_r subgroup. We prove alpha_ind(H) <= delta_mc(H), SymSize_H(det_n) <= n^{O(1)} 4^{delta_mc(H)}, and SymSize_H(det_n) >= 2^{Omega(alpha_ind(H))}, where the lower bound applies once the independent alternating core is sufficiently large. For the natural family H_{n,q} preserving a partial matching of size n-q, we show H_{n,q} ~= S_{n-q} x D_q, delta_mc(H_{n,q}) = alpha_ind(H_{n,q}) = q for q >= 3, and hence 2^{Omega(q)} <= SymSize_{H_{n,q}}(det_n) <= n^{O(1)}4^q. Consequently polynomial-size H_{n,q(n)}-symmetric determinant circuits exist exactly when q(n)=O(log n). We also classify subdirect subgroups H <= D_n with surjective projections for n >= 7: they are either D_n itself or graphs of inner automorphisms of S_n, yielding a hard/easy dichotomy between independent parity coupling and complete row–column coupling. The results concern symmetry-constrained arithmetic circuits only; they do not imply lower bounds for unrestricted arithmetic circuits or resolve VP versus VNP.