A Matrix Amari–Schwarzian Identity, with Witnesses at Genus 1, Genus 2, and the Mirror Quintic

The scalar Amari-Schwarzian residual identity (A + S')^2 = 2(D + S)(D + 2S)^2 established in Paper 5 (Maino, 2026; DOI 10.5281/zenodo.20031926) holds for the logarithmic derivative u = omega'/omega of any solution of a scalar second-order linear ODE omega'' + q(z) omega = 0. We lift this identity to matrix-valued U and Q associated with rank-r second-order linear ODEs Omega'' + Q(z) Omega = 0. With U := Omega' Omega^{-1} satisfying the matrix Riccati equation U' = -U^2 - Q, the lifted residual admits the closed form R = 16 U [U, Q] U'. The proof is four lines from the matrix Riccati equation alone (Theorem 1). The two natural scalar reductions are tied at every rank by tr(U [U, Q] U') = (1/2) tr([U, Q]^2) (Lemma 3), so the genuine matrix-rank content of R at higher rank lives in tr([U, Q]^k) for k >= 3. We illustrate the framework on four explicit families: (1) Legendre family of elliptic curves at genus 1 (rank 2), with symbolic and 50-digit numerical verification. (2) Genus-2 hyperelliptic family at rank 4 via Griffiths-Dwork reduction, where tr([U,Q]^3) = 0 while tr([U,Q]^2) and tr([U,Q]^4) are non-zero rational functions with clean rational residues antisymmetrising under the family involution. (3) Genus-3 sextic hyperelliptic at rank 6, extending the odd-trace vanishing to a third rank class. (4) Mirror quintic Calabi-Yau 3-fold at rank 4, providing the first non-hyperelliptic confirmation of both the odd-vanishing observation and the rank-4 lower bound of the genus-stratification conjecture. Lemma 5 proves odd-trace vanishing in any symplectic gauge of the Gauss-Manin connection, using the paired spectrum of the symplectic Lie algebra. Version 3 adds three results. First, Theorem 4 (companion-matrix odd-trace vanishing): for any r x r companion matrix U with Q = -(U' + U^2), tr([U,Q]^{2k+1}) = 0 identically. Verified symbolically at ranks 2 and 4 with generic coefficient functions a_0(z),...,a_{r-1}(z), numerically at rank 6. This is a structural algebraic property of the companion matrix form, independent of symplectic gauge, closing Open Question (ii) of the previous version. Second, Proposition 8 (differential closure at order 4): both trace invariants f_2 = tr([U,Q]^2) and f_4 = tr([U,Q]^4) of the mirror quintic admit quadratic differential polynomial representations in the Yukawa coupling W(z) = 1/(z^3(1-3125z)) and its derivatives through W^(4), but not through W'''. The 21-dimensional symmetric jet space Sym^2(J^4(W)) has rank 20, with a 1-dimensional syzygy kernel verified as a rational identity by SymPy (S(z) = 0 identically). The syzygy has support at differential weight omega = 5 are therefore gauge-rigid, with prime factorisations 10204620 = 2^2 * 3 * 5 * 53 * 3209 and 14788040 = 2^3 * 5 * 173 * 2137 at weight 5. A structural bifurcation emerges: f_2 has nonzero linear terms spanning the low-weight sector (omega = 2, 3), while f_4 is purely quadratic spanning 0