A recursive counterexample to a complex extension of the Goreinov–Tyrtyshnikov–Zamarashkin conjecture

Weighted orthogonal-complement duality proves the real square-root bound of the
Goreinov–Tyrtyshnikov–Zamarashkin conjecture for Parseval frames (matrices with
orthonormal columns) having at most $r+2$ nonzero row directions, using the
established real two-column theorem. The unrestricted real conjecture is not
resolved here. Separately, an explicit sequence of complex matrices with
orthonormal columns disproves a dimension-independent complex extension of the
conjecture. The matrices $U_k$ have $2\cdot3^k$ rows and $3^k$ columns, every
row has squared norm $1/2$, and every nonsingular square row submatrix satisfies
$\|U_{k,I}^{-1}\|_2^2\ge(3\cdot4^k+1)/2$. Consequently, no absolute constant
$\alpha$ can bound all these inverse norms by $\alpha\sqrt n$ after optimal row
selection. The proof uses an orthonormal three-phase lift, an exact
classification of all nonsingular selections, and one Rayleigh quotient. A
stronger analysis gives a scalar recurrence for the exact best inverse norm, the
entire singular spectrum of every basis, and an exact count of the nonsingular
bases.