A Direct Smooth-Label-Cover Reduction for Fixed-Rank Block PSD Optimization

For fixed 1 <= d <= r and p >= 2, we study positive semidefinite block quadratic maximization over d by r matrices with Schatten-p norm at most one. We identify the exact ratio of its natural convex spectral relaxation as the square of a Gaussian singular-profile minimum. For fixed rational p, this ratio is attained to arbitrary constant accuracy by a polynomial-time randomized algorithm and is matched by a deterministic NP-hardness reduction. The algorithm realizes prescribed first Gaussian moments through finite convex calibration. The reduction uses weighted finite matrix tests and preserves the column-module structure of the objective. A separate Gaussian Gram construction establishes the exact relaxation gap over rational instances. The constant recovers the scalar rank-r and row-isometry endpoints. When d >= 2, near the Frobenius endpoint every minimizing profile has unequal positive singular values, so minimizing over uniform active ranks does not determine the intermediate thresholds. Version 2.0 extends the row-isometry result in version 1.0 to fixed Schatten-p constraints and includes the full proofs and quantitative appendices.