Product-Aware Deterministic Rounding for Quantized Matrix Multiplication

Scalar rounding decisions interact through matrix multiplication. We study deterministic product-aware rounding after scales, clipping bounds, and grids are fixed, with each active scalar choosing between adjacent levels. For dynamic activation rounding, null-space reduction preserves the relaxed product while leaving at most r fractional decisions, where r is the rank of the active gap-weighted weight block. Conditional-expectation completion gives a deterministic polynomial-time algorithm with squared product error at most OPTdyn+rnumax2/4, where OPTdyn is the best admissible error and numax is the largest row norm of that block. For reusable static weights, the exact expected product-loss metric is the uncentered input second moment with fixed output bias; free bias recalibration yields the centered covariance. Exact optimization is NP-hard even at rank one. In balanced blocks with K=1024 and r=16, conditional- expectation completion attains a dither-normalized median error of 0.010, compared with 0.899 for round-to-nearest. Clipping-aware initialization reduces median normalized error by a factor of 43.4 at ten-percent clipping. Held-out Digits experiments show that retaining the input mean or correcting the output bias improves median product error over round-to-nearest in all four tested bit-width and calibration-size settings.