arXiv:2610.30497v1 Announce Type: new Abstract: An F-type normal-reference calibration is developed for testing equality of two high-dimensional covariance matrices with unknown means. The test statistic is a ratio whose numerator measures the Frobenius covariance discrepancy and whose denominator estimates its null expectation. Their difference is the complete Li-Chen statistic. Under suitable moment and trace conditions, we establish a Gaussian-reference approximation. After standardization, the statistic and its reference share the normal or nonnormal limits determined by a convergent normalized spectrum. The reference is a ratio of independent chi-square-type mixtures. Applying Welch-Satterthwaite matching to its two components gives an ordinary F approximation, with degrees of freedom estimated from independent four-observation blocks. The effect of estimating the population means is controlled by an exact expansion, and local-power and consistency results are established. The ordinary F approximation is asymptotically valid under a diffuse spectrum, whereas a covariance spike can produce a nonvanishing approximation error. Simulation studies examine empirical size and power under normal and nonnormal sampling. Averaging the trace estimates over fixed partitions improves their stability, but does not remove size inflation in the strongly correlated settings.