arXiv:2611.14598v1 Announce Type: new Abstract: We study noncentral normal-reference inference for high-dimensional two-sample Behrens-Fisher problems. Within the Gaussian reference, a shifted null law is accurate if and only if covariance-weighted signal energy is negligible relative to the trace of the squared covariance. We establish the sharp uniform fourth-root error order over unrestricted spectra and a linear bound under spectral spread. A separate spectral-range criterion determines when signal magnitude alone suffices. An augmented spherical reference gives finite-sample calibration for Gaussian samples and asymptotically recovers Gaussian-reference local power under projection fourth-moment bounds and control of diagonal fluctuations, without a dimension-to-sample-size restriction. For unknown Gaussian covariance, explicit trace and hollow-Gram confidence bounds turn the linear approximation bound into a simultaneous, computable certificate over local effect ranges. Eigenspace signal energies describe directional power, while a two-point argument limits estimation of unknown local power. Simulations with 10,000 evaluation replications per configuration examine unequal and noncommuting covariances, severe sample imbalance, non-Gaussian observations and the informativeness of the certificate.
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