Independently Verified Causal Spatio-Temporal Covariance Under Correlated Amplitude and Pointing Noise in Fractional Nonlocal Photo-Thermoelastic Semiconductors

A two-dimensional causal benchmark is developed for a normally incident finite-width Gaussian beam that jointly generates Beer–Lambert carrier and heat sources in a fractional nonlocal photo-thermoelastic semiconductor. Colored amplitude noise and beam-centre jitter are treated as correlated properties of the same optical field. Fourier transformation of the physical transverse profile generates the continuum of transverse wavenumbers, while unilateral Laplace transformation retains the frequency dependence of the fractional heat law. The two-input spectral matrix is propagated through a seven-output transfer operator for T, N, u, v, σxx, σxy, and σyy. Independent time-domain verification uses correlated Ornstein–Uhlenbeck paths generated by their exact discrete transition and filtered through the full frequency-dependent transfer, rather than sampled from the analytical output covariance. This test exposed and corrected a conjugation-order error in the correlated cross-spectral projection. After correction, 256 paths containing 12,582,912 retained samples reproduce every field variance within 1.274% and every selected cross-field correlation within 0.00365. A 180 × 101 frequency-wavenumber grid differs from a 260 × 161 reference by at most 0.944% in variance and 0.00226 in correlation. Exact signed attribution shows constructive amplitude-pointing interaction for five fields but destructive interaction for v and σxy. The innovation-correlation sweep is affine to machine precision. The work is positioned as a dimensionless verification and mechanism study, not a calibrated device reliability prediction.