White-light scanning interferometry (WLSI) is a high-precision three-dimensional surface metrology technique widely used in nanometrology, semiconductor advanced packaging, additive manufacturing, and thin-film characterization. In practical measurements, however, the attainable accuracy is limited not only by detector additive noise but also by mechanical vibration, which randomly perturbs the actual scanning position and simultaneously affects the interferometric envelope and carrier phase. Most existing Cramér-Rao lower bound (CRLB) analyses for WLSI consider additive noise alone, and a unified theoretical framework for jointly characterizing additive noise and correlated mechanical vibration is still lacking. In this work, we develop a unified complex observation model for WLSI under degraded scanning conditions. The model incorporates exposure-induced fringe attenuation, additive circularly symmetric complex Gaussian noise, and scanning-direction mechanical displacement perturbations. Based on this model, we derive the Fisher information matrix and the CRLBs for the envelope peak position z_M and the absolute phase \varphi_0. For the pure additive-noise case, it is shown analytically that z_M and \varphi_0 are orthogonal in the Fisher-information sense, because envelope-position perturbations and phase perturbations correspond to radial and tangential directions in the complex observation space. For correlated mechanical vibration, a hybrid CRLB formulation is introduced by treating z_M and \varphi_0 as deterministic unknown parameters and the vibration sequence as a random nuisance process with a known covariance structure. Using the Schur complement and the Woodbury matrix identity, we obtain closed-form CRLB expressions involving an equivalent residual vibration covariance matrix \mathbfT. The resulting CRLB matrix reveals that mechanical vibration not only raises the marginal lower bounds of z_M and \varphi_0, but may also introduce statistical coupling between them through the common vibration process.
Furthermore, we analyze relative topography measurement and show that common vibration components can be suppressed by differential measurement, with the residual vibration contribution determined by the difference between the vibration-sensitivity vectors of two pixels. For constant drift, the common displacement is strictly cancelled in ideal synchronous relative measurement. For narrowband single-frequency vibration, analytical frequency-sensitivity expressions are derived, showing how the vibration contribution depends on the envelope spectral overlap, the envelope-position response, the scanning speed, and the height difference between pixels. The proposed framework provides a quantitative theoretical tool for accuracy-limit evaluation, error budgeting, parameter optimization, and vibration-resistant design in WLSI systems.