Frame Degeneracy and Sine-Quadrature Residuals in Lunar Tidal Modeling

Abstract

The dominant Earth-Moon tide is accurately described by the Newtonian quadrupolar tidal tensor and by standard terrestrial tide modeling. In a local principal frame aligned with the Earth-Moon direction, the leading two-dimensional lunar tidal block is diagonal and produces a plus-type angular dependence proportional to cos(2 beta). A symmetric off-diagonal residual would instead enter the sine-quadrature channel proportional to sin(2 beta). This separation defines a useful residual basis, but it does not by itself establish either a detection or a unique physical origin. The key difficulty is that the sine-quadrature coefficient is frame-degenerate. A small misregistration delta of the ordinary Newtonian principal frame produces an apparent sine coefficient chi_app = (3/2) sin(2 delta), approximately 3 delta. Thus target coefficients of order 10^-3 and 10^-2 correspond to frame-control scales of about 69 arcsec and 688 arcsec, respectively. We reformulate the problem as a frame-degeneracy-aware residual-estimation framework. The Halilsoy cross-polarized cylindrical-wave sector is used only as a local morphology template, not as a global Earth-Moon spacetime or as an absolute lunar acceleration prediction. The paper derives the Newtonian plus channel, the off-diagonal sine-quadrature residual basis, the frame-misregistration degeneracy, and the generalized least-squares form of a residual gravimetric estimator. The result is not a replacement for standard lunar tidal theory and not a detection claim; it is a framework for estimating or bounding a possible off-diagonal lunar tidal residual.