Frame Degeneracy and Sine-quadrature Residuals in Lunar Tidal Modeling

The dominant Earth–Moon tide is accurately described by Newtonian theory, but interpreting a possible off-diagonal residual requires separating its signature from errors in the standard tidal model. In a local lunar principal frame, the Newtonian plus channel varies as cos ( 2 β ) , whereas a symmetric off-diagonal residual enters the sin ( 2 β ) quadrature. We extend this separation to a globally defined three-dimensional symmetric trace-free tensor with two transverse coefficients projected onto each station vertical. This formulation establishes an exact first-order identifiability limit: lunar-frame errors generate responses indistinguishable from the cross-residual signal, making finite constraints intrinsically prior limited. A joint-prior GLS/Fisher forecast incorporates frame, ephemeris, projection, terrestrial-response, loading, environmental, instrumental, and correlated-network uncertainties. The dimensionless worst-principal-axis uncertainties are 9.513 × 10 −3 , 4.490 × 10 −4 , and 4.182 × 10 −5 for conservative, moderate, and optimistic synthetic networks. Monte Carlo injection–recovery experiments assess robustness using independent calibration and validation halves. Under the declared response-expanded Stress budget, optimistic formal all-draw coverage falls to 0.317, with a false-positive rate of 0.685; calibrated robust covariance restores held-out coverage to 0.949 and the false-positive rate to 0.051. These results remain conditional on the synthetic assumptions. The Halilsoy cross-polarized sector provides only a local morphology template; no detection or constraint on γ H is claimed. The physical contribution is to establish how a generic lunar cross-residual would enter gravimetric observations and what independent constraints are required to distinguish it from standard tidal-model errors, providing a quantitative foundation for future observational tests.