Spectral Structure and Logarithmic Fluctuations in Constrained Null Geometry

This paper develops the spectral formulation of the reduced fluctuation sector in constrained null geometry. Building on the preceding paper, where the reduced shape space, flat measure, and exact local coefficient were established, the present work shows why the logarithmic fluctuation law follows necessarily from the spectral structure of the reduced operator itself. The reduced complex fluctuation mode is governed locally by the two-dimensional Laplacian, so individual modes carry inverse-eigenvalue covariance proportional to 1 over kappa times q squared. Because the density of modes in two dimensions grows linearly with q, the variance accumulated over momentum shells is constant per logarithmic interval. The logarithmic law is therefore not introduced as a heuristic ansatz, but derived as a direct consequence of reduced two-dimensionality and inverse-Laplacian covariance. The paper further shows the equivalence between three formulations of the same fluctuation structure: the spectral sum over eigenfunctions, the local Fourier representation, and the logarithmic Green kernel in position space. In this way, the result clarifies the mathematical meaning of the logarithm established in the earlier paper. The observable scalar mode inherits the variance law with coefficient 1 over 4 pi kappa, reproducing C = 1 over 4 pi in canonical normalization. What remains open is not the existence of the logarithm or its coefficient, but the physical determination of the infrared scale L, which belongs to the global closure structure and is left to subsequent work. More broadly, the paper sharpens the status of the fluctuation sector within constrained null geometry. What previously appeared as a plausible logarithmic scaling law can now be stated as a strict structural consequence: reduced null geometry leads to a two-dimensional isotropic fluctuation operator, that operator leads to inverse-Laplacian modal covariance, and two-dimensional mode counting leads to logarithmic accumulation of variance. The paper therefore closes the spectral side of the fluctuation problem while keeping the global infrared completion as the next step of the program.