Functional Measure and Mode Reduction in Constrained Null Geometry

This paper develops a rigorous reduced configuration-space treatment of a system of three future-directed null vectors subject to a timelike closure condition. Working in the rest frame of the closure vector, it shows that the space of constrained ordered null triples, after quotienting by global spatial rotations, is exactly two-dimensional and can be represented as an open triangular shape space in normalized weight coordinates. Starting from the Lorentz-invariant cone measure and the closure constraint, the paper performs the rotational reduction explicitly and derives the corresponding reduced measure in exact form, showing that it is flat on the reduced shape space. The paper then studies fluctuations around the symmetric configuration. Using the residual discrete symmetry of the reduced system, it shows that the local quadratic fluctuation sector is necessarily isotropic and therefore admits a natural complex representation. A scalar observable probes only one real projection of this isotropic mode, which yields the geometric projection factor entering the observable variance. The logarithmic coefficient is then derived from the Green function of the two-dimensional Laplacian on the local reduced fluctuation plane. In this way, the coefficient governing the logarithmic fluctuation sector is not introduced as a free normalization constant, but follows directly from the exact reduced geometry, the isotropic local fluctuation structure, and the observable projection. The result provides a rigorous mathematical closure of the functional-measure and coefficient problem within constrained null geometry, while leaving broader dynamical and phenomenological developments to subsequent work.