Global Closure and Emergent Infrared Scale in Constrained Null Geometry

This paper completes the fluctuation program of constrained null geometry by deriving the global origin of the infrared scale that enters the logarithmic fluctuation law. The preceding two papers established the exact reduced shape space, the flat reduced measure, the isotropic local fluctuation structure, the exact coefficient of the logarithmic variance, and the spectral necessity of logarithmic accumulation. What remained open was the physical meaning of the infrared scale L. The present paper shows that this scale is not an external cutoff or a phenomenological insertion, but a global spectral datum of the constrained null system itself. The paper introduces a global closure defect that measures whether a locally admissible reduced configuration belongs to the globally coherent closure class. Linearizing this defect around an exactly closed background yields a positive global closure operator. Its lowest nonzero eigenvalue defines the softest admissible global incoherence mode, and the infrared scale is identified structurally as the inverse of this spectral gap. In this way, the logarithmic fluctuation law acquires its last missing ingredient: the scale entering it is derived from global closure rather than inserted by hand. The same global spectral datum also determines the characteristic rate at which closure defects relax. Under the associated linear semigroup, each nontrivial mode decays exponentially, and the slowest mode sets a characteristic closure rate Omega equal to c times the lowest nonzero eigenvalue, equivalently c divided by the infrared length. The infrared scale and the closure rate are therefore not two independent parameters, but two forms of the same global mode: one spatial and one temporal. This gives the constrained-null framework a structurally closed mathematical chain from reduced geometry and local fluctuations to global infrared completion.