The measurable first cohomology of the tail relation: turbulence and the arithmetic embedding

The obstruction group of a localized Bost-Connes system was previously identified as the kernel of the natural map from the dual symmetry group to the measurable first cohomology of the tail equivalence relation. This paper determines the receiving group in the correct sense: the cocycle group is Polish, the coboundaries form a dense meager Borel subgroup, the quotient Borel structure is trivial, and the coboundary action is turbulent in the sense of Hjorth, so no classification of the cohomology by countable structures exists - the exact upper bound on what computing it can mean. The complete smooth residue is the Mackey-range spectrum. The arithmetic embeds canonically: the quotient of the dual by the obstruction group injects into the equicontinuous classes, while by Dye-Krieger the ambient group depends only on the Krieger type of the relation - the ambient cohomology is amnesiac, and the arithmetic survives as the embedded subgroup and its position. The verdict states the series' philosophy finally: the arithmetic lives in the measure class, the only equicontinuous thread in a turbulent sea, and the turbulence is the measure-theoretic root of the collective blindness of every smooth invariant. Two honest gaps are recorded: the literature localization of the third turbulence condition, and noncompact coefficients.