BENFORD'S LAW AS GEOMETRIC THEOREM: From Statistical Regularity to Geometric Necessity via the S⁵ Manifold and SO(6) Projection Geometry

Benford's Law — the observation that leading digits in naturally occurring datasets are distributed logarithmically rather than uniformly — has been explained as a consequence of scale invariance and base invariance. These explanations are correct at the level of statistical description but they do not constitute theoretical understanding. They explain what Benford's Law is without explaining why naturally occurring processes should be scale invariant in the first place. We show that this 'why' is answered by the geometry of the S⁵ manifold with SO(6) rotation symmetry, the compact five-dimensional ground from which the three-dimensional baryonic regime is projected in the One-Object Universe (OOU) framework. Benford's Law is not a statistical regularity about the baryonic regime. It is a geometric theorem: the necessary shadow cast by a uniform distribution on the compact S⁵ manifold onto the logarithmic real line of the baryonic projection. The 3D baryonic regime is not itself compact — it is a projection, and projections are non-compact shadows. The compactness resides in the ground, the S⁵ manifold, not in its shadow. The baryonic regime inherits the statistical consequences of the ground's compactness through the projection map, even though the shadow itself extends without bound in the Archimedean metric. The irrationality of log₁₀(1 + 1/d) for every digit d is not an incidental feature. It is the diagnostic fingerprint of dimensional reduction — the mathematical signal that the phenomenon being described is native to the compact S⁵ ground and has been projected into the non-compact 3D baryonic shadow. The distribution is irrational in the shadow because it is rational in the ground. The framework provides not only a proof of Benford's Law but a principled account of when it holds and when it fails, replacing probabilistic expectation with geometric necessity.