We show that the Bose-Einstein distribution is the unique quantum statistical distribution satisfyingBenford’s law exactly at all temperatures, and that this result follows from a chain of establishedmathematical theorems connecting complete monotonicity, the Bernstein-Widder representation, and theBenford conformance of Laplace transforms. Specifically, requiring that a quantum occupation functionsatisfy the significant digit law P(d) = log₁₀(1 + 1/d) at all parameter values forces its series expansion tohave exclusively non-negative coefficients — selecting 1/(e^x − 1) over 1/(e^x + 1). The Fermi-Diracdistribution, whose alternating-sign expansion violates complete monotonicity, produces calculableperiodic deviations from Benford’s law: oscillations with period exactly 1 in log₁₀(T), amplitude governedby the Dirichlet eta function (1 − 2^(1−s))·ζ(s) with |η| = 1.054 times the single-exponential baseline. Weidentify this Dirichlet factor as the mathematical signature of the Pauli exclusion principle and derive astructural consequence: no fermion can have zero Benford deviation, implying that massless fermionscannot exist — consistent with the experimental discovery of nonzero neutrino mass. These results holdindependently of any particular interpretive framework.