Complete Monotonicity and Benford's Law: Deriving Quantum Statistics from the Significant Digit Distribution

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