THE 2-ADIC VALUATION OF SPECTRAL BERNOULLI NUMBERS: A DIGIT-SUM CANCELLATION LAW

For n = 2r with r ≥ 1, the spectral Bernoulli numbers Bm(n) defined through the logarithmic derivative of Wn = ∏k=0n−1 Enk satisfy v2(Bm(n)) = 4m + 2r − 3 for all m ≥ 1. The 2-adic valuation is exactly linear in m with universal slope 4 and intercept 2r − 3, independent of the binary expansion of m. This uniformity arises from a digit-sum cancellation: the factorial v2((nm)!) and the Taylor coefficient v2(bm) each depend on the binary digit sum s2(m), but with opposite signs, so the dependence cancels in the product Bm(n) = (nm)! · bm. We prove the closed form W4(x) = (sinh4x − sin4x)/16 and establish the valuation law via a Kummer-theoretic argument on the 2-adic inverse of a rescaled factorial series: the even-index step uses a direct parity argument, while the odd-index step employs an even/odd decomposition in which the dominant term is controlled by unshifted Kummer and a cross-convolution remainder is proved subdominant via a parity argument on carry-free decompositions. The general conjecture is verified computationally for n = 2, 4, 8, 16.