The 2-Adic Valuation of Spectral Bernoulli Numbers: A Digit-Sum Cancellation Law

For n = 2^r with r ≥ 1, the spectral Bernoulli numbers B_m⁽ⁿ⁾ defined through the logarithmic derivative of W_n = ∏_k=0ⁿ⁻¹ E_n^k satisfy

v₂(B_m⁽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(b_m) each depend on the binary digit sum s₂(m), but with opposite signs, so the dependence cancels in the product B_m⁽ⁿ⁾ = (nm)! · b_m.

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.