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

We study the 2-adic valuation of the spectral Bernoulli numbers B_m^{(4)}, which arise as Taylor coefficients of the logarithmic derivative of W_4(x) = (sinh^4 x - sin^4 x)/16.

The main results are:

1. Closed form: W_4(x) = (sinh^4 x - sin^4 x)/16 (proved).
2. Logarithmic derivative identity: The generating function for the spectral cotangent S_4 satisfies G(v) = 4 + 4v d/dv log D(v), yielding g_m = 4m c_m where c_m are the coefficients of the formal logarithm of D(v)/delta_0. This separates the valuation into a smooth factor 4m and an arithmetic factor c_m (proved).
3. Digit-sum cancellation: The 2-adic valuation v_2(B_m^{(4)}) = 4m + 1 for all m ≥ 1. This arises from the exact cancellation v_2((4m)!) + v_2(b_m) = (4m - s_2(m)) + (1 + s_2(m)) = 4m + 1, where the binary digit sum s_2(m) appears with opposite signs in the factorial and Taylor coefficient contributions. The formula is proved rigorously conditional on v_2(c_m) = s_2(m) - 1 - v_2(m), which is verified computationally for m ≤ 100.

We conjecture the unified law v_2(B_m^{(2^r)}) = 4m + 2r - 3 for all powers of 2, with universal slope 4 and intercept depending only on r.

This is part of a series on the generalized exponential functions E_n^k. The series is available at https://zenodo.org/communities/nabil-el-mahyaoui/