Wasserstein Distance of Divisor Measures to Benford's Law: Dirichlet-Kernel Spectral Bounds, Dyadic Counterexamples, and Wintner-PNT Divergence for Highly Composite Numbers

For a positive integer M, the base-2 mantissas of the divisors of M define an empirical probability measure on the multiplicative cell [1, 2) that depends strictly on the odd kernel M_odd and has K = tau(M_odd) atoms of uniform weight 1/K. This preprint investigates the Monge-Kantorovich Wasserstein-W1 convergence of divisor measures to the scale-invariant Benford measure d mu_infty(u) = du / (u ln 2) on [1, 2) and to the uniform Haar measure on [0, 1) and on the unit circle S^1. Core Mathematical Results: Uniform Quantization Benchmark (Theorem 2.1): The minimum W1 distance achievable by any uniform K-atom measure on [1, 2) is tanh(ln 2 / (4K)) / ln 2 = 1/(4K) - (ln 2)^2 / (192 K^3) + O(K^-5), while on [0, 1) and S^1 it is 1/(4K). This recovers, for base b = 2, the uniform quantization bound of Berger & Xu (J. Theor. Probab. 32 (2019), Cor. 4.2). Because prime factorization forces all K divisor mantissas to be distinct with equal weights 1/K, this uniform bound is the exact sharp benchmark for divisor measures. Dirichlet-Kernel Spectral Factorization (Theorem 2.2): On the circle S^1, the empirical phase measure is the circular convolution of uniform arithmetic progressions generated by prime powers p_j^(a_j) || M_odd. Its Fourier coefficients factor into normalized Dirichlet kernels |nu_hat(m)| = (1/K) prod_j D_(a_j)(m log_2 p_j), yielding explicit Kantorovich-Rubinstein lower bounds and Parseval upper bounds on W1(S^1). Dyadic Counterexample to General Weyl Divergence (Proposition 3.1): We construct square-free integers n_r = prod_(j=1)^r p_j with 71-bit primes near 2^(k_j + 2^-j). Although {log_2 p_j} are Q-linearly independent, the divisor phases approximate the dyadic grid and satisfy |K * W1(S^1) - 1/4| <= 2.2e-13 for r <= 18. Thus Q-linear independence alone cannot force divergence. Wintner-PNT Asymptotic Divergence for HCNs (Theorem 4.4): Using the Prime Number Theorem in dyadic octaves, the prime logarithms converge to Wintner's tilted density rho_u(s) = (2 ln 2) 2^-{u-s}. The resulting drift h_m(u) satisfies an exact first-order ODE and has a certified strictly positive envelope min_u max_(1<=m<=32) h_m(u) >= 0.00195, certified via adaptive subdivision in 64-bit Arb ball arithmetic (python-flint). Combined with a self-contained variational bound on prime exponents in Highly Composite Numbers (HCNs), this proves unconditionally that along odd primorials and all HCNs: K * W1 >= K^(delta/ln 2 - o(1)) -> infinity, with certified exponent delta/ln 2 >= 0.00281. Exhaustive Record-Holder Enumeration: Complete census of all 100,000 odd integers 3 <= n <= 200,000 reveals 25 record holders in W1([1, 2)), 27 in W1([0, 1)), and 26 in W1(S^1); in every metric, exactly 10 of the 17 odd HCNs (OEIS A053624) are global record holders. Included Assets in Package: Complete manuscript PDF (publication grade, compiled with MathJax); Full Markdown source code; Standalone Python certification script (certify_divisor_w1_wintner_hcn.py) reproducing all theorems, bounds, Arb ball certifications, and record-holder tables; Cryptographic SHA-256 integrity manifest.