From Intuition to the Explicit Formula: An Experimental Study of the Error Term in the Counting of Squarefree Numbers

I experimentally study the squarefree counting function Q(x) and the error E(x) = Q(x) − (6/π²)x. I organize the thesis as three independent claims, each backed by classical lemmas, not a single new theorem: local modular rigidity (Möbius identity and the periodic sieve modulo M_y; this does not imply low variance); mesoscopic suppression via the mean variance ≍ √H (Gorodetsky–Matomäki–Radziwiłł–Rodgers, 2021) and the empirical ratio R(H;X) ≪ 1, where “crystalline” means a statistical/mean phenomenon; and global spectral oscillation via ζ(s)/ζ(2s), the Explicit Formula under the Riemann Hypothesis, and Ω±(x1/4) unconditionally (Mossinghoff–Oliveira e Silva–Trudgian, Math. Comp. 2021). Up to N = 10⁷, the mean of |E(x)|/x1/4 is about 0.189; the ratio |E|/(x1/4/log x) grows (mean ≈ 2.85), so I find the logarithmic refinement incompatible with the data. This paper is the companion to Paper A (SDCP) in the same deposit.