Muazzam Theorem: An Empirical 1/log(N) Scaling Law for the Average of |Sha| * L(E,1) over Elliptic Curves

We present the Muazzam Theorem v1.0, an empirical scaling law for the arithmetic invariants of elliptic curves over Q. Based on statistical analysis of 1000 elliptic curves of rank 0 from the LMFDB database, we observe that the average value of the product |Sha(E)| * L(E,1) decays logarithmically with the conductor N. The proposed empirical law is:avg(|Sha(E)| * L(E,1) | N) = C / log(N)where C = 0.8135 is defined as the "Muazzam Constant". This result offers a new statistical perspective related to the Birch and Swinnerton-Dyer conjecture. CONTENTS:1. muazzam_paper.pdf - Full preprint with theorem, proof sketch, and discussion2. muazzam_data.csv - Processed dataset of binned averages used for the fit3. fig1_muazzam_theorem_scaling_law_for_elliptic_curves.pdf - Figure 1: Vector format for publication4. fig1_muazzam_theorem_scaling_law_for_elliptic_curves.png - Figure 1: Image format for presentations KEYWORDS: Elliptic Curve, BSD Conjecture, Tate-Shafarevich Group, L-function, Number Theory, Empirical Law, Statistical Number TheoryLICENSE: CC-BY-4.0AUTHOR: Haqqi Muazzam MulhimYEAR: 2026