extra;;"Symmetry Restoration in Elliptic Curves: The Kenuli–Barlow–Inverse Law and Stable Null Points" ✅

This research presents a numerical symmetry framework for elliptic curves, introducing the concepts of Kenuli Stable Null Points and the Kenuli–Barlow–Inverse Law. The framework formalizes how the algebraic rank ($r$) and the order of the Tate–Shafarevich group ($|\Sha(E)|$) act as symmetry modulators, influencing the convergence of curve coefficients toward palindromic, low-entropy configurations.

Rank-zero curves naturally achieve a stable configuration (Kenuli Stable Null Point) with minimal iterations, while curves with higher rank or larger $|\Sha|$ values require discrete logarithmic shifts and middle-digit corrections (Kenuli Nudge) to restore symmetry.

The study provides:

• A deterministic, heuristic tool for analyzing elliptic curve rank and the influence of $|\Sha(E)|$.

• Empirical validation across multiple curve classes, showing predictable step counts and resonance behaviors.

• A BSD-compatible conceptual interpretation, linking numerical symmetry patterns to structural arithmetic invariants, without claiming analytic proof of the Birch–Swinnerton–Dyer conjecture.

This dataset and methodology offer a new perspective on structural and numerical symmetry in elliptic curves, providing a bridge between arithmetic invariants and computational analysis. Python code for automated detection of Kenuli Stable Null Points is included in the appendix.

Keywords: Elliptic Curves, BSD Conjecture, Numerical Symmetry, Palindrome Patterns, Kenuli Stable Null Point, Barlow–Inverse Law, Tate–Shafarevich Group, Rank Analysis, Python Code