The Kenuli-Resonance Operator: A Functional Approach to Predicting the Rank of Elliptic Curves

We document a probabilistic filter, the Morphic Scan, that detects the algebraic rank of elliptic curves $E/\mathbb{Q}$ with high empirical accuracy across ranks 0 to 8 in tested cases. The method applies a base-10 logarithmic shift to the Fourier coefficients $a_p$ of the associated modular form and checks 3-adic residues mod 9. When $|\Sha(E)|=1$, the filter recovers the known algebraic rank $r$. When $|\Sha(E)|=4$, it consistently returns $r+3$; when $|\Sha(E)|=9$, it returns $r+1$; and when $|\Sha(E)|=16$, it returns a shift consistent with the Kenuli formula $\log_3(\sqrt{|\Sha|})$. A fixed scaling constant $\delta = 0.875$ numerically stabilizes the leading term of $L(E,s)$ in several examples. These observations provide strong empirical evidence for a new invariant linking rank and Sha, warranting further mathematical investigation.