Morse Topology, Zero Products, and Elliptic Prime Crossings

Abstract

We study how the noncentral zeros of an elliptic-curve $L$-function enter the Birch--Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed $L$-function yields an exact Hadamard decomposition, and the zero statistics $J_E$ and $V_E$ satisfy the positive identity $J_E=V_E/8+Δ_E$. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average $82.4%$ of $Δ_{E,20}$, while independent edge values reveal a discrete $35/36$ zero-count split beyond height $20$. For Haar $SO(2N+1)$ we prove a fixed-dimensional hard-edge law $\Pr{J_N>y}\sim C_Ne^{-3y/2}$. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.