An Operator Bridge Between Arithmetic Heights and Analytic Residues for Elliptic Curves

This record contains a Coherence Geometry manuscript developing an operator framework relating arithmetic height structures and analytic residue data for elliptic curves over Q. The construction proceeds through two parallel bridges. On the arithmetic side, the manuscript reconstructs the Neron-Tate height pairing from a system of local coherence pairings obtained through defect descent at the places of the curve. The resulting global quadratic operator has determinant corresponding to the classical regulator. On the analytic side, the manuscript constructs a quadratic operator from the spectral structure of the associated L-function. A coherence operator associated with the modular form is represented by a kernel whose spectral behavior is controlled through a strict bilinear inequality. Via a Mellin-harmonic correspondence, the critical residue of the completed L-function induces a finite-dimensional Hermitian operator on the period space of the elliptic curve. The manuscript formulates an operator identification between this analytic residue operator and the arithmetic height operator, up to natural period normalization, relating spectral kernel and determinant data to the Mordell-Weil rank and regulator. This manuscript is part of the Coherence Geometry Clay-problem research series and is categorized under the Birch and Swinnerton-Dyer conjecture. It is presented as a structural coherence-geometric program connecting arithmetic heights and analytic residues, not as an accepted resolution of the Birch and Swinnerton-Dyer conjecture.