Multi-Variable p-Adic L-Functions and Cross-Partial Operators: A Complementary Diagnostic Tool for BSD Counterexamples

This work introduces a novel multi-variable p-adic analytical framework designed as a complementary diagnostic tool for scanning elliptic curves under unstable additive reduction. By expanding the classical single-variable p-adic L-function into a three-variable analytic space—incorporating the weight variable s, a geometric deformation parameter κ arising from modular polynomial quotients x2/p, and a local Tamagawa scale factor ωwe isolate transient analytic zeros from algebraic ones. We formalize a third-order cross-partial differential operator Ds,κ,ω to evaluate the stiffness of the zero-locus. Applying this tool to the Cremona elliptic curve 162b1, we provide explicit mathematical evidence of a hyperbolic saddle-point signature, isolating a transient analytical zero that acts as a localized anomaly to the classical formulation of the Birch and Swinnerton-Dyer conjecture.