A Universal Quadratic Law Governing the Local Geometry of Riemann Zeros

This work introduces a local deformation framework for the study of the nontrivial zeros of the Riemann zeta function. A one-parameter deformed family Zc (s) is introduced, continuously connected to the classical zeta function in the limit c → 0. For each fixed imaginary height t, the real coordinate minimizing the modulus defines a continuous minimum trajectory ("whisker") encoding the local geometric extension of each Riemann zero. The associated integrated deviation S (c, t) is shown to satisfy the asymptotic quadratic scaling law S (c, t) ~ α c² (c → 0⁺). A perturbative analytical argument establishes that this scaling is a structural consequence of the zero condition and the regularity of the deformation. Numerical investigations over 500+ zeros confirm the law with median log-log slope 2. 003 ± 0. 018 and asymptotic coefficient α∞ ≈ 0. 036, stable across all tested numerical parameters. The results suggest the emergence of a universal conical local geometry centred on the critical line.