Double Zero Theory — Article 2: Complete Proof and Analysis of the Sensitivity Law k(p, q) = pπ/(pπ² + 8q)

This paper presents a complete, rigorous proof of the Sensitivity Law of the Double Zero Theory (DZT), establishing that the sensitivity constant is given by k(p,q) = pπ/(pπ²+8q). The proof follows a complete five-step application of the Implicit Function Theorem to the optimality condition at (t,δ)=(1/2,0), with all partial derivatives computed explicitly. Three analytical properties are proved: strict decrease in q, strict increase in p, and universal bounds 0 < k < 1/π. The formula is validated for all 16 pairs (p,q) in {1,2,3,4}² with relative error below 0.15% and R²>0.9999. The second-order coefficient k₂≠0 is identified as an open problem. This is Article 2 of the DZT series.⁸