What question did this study set out to answer?

The aim is to rigorously prove the sensitivity law of the Double Zero Theory.

June 1, 20260 citationsOpen Access

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

Key Points

The aim is to rigorously prove the sensitivity law of the Double Zero Theory.
Applied the Implicit Function Theorem in five steps
Computed all partial derivatives explicitly at the point (1/2, 0)
Validated the formula for 16 pairs of values (p, q).
Proved the sensitivity constant k(p,q) = pπ/(pπ² + 8q)
Found k strictly decreases in q and increases in p
Achieved validation with a relative error below 0.15% and R²>0.9999.

Abstract

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 0.9999. The second-order coefficient k₂≠0 is identified as an open problem. This is Article 2 of the DZT series.⁸

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