A counterexample on two kinds of positive dependence

We construct a strict bivariate Archimedean copula that is positively quadrant dependent but is not positive measure inducing. This disproves the forward implication in the equivalence conjectured by Sebastian Fuchs and Marco Tschimpke. Their work introduced the positive measure inducing property through a reflected copula sum and established the equivalence for the Frank family. Our example separates the two properties in the full Archimedean class. Its inverse generator is the maximum of two exponential functions. Convexity gives copula validity, and a piecewise linear inequality in logarithmic coordinates gives positive quadrant dependence throughout the unit square. A rectangle strictly inside the open square $(0,1/2)^2$ has a negative increment for the reflected sum, excluding the positive measure required by the definition. All corner estimates use rational arithmetic and squaring, without a density assumption. The complete argument, including generator admissibility, boundary values and the measure-theoretic contradiction, is formalized in Lean. The converse implication and the separate negative-dependence equivalence are not resolved by this example. This work has not yet been reviewed by experts. The proofs, the Lean formalization and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Checked in Lean from the definitions; not yet reviewed by experts. Produced by AI models under the author's direction.