This paper establishes a novel optimization theoretical framework—Entropy-Regularized Measure Conic Programming (ERMCP). This theory does not simply nest classical cone programming with sparse Brull bar optimization, but rather achieves a deep integration at the geometric and dual levels. We construct the concept of a "measure cone," embedding the probability measure space into the cone structure, and interpreting the relative entropy constraint as an information geometric boundary, thus forming a composite cone structure under a unified convex analysis system. This paper proves that sparse Brull bar cone programming based on the relative entropy uncertainty set is equivalent to a smooth exponential cone programming problem; further, it establishes its strong duality, zero duality gap condition, KKT optimality system, and strict convexity and unique solution theorem. This theory geometrically realizes a structural leap from linear cones to information cones, providing a unified mathematical foundation for sparse Brull bar optimization, risk control, and machine learning.