ECF IV: The Principle of Correlation Extrema — The Triple Convergence of Mathematics, Physics, and Information

The Emergence-Convergence Framework (ECF) derives from a unique ontological postulate: existence is correlation. Its three predecessor papers independently established the inferential unfolding (construction of mathematical structures), the metric convergence (gradient-flow dynamics of the correlation strength λ towards 1/2), and the role of information. However, a deep unification remained open: at λ=1/2, mathematical richness, physical self-consistency, and information capacity simultaneously attain their optimum. Is this a coincidence of three independent facts, or the necessary projection of a single principle? This paper provides the closure. We construct a unified functional Ω(λ)=S(λ)+1/I(λ), where S is the self-consistency functional and I is the information capacity, rigorously defined from the positive-definite kernel of the correlation network. We prove the Triple Extremum Theorem: at the unique self-dual fixed point λ=1/2, Ω achieves its unique global minimum. This minimum simultaneously implies (I) optimal physical self-consistency (S=0), (II) maximal information capacity (I=Imax), and (III) maximal logical-category richness (R=Rmax). Each extremum is independently forced by self-duality; their coincidence is the inevitable expression of a single extremum principle. From this theorem, we derive seven corollaries, including a resolution of Wigner's "unreasonable effectiveness" puzzle, an explanation of physical criticality as a structural necessity, the ontological status of information, and a mechanism for theory innovation in science. We further map the self-referential limit of the framework: Ω possesses an irremovable singularity in its self-referential directions precisely at λ=1/2, marking the boundary where any coding system must expose its own blind spot. This singularity is not a defect but the structural condition for the birth of new theories. The conclusion is: the trinity of mathematics, physics, and information is not a random coincidence but the three-fold projection of the self-dual fixed point of the correlation network. Wigner's puzzle is transformed from an epistemological mystery into a mathematical necessity.