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=I_max), and (III) maximal logical-category richness (R=R_max). 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 trinity of mathematics, physics, and information is the three-fold projection of the self-dual fixed point of the correlation network. V2.0 Version Update NotesMain revisions relative to V1.0: (1) Unified ontological presupposition narrative. The ECF I starting point is now expressed as a single presupposition ("existence is correlation") with two dimensions (P1, P2), and the individuality theorem follows as a B-tier consequence. (2) Cross-paper section references updated to align with the final section numbering of ECF I and ECF III. (3) Reference to CDUFD Supplementary Material IV added with formal citation. (4) Declaration format unified with the CDUFD series standard. DOIs updated to full Zenodo DOIs. All core technical content—definitions, theorems, proofs, corollaries, and open problems—remains unchanged from V1.0.