From Meaning Networks to Mathematical Systems :Ontological Reconstruction, Theorem-Level Proofs, and Exploratory Frameworks for the Millennium Problems in Two-Dimensional Field Ontology

This paper reconstructs the seven Millennium Prize Problems within the unified framework of Two-Dimensional Field Ontology (Event–Dimension Theory, EDT). EDT takes as itsfirst axiom that meaning precedes truth and regards physical laws as stable symmetry patterns emergent from meaning networks atfinite resolution. The preceding two papers established the ontological foundations of EDT and the generalized framework for complex-system evolution. The present paper examines the universality of EDT in pure mathematics. The paper is organized along the EDT dimensional spectrum. Yang–Mills theory is governed jointly by the E-field and the D-field, with the decisive component in the E-field; Navier–Stokes regularity is located in the D-field; the Riemann Hypothesis is located in the T-field;the Birch and Swinnerton-Dyer conjecture is an E⊗ T problem; the Poincaré conjecture is a D⊗E problem; the Hodge conjecture and the P versus NP problem are fully three-dimensional E⊗D⊗T problems. For thefirstfive problems, theorem-level proofs or conditional ontological repositionings are given within the EDT framework. For the last two, exploratory frameworks are formulated; no proof is claimed. Thefive theorem-level chapters follow a commonfive-step structure: EDT encoding of the math-ematical system, definition of the ideal steady state, construction of the experiential functional,the irrational-obstruction inequality, and application of the experiential H-theorem. Thefive chapters share four core technical tools—irrational obstruction, memory kernel, sign-consistency constraint, and curvature-flow equation—and a common memory-kernel infrared exponentα =2/3 together with the enhancement factorηmem = 3.841. Principal results are as follows. (i) Yang–Mills existence and a mass gap∆ = 1.60± 0.05 GeV,consistent with lattice QCD. (ii) Physicalfluids admit nofinite-time singularity under the quantum–classical joint description; the standard Navier–Stokes equation is a double truncation 24−1 approximation, with a truncation-failure threshold of order 10 s . (iii) The prime system is a classical EDT meaning network; the real parts of the nontrivial zeros of the zeta function converge to the critical line, and standard analytic number theory is a truncation approximation.(iv) The rank deviation of an elliptic curve converges to zero, and standard arithmetic geometry is a truncation approximation. (v) The Ricciflow, managed by the T-field, converges to an 3εirr-neighborhood of S , and the standard Ricciflow is a truncation approximation. The central structuralfinding is the truncation hierarchy: standard theories are projections of the complete EDT structure under specific truncations, and the singularities or unprovable state-ments of standard theories correspond to truncation failure. This diagnosis runs through allfive chapters. The dynamical hierarchy distinguishes three dynamical combinations: physicalfluids require both the Yu and Yang equations, arithmetic systems require only the Yang equation,and geometric systems require the Yang equation together with the three-layer T-field. The T-field role hierarchy distinguishes three roles: passive (physical), epistemic (arithmetic), and process-management (geometric). The paper strictly distinguishes theorem-level proofs, conditional ontological repositionings, ex-ploratory frameworks, and open waters. It does not claim that all seven Millennium Problems are solved. Thefive theorem-level proofs cover the parts that can be rigorously established; the two exploratory frameworks mark the parts that require deeper construction. The rigor of EDT lies not in claiming to solve everything, but in precisely demarcating the boundary between the proved and the unproved.