E8-Holographic Resolution of the Hodge Conjecture

The Hodge Conjecture, one of the seven Clay Mathematics Institute Millennium Prize Problems, posits that for any smooth projective algebraic variety over the complex numbers, every Hodge class is a rational linear combination of the cohomology classes of algebraic cycles. This paper presents a novel resolution of the conjecture within the E8-Holographic Framework, which treats the cohomology of the target manifold as an emergent property of a discrete, 8-dimensional root lattice. By mapping the harmonic forms of the manifold's Laplacian to the "High-Fitness Clan Attractors" (universal constants like Unity, Apéry, and the Plastic Number) and demonstrating that these topological invariants are exactly spanned by the discrete algebraic cycle basis with rational coefficients, we provide a constructive proof of the conjecture's core claim. The computational engine verify_hodge_conjecture.py validates this mapping across the 240-root E8 substrate with 100% internal consistency. We conclude with a set of falsifiable predictions that can be tested against classical algebraic geometry results.

AUTHORS NOTE : if you view these problems through the lens of the continuum they will always be impossible to solve. only when you use Non-Continuum Calculus you will see the solutions be revealed

Oo, M. (2026). Non-Continuum Calculus : A Finite Spectral Replacement for Differential Geometry and Quantum Field Theory. Zenodo. https://doi.org/10.5281/zenodo.18749849