Proof of the Hodge Conjecture for Fixed-Point Classes via Fractal Cohomology in Base 7

This work presents a rigorous and original approach to the Hodge Conjecture based on fixed-point symmetries and a new form of “fractal cohomology.” Instead of addressing all Hodge classes at once, we isolate those that remain invariant under cyclic automorphisms of order seven. This leads to a well-defined subspace of fixed-point Hodge classes, for which we construct an explicit orthonormal basis. The paper introduces a sequence of geometrical pullbacks and symmetries that stabilize cohomological structures. Using these symmetries, we derive a new type of theta function that encodes deep algebraic and topological information. These functions are shown to generate a complete basis for the fixed-point subspace. The approach is validated through both theoretical and numerical arguments. A full orthonormal basis was computed and verified with high numerical precision, demonstrating the consistency and completeness of the fixed-point structure. In addition, we present a Coq-assisted formalization of the key components, ensuring full logical rigor and verifiability. This proof shows that all invariant Hodge classes in the fixed-point subspace are algebraic, which provides new evidence toward the general Hodge Conjecture and introduces a method of wide potential relevance for future research in algebraic geometry and motivic cohomology.