Modular Ricci Flow and the General Theory of Singularity: Toward a Torsion-Constrained Resolution of the Hodge Conjecture

Abstract

Josef Edwards

We present a modular geometric framework that bridges recent advances in Ricci flow dynamics with the General Theory of Singularity (GTS), aiming to tackle the longstanding Hodge Conjecture in algebraic geometry. By formulating modular recurrence relations on topological quantum geometries, we incorporate discrete (modular) flux quantization into the Ricci flow equations. This yields a definition of a Modular Quantum Ricci Tensor (QRT) on four-dimensional curved manifolds, which includes contributions from torsion and quantized flux. Using GTS – an extension of Einstein’s gravity that introduces an intrinsic spacetime torsion to regularize singularities – we impose torsion-constrained conditions on cohomology cycles.

These torsion cycles are given explicit geometric interpretation as finite-order (modular) elements in homology, providing a novel mechanism by which certain cohomology classes become “algebraic.” We link this framework to Calabi–Yau manifolds and mirror symmetry, showing how modular flux constraints naturally align with discrete invariants like Hodge numbers and how algebraic cycles might be captured via derived categorical structures. Figures illustrate toroidal (donut- like) embeddings that visualize the chained modular structure of the flow in discrete time. The results suggest that combining modular Ricci flows with torsion physics can smooth out singularities while enforcing integrality conditions on curvature and flux – a synergy that could support a constructive approach to proving the Hodge Conjecture. We conclude with implications for string theory compactifications and outline future work needed to rigorously validate this approach in both mathematics and physics.

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