From Force to Fractal: A Structural Realist Synthesis of Riemannian Manifolds and Invariant Set Theory

The transition from force-based epistemic placeholders to a purely geometric ontic bedrock represents the defining trajectory of modern theoretical physics. By synthesizing Ontic Structural Realism with contemporary developments in Invariant Set Theory, we propose a unified framework where reality is modeled as a hyperdimensional fractal manifold. We utilize numerical simulations of spectral dimensions and p-adic distance metrics to demonstrate the “gappy” nature of the ontic bedrock at the Planck scale. This approach allows for a rigorous formalization of the “Flatlander” constraint, characterizing quantum indeterminacy not as a fundamental blurriness, but as a geometric boundary of the observer. Our findings demonstrate that the spectral dimension of spacetime reduces from four to approximately 1.58 at the Planck scale, a result consistent with fractal-constrained manifolds such as the Sierpinski gasket. This dimensional reduction provides a natural resolution to the renormalization problem, as the “gappy” nature of the invariant set renders quantum field theory integrals finite without the need for arbitrary cut-offs. The implications of this synthesis are profound for the search for a Unified Field Theory and our understanding of epistemic humility. By addressing the methodological gap between smooth manifolds and discrete fractals, we provide a continuous mathematical bridge that preserves structural invariants across scales. This research suggests that the “Information Horizon” is a hard geometric limit, and our current physical laws are high-resolution maps of a territory we can only partially resolve.