Topological Saturation and the End of Infinity: Resolving the Peano and Menger Paradoxes via Computational Finitism

Classical topology presents profound paradoxes when taken to the infinite limit: the Peano curve achieves 2D space-filling with a 1D line of infinite length, while the Menger Sponge possesses infinite surface area but zero volume. These paradoxes rely entirely on the assumption of a continuous, infinitely divisible manifold. In this paper, we resolve these paradoxes through the lens of Computational Finitism. By imposing a "Hardware Ceiling" (the Planck voxel) and a finite computational alphabet (Modulo-9), we demonstrate that infinite recursion is physically impossible. We prove that space-filling curves achieve 100% topological mapping at finite depths via Structural Completion, providing a discrete mechanism for the Holographic Principle. Furthermore, we show that the Menger Sponge does not vanish into zero volume or infinite area; rather, it reaches a bounded "Topological Mass" (), perfectly modeling the porous, holographic nature of the quantum vacuum. We conclude that dimensionality and fractal complexity are not properties of infinity, but emergent data structures of a finite, pixelated universe. The universe does not need infinite precision to create complex, space-filling, or highly porous structures. It only needs a finite alphabet, a recursive rule, and a hardware ceiling.