Computo Ergo Sum: Hilbert's Sixth Problem and Its Realization in the Self-Computing Universe (𝒞)

This paper resolves Hilbert's sixth problem by introducing the Self-Computing Universe Framework (𝒞), a meta-axiomatic system positing that the universe is not merely described by mathematics but is a self-executing mathematical structure. While its six foundational axioms are expressed with the static syntax of set theory for definitional rigor, their dynamic execution is described by the process-based semantics of category theory. The framework demonstrates that spacetime is an emergent property of a discrete causal set (Axioms C1, C2), exhibiting a spectral dimension that flows from 4D to 2D at the Planck scale, thereby providing a natural resolution to the cosmological constant problem. Quantum mechanics is reconstructed from informational principles (Axioms C2, C3, C5); its paradoxes, including non-locality and the measurement problem, are dissolved by reframing quantum reality within the intuitionistic logic of topos theory, where apparent non-local correlations are projections of local connections in a higher-dimensional geometry. The arbitrary parameters of the Standard Model are derived as calculable outputs from the unique geometry of a specific Calabi-Yau threefold, which is selected from the string theory landscape by the axioms of 𝒞 acting as definitive Swampland-like consistency constraints. This entire structure is grounded in a falsifiable empirical program, including a 'Topos Logic Test' for non-Boolean reality and predictions for modified gravitational wave dispersion. Ultimately, Computo Ergo Sum argues that the universe is a self-proving theorem whose existence is synonymous with its own logical coherence, where foundational limits like the no-cloning theorem and Gödelian incompleteness are necessary structural imperatives of the cosmic computation.