The Ontological Singularity Interpretation: A Heuristic Geometric Toy Model for the Emergence of Quantum Mechanics and Gravity

We present a conceptual "toy model" that attempts to unify Quantum Mechanics and General Relativity under a single geometric principle: the prohibition of the "Ontological Singularity" ($\mathcal{S}$). We define $\mathcal{S}$ as a state of absolute localization ($\Delta x \to 0$), corresponding to the cessation of physical existence. In this heuristic framework, we postulate that the vacuum possesses a fundamental "geometric stiffness" that resists collapsing into $\mathcal{S}$. We demonstrate that the Schrödinger equation can be derived not as a fundamental law, but as a condition for minimizing the geometric curvature of the probability amplitude. Furthermore, we suggest that gravity emerges as the macroscopic tension resulting from this resistance. \textbf{Crucially, we strip away previous thermodynamic analogies (e.g., heat release) and mechanical constructs (e.g., tethers), proposing instead a purely topological view where particles are local deformations of the vacuum manifold.} This work serves as a preliminary proposal for a background-independent geometry of existence.