Ontological Bedrock: Persistence, the Void, and the Fine-Structure Constant

We establish a minimal, defect-free foundation for determinate existence and apply it to two problems: the structural impossibility of absolute non-existence, and the derivation of the fine-structure constant. Beginning from the single premise something exists determinately, we give a precise four-condition definition of "determinately": not vague, not self-contradictory, not flux, and—crucially—not vacuous. From this definition we prove that exactly three structural primitives are forced: a state space, a selective criterion, and an admissible transformation family. Two independent persistence conditions—Idempotent Closure (IC) and Energetic Viability (EV)—are derived and shown to be necessary and sufficient for any form to persist determinately. The framework is machine-verified in Lean 4 (DOI: 10.5281/zenodo.19023530). We apply the filter to its own limiting case: the Void. Absolute non-existence with total potentiality and total absence simultaneously contains a Globally Coupled Contradiction (GCC) and fails both IC and EV. Leibniz's question "Why is there something rather than nothing?" presupposes a coherent alternative that does not exist. Existence is structural residue; the first distinction is a logical fixed point guaranteed by Lawvere and Kleene independently. Finally, we apply the filter to the electromagnetic vacuum in a consistent five-dimensional geometric framework. The EV condition forces a solvency equation whose unique solution is \alpha = \frac{9}{16\pi^4}\left(\frac{\pi^5}{120}\right)^{1/4} \approx 1/137.036, derived without assuming its value. The relative error against CODATA 2022 is 6 \times 10^{-7}. The filter does not merely permit or forbid—it is the difference between form and formlessness, where formlessness, as proved, was never an option.