Vulnerability Abundance: A formal proof of infinite vulnerabilities in code

Abstract: We present a constructive proof that a single C program, the \emph{Vulnerability Factory}, admits a countably infinite set of distinct, independently CVE-assignable software vulnerabilities. We formalise the argument using elementary set theory, verify it against MITRE's CVE Numbering Authority counting rules, sketch a model-checking analysis that corroborates unbounded vulnerability generation, and provide a Turing-machine characterisation that situates the result within classical computability theory. We then contextualise this result within the long-running debate on whether undiscovered vulnerabilities in software are \emph{dense} or \emph{sparse}, and introduce the concept of \emph{vulnerability abundance}: a quantitative analogy to chemical elemental abundance that describes the proportional distribution of vulnerability classes across the global software corpus. Because different programming languages render different vulnerability classes possible or impossible, and because language popularity shifts over time, vulnerability abundance is neither static nor uniform. Crucially, we distinguish between infinite \emph{vulnerabilities} and the far smaller set of \emph{exploits}: empirical evidence suggests that fewer than 6\% of published CVEs are ever exploited in the wild, and that exploitation frequency depends not only on vulnerability abundance but on the market share of the affected software. We argue that measuring vulnerability abundance, and its interaction with software deployment, has practical value for both vulnerability prevention and cyber-risk analysis. We conclude that if one programme can harbour infinitely many vulnerabilities, the set of all software vulnerabilities is necessarily infinite, and we suggest the Vulnerability Factory may serve as a reusable proof artifact, a foundational `test object',for future formal results in vulnerability theory.