Vulnerability Abundance: A formal proof of infinite vulnerabilities in code

2026-04-08

AI Summary

This study investigates whether software vulnerabilities are densely or sparsely distributed within programs and formally proves, for the first time, that a single program can contain a countably infinite number of distinct vulnerabilities, each eligible for its own CVE identifier. By constructing a “vulnerability factory” in C and integrating set theory, Turing machine models, and CVE naming conventions, the work introduces the concept of “vulnerability abundance” to quantify the distribution of vulnerability classes across the software ecosystem. The research carefully distinguishes between theoretically infinite vulnerabilities and their finite practical exploitability, thereby demonstrating that the global set of software vulnerabilities must be infinite. This result establishes a reusable formal framework for vulnerability theory grounded in rigorous mathematical reasoning.

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.

Problem

• vulnerability abundance
• infinite vulnerabilities
• CVE
• software vulnerabilities
• vulnerability density

Innovation

• Vulnerability Abundance
• Constructive Proof
• CVE-assignable Vulnerabilities
• Vulnerability Factory
• Computability Theory