Dept. of Computer Science, The University of Manchester, UK. · Dept. of Electrical Engineering, Federal University of Amazonas (UFAM), Brazil.
The integration of Formal Verification tools with Large Language Models (LLMs) offers a path to scale software verification beyond manual workflows. However, current methods remain unreliable: without a solid theoretical footing, the refinement process acts as a black box that may oscillate, loop, or diverge. This work bridges this critical gap by developing an LLM-Verifier Convergence Theorem, providing the first formal framework with provable guarantees for termination in multi-stage verification pipelines. We model the interaction not as a generic loop, but as a sequential absorbing Markov Chain comprising four essential engineering stages: \texttt{CodeGen}, \texttt{Compilation}, \texttt{InvariantSynth}, and \texttt{SMTSolving}. We prove that for any non-zero stage success probability (
δ>0), the system reaches the \texttt{Verified} state almost surely. Furthermore, because of the sequential nature of the pipeline, we derive a precise latency bound of
E[n]≤4/δ. We stress-tested this prediction in an extensive empirical campaign comprising over 90,000 trials. The results match the theory with striking consistency: every run reached verification, and the empirical convergence factor clustered tightly around
Cf≈1.0, confirming that the
4/δ bound accurately mirrors system behavior rather than serving as a loose buffer. Based on this data, we identify three distinct operating zones -- marginal, practical, and high-performance -- and propose a dynamic calibration strategy to handle parameter drift in real-world environments. Together, these contributions replace heuristic guesswork with a rigorous architectural foundation, enabling predictable resource planning and performance budgeting for safety-critical software.