LLM-Verifier Convergence Theorem

• LLM-Verifier Convergence Theorem is a formal framework that guarantees almost-sure termination and bounds the expected iterations (4/δ) in multi-stage LLM-driven verification pipelines.
• It models each verification stage as a geometric process within an absorbing Markov chain, ensuring predictable resource usage and latency through provable exponential tail bounds.
• Empirical validation from large-scale trials confirms its accuracy, supporting dynamic calibration and safe resource planning in safety-critical software verification.

The LLM-Verifier Convergence Theorem formalizes the convergence properties of LLM-driven verification systems by establishing provable guarantees for the termination and resource cost of multi-stage verification pipelines. These pipelines, central to transformer-based formal verification architectures, interleave LLM reasoning (“solver”) with symbolic verification components, thus enabling the scalable automation of correctness proofs for safety-critical software. The theorem provides both a quantitative upper bound on latency and a strong almost-sure guarantee of eventual verification, replacing empirical guesswork with a foundation grounded in absorbing Markov chain theory and extensively validated through large-scale empirical trials (Dantas et al., 30 Nov 2025).

1. Formal Definition and Markov Chain Model

The LLM-verifier pipeline is rigorously modeled as a discrete-time absorbing Markov chain with five states. Four transient states correspond to canonical workflow stages: CodeGen, Compilation, InvariantSynth, and SMTSolving. The terminal Verified state is absorbing. State transitions are determined by a fixed per-stage success probability $\delta \in (0,1]$, independent across stages and iterations. The transition structure is:

• Progress to next stage: $P_{i,i+1} = \delta$
• Retry current stage: $P_{i,i} = 1-\delta$
• No other transitions allowed

The system begins in CodeGen and advances sequentially. These Markovian dynamics ensure that, so long as $\delta > 0$, the absorber is almost surely reached (Dantas et al., 30 Nov 2025).

2. The LLM-Verifier Convergence Theorem

The main theorem quantifies the fundamental performance guarantees of the system:

LLM-Verifier Convergence Theorem:

1. Almost-sure absorption: $\Pr(\tau < \infty \mid X_0 \in T)=1$. Every verification attempt eventually concludes in the absorbing Verified state.
2. Latency bound: The expected number of overall iterations is $\mathbb{E}[\tau \mid X_0=s_1] \leq 4/\delta$.
3. Tail behavior: There exist $\alpha > 0$ and $\lambda_Q = 1-\delta < 1$ so that $$\Pr(\tau > k \mid X_0=s_1) \leq \alpha (1-\delta)^k$$.

Intuitively, each stage is repeated an independent, geometrically-distributed number of times with success probability $\delta$, resulting in a total expected iteration count that is additive across stages. The $P_{i,i+1} = \delta$0 result is tight and interpretable: higher $P_{i,i+1} = \delta$1 yields faster convergence, while lower $P_{i,i+1} = \delta$2 creates a predictable slowdown but not non-termination (Dantas et al., 30 Nov 2025).

3. Derivation and Empirical Validation of the 4/$P_{i,i+1} = \delta$3 Bound

The derivation proceeds by modeling each stage as a geometric random variable $P_{i,i+1} = \delta$4, corresponding to the number of attempts required for success at that stage. Linearity of expectation yields the aggregate bound:

$P_{i,i+1} = \delta$5

Empirical validation on over 90,000 Monte Carlo trials (for $P_{i,i+1} = \delta$6, 10,000 trials per $P_{i,i+1} = \delta$7) demonstrated that:

• All runs reached the absorbing state.
• Observed mean iteration counts (μ) tracked $P_{i,i+1} = \delta$8 with a convergence factor $P_{i,i+1} = \delta$9 tightly distributed around 1.0 (min $P_{i,i} = 1-\delta$0, max $P_{i,i} = 1-\delta$1).
• The observable latency tails conformed to the theoretical exponential decay.

These results empirically validate both the almost-sure convergence and the sharpness of the derived bound (Dantas et al., 30 Nov 2025).

4. Operating Zones and System Calibration

Statistical analysis of the latency and variance across $P_{i,i} = 1-\delta$2 partitions operational behavior into three zones:

Zone	$P_{i,i} = 1-\delta$3 Range	Mean Iterations	Latency Variance
Marginal	$P_{i,i} = 1-\delta$4	up to $P_{i,i} = 1-\delta$5	wide, $P_{i,i} = 1-\delta$6 up to 19
Practical	$P_{i,i} = 1-\delta$7	$P_{i,i} = 1-\delta$8	sharply falling
High-Performance	$P_{i,i} = 1-\delta$9	$\delta > 0$0	$\delta > 0$1 (at $\delta > 0$2)

The practical implication is that system architects can estimate the effective $\delta > 0$3 during deployment and adapt operational budgets (e.g., by tuning timeouts or fallback strategies) to maintain operation within the practical or high-performance regimes. Explicit calculation of $\delta > 0$4 from a sliding window of step-advance ratios enables dynamic calibration, ensuring robust performance despite parameter drift (Dantas et al., 30 Nov 2025).

5. Broader Context: Sample Complexity and Learning Verifier Dynamics

The convergence theorem addresses the engineering scale-up of formal verification, but related literature investigates the feasibility of learning reliable verifiers for natural language reasoning within PAC frameworks (Balcan et al., 28 May 2025). Sample complexity analysis shows that, for finite verifier hypothesis classes $\delta > 0$5, learning an $\delta > 0$6-accurate verifier with probability $\delta > 0$7 requires $\delta > 0$8 samples. In “trustable” regimes, completeness and soundness criteria—often captured by TVPAC learning—impose additional hardness, with impossibility results establishing that linear sample lower bounds are necessary when $\delta > 0$9 is large.

In the theoretical modeling of self-improvement, the convergence of solver and verifier capabilities is modeled by coupling their “uncertainties” $\Pr(\tau < \infty \mid X_0 \in T)=1$0 and $\Pr(\tau < \infty \mid X_0 \in T)=1$1 with a gap $\Pr(\tau < \infty \mid X_0 \in T)=1$2 evolving according to a linearized ODE:

$\Pr(\tau < \infty \mid X_0 \in T)=1$3

yielding exponential convergence of $\Pr(\tau < \infty \mid X_0 \in T)=1$4 and ultimate stagnation at a finite capability gap (Sun et al., 29 Jun 2025). This provides a complementary, continuous-time perspective on how LLM-verifier policies may converge during iterative refinement cycles.

6. Significance and Applications

The LLM-Verifier Convergence Theorem delivers a first-principles justification for the observed empirical predictability and reliability of LLM-driven formal verification. Its formal and empirical convergence guarantees supplant previous black-box heuristics, supporting engineering practices such as:

• Predictable resource planning, by bounding worst-case verification costs in terms of $\Pr(\tau < \infty \mid X_0 \in T)=1$5.
• Dynamic adaptation to environment or prompt drift, via continual $\Pr(\tau < \infty \mid X_0 \in T)=1$6 estimation and pipeline adjustment.
• Safety-critical budget allocation, with risk quantification via explicit exponential tail bounds.

The result generalizes across LLM-verifier architectures supporting sequential, Markovian error-reduction at each step. A plausible implication is the potential of this framework as a modular component for more sophisticated software verification workflows, so long as their stage transitions can be abstracted as geometric with known or estimable $\Pr(\tau < \infty \mid X_0 \in T)=1$7 (Dantas et al., 30 Nov 2025, Sun et al., 29 Jun 2025, Balcan et al., 28 May 2025).