Martingale Majority Certificate (MMC)

• Martingale Majority Certificate (MMC) is a framework that uses sequential martingale analysis to certify majority vote accuracy with non-asymptotic guarantees.
• It leverages concentration inequalities, PAC-Bayes theory, and e-processes to provide finite-sample, anytime-valid error bounds.
• MMC underpins reliable LLM inference, distributed protocols, and control verification by quantifying sample efficiency and decision reliability.

The Martingale Majority Certificate (MMC) is a statistical framework for certifying aggregate decisions made via majority vote or weighted voting in settings where outcomes are modeled by sequential stochastic processes, typically martingales or supermartingales. Originally introduced in the context of reliable inference for LLMs, MMC generalizes to a variety of sequential decision-making, verification, and protocol design scenarios. The certificate leverages martingale concentration inequalities, PAC-Bayes theory, and sequential e-processes to provide non-asymptotic, anytime-valid guarantees that the majority vote is correct or, more generally, that the group-level decision remains within a specified error envelope.

1. Mathematical Formulation and Core Principles

MMC is fundamentally based on constructing a sequential stopping rule and a family of nonnegative supermartingales (e-processes) that monitor the empirical margin between the “leader” (majority answer) and the runner-up (second-most frequent answer) across independent samples. Let $X_{1}, X_{2}, \ldots, X_{n}$ be i.i.d. draws from the terminal distribution $p$ of an LLM or, more generally, from the output of a stochastic process on a hypothesis space $H$. The majority vote at time $n$ is $\hat{c}_n = \arg\max_{c} N_c$ where $N_c = \sum_{i=1}^{n} \mathbf{1}\{X_i = c\}$. MMC seeks to answer, for a given sample size $n$, whether $\hat{c}_n$ coincides with the mode $c^*$ of $p$.

The core idea is to construct e-processes $E_n$ such that the maximum value over all $n$ is sub-exponential under the null hypothesis (i.e., the majority vote is not the true mode). Ville’s inequality then gives anytime-valid probabilistic guarantees that, upon crossing a preset threshold $1/\epsilon$, the certificate guarantees with high probability ($1-\epsilon$) that the majority decision matches the correct label. The construction typically uses cumulative log-margins or sign-margins (e.g., $\sum_{i=1}^{n} [1\{X_i = c^*\} - 1\{X_i = j^*\}]$).

2. Concentration Inequalities and Finite-Sample Certification

MMC leverages concentration bounds to quantify the sample complexity and reliability of the certification. For finite $n$ and $k$ candidate answers, Hoeffding’s inequality provides a basic guarantee: $$\mathbb{P}\left[\hat{c}_n \neq c^*\right] \leq (k-1)\exp\left(-\frac{n}{2}\delta^2\right),$$ with $\delta := p_{c^*} - \max_{j \neq c^*} p_j$. Bernstein and Chernoff–Markov bounds yield sharper certificates incorporating variance:

• Bernstein: uses $\sigma^2_j := p_{c^*} + p_j - (p_{c^*} - p_j)^2$
• Chernoff–Markov: $$\mathbb{P}[\hat{c}_n \neq c^*] \leq \sum_{j} \exp\{n\log[1 - (\sqrt{p_{c^*}} - \sqrt{p_j})^2]\}$$

The bounds ensure that error probabilities decay exponentially with $n$, with rates governed by the signal-to-noise ratio (SNR) of the margin variable and the separation $\delta$.

3. Sequential Decision Making and Anytime-Validity

MMC formalizes adaptive sample sufficiency through sequential analysis. Each new sample updates the e-processes, and the system monitors whether thresholds have been surpassed, thus triggering certification:

• The stopping rule is governed by the martingale property of the e-processes: nonnegative supermartingales such that expected future increments, conditioned on the present, are non-increasing.
• By Ville’s inequality, for any $\epsilon$, the probability that the running maximum of the e-process exceeds $1/\epsilon$ is at most $\epsilon$.
• This mechanism yields an “anytime” guarantee: one can halt sampling as soon as certification is achieved, balancing computational cost and reliability.

In LLMs, this approach certifies self-consistency: the majority vote over rollouts matches the modal output with high probability (Cordero-Encinar et al., 20 Oct 2025).

4. Connection to Self-Consistency, Test-Time Training, and Post-Training Objectives

MMC analyzes and unifies the statistical justification for two widely used LLM reliability strategies:

• Self-consistency sampling: Aggregates answers from independent chain-of-thought trajectories; MMC certifies the mode.
• Test-Time Reinforcement Learning (TTRL): Adaptively modifies the terminal distribution via KL-regularized objectives to sharpen the mode.

Label-free post-training methods exponentially tilt the answer distribution, increasing the margin $\delta$ and SNR, thus reducing sample requirements for certification. Proposed test-time objectives directly optimize SNR:

• SNR-based reward: $$r^1_n(\mathcal{Y}) = \frac{(N_{\text{leader}} - N_{\text{runner-up}})^2}{n(N_{\text{leader}} + N_{\text{runner-up}}) - (N_{\text{leader}} - N_{\text{runner-up}})^2}$$, converging to SNR($\Delta$).
• Entropy-based reward: $$r^2_n(\mathcal{Y}) = \sum_j \frac{N_j}{n} \log \frac{N_j}{n}$$ (negative Shannon entropy).

These objectives are optimized under a KL-regularization constraint, balancing sharpness and bias.

5. Extension to Martingale Mixtures and General Sequential Protocols

MMC generalizes to ensemble decision-making and sequential protocols where aggregated behavior is modeled via martingale mixtures. Consider a hypothesis space $H$ with discrete-time martingales $\{M_t(h)\}$ and mixing distribution $\rho$ on $H$, the mixture process being $\mathbb{E}_\rho[M_t]$. PAC-Bayes bounds provide uniform error guarantees over all mixtures: $$\mathbb{E}_{\rho}[M_t] \leq \sqrt{6(e-2)\mathbb{E}_\rho[V_t]\left(2\ln \ln \frac{3(e-2)\mathbb{E}_\rho[V_t]}{\mathbb{E}_\rho[M_t]} + \ln(2/\delta) + \text{KL}(\rho\|\pi)\right)}.$$ Here $V_t$ is the cumulative conditional variance, and the iterated logarithm (ln ln) factor ensures optimal scale-adaptation, especially in low-variance regimes (Balsubramani, 2015). This approach enables majority certificate guarantees in distributed or ensemble protocols, including cryptographic coin-tossing, where security bounds can be sharply quantified in terms of martingale gap functions (susceptibility curves) and stopping times (Khorasgani et al., 2019).

6. Quantitative Supermartingale Certificates and Control Synthesis

The supermartingale certificate methodology generalizes MMC to quantitative model checking and control. Given a shift-invariant specification $L$, one synthesizes a stochastic invariant $I$ and value function $V_0$ such that:

• $PV_0(s) \leq V_0(s)$ for $s \in I$
• $V_0(s) \geq 1$ for $s \notin I$

The probability of remaining in $I$ forever, $P(\Phi \in I^\omega)$, lower-bounds the satisfaction probability of $L$. In finite-state systems, the bound is tight. The decomposition strategy supports both quantitative verification and control, applying certificate search algorithms (using polynomial templates and SMT/Positivstellensatz tools) to synthesize invariants and supermartingales (Abate et al., 7 Apr 2025). This framework can be viewed as providing a “majority certificate” for the satisfaction of sequential (often temporal) specifications in both stochastic verification and control synthesis.

7. Significance, Applications, and Generalizations

The Martingale Majority Certificate framework has broad implications:

• In LLM inference, MMC specifies when majority voting is statistically reliable and quantifies sample sufficiency.
• In distributed randomness generation and cryptographic protocols, MMC-style bounds quantify adversarial susceptibility and optimize protocol fairness/security.
• In verification and control, supermartingale certificates ensure satisfaction probabilities for complex temporal specifications.
• The statistical unification of self-consistency and test-time adaptation connects heuristic aggregation strategies to formal martingale theory and PAC-Bayes analysis, enabling the design of certificate-based adaptive algorithms for reliability.

A plausible implication is that MMC methodologies will deepen the connection between sequential statistical guarantees, algorithmic adaptivity, and ensemble decision reliability across AI, cryptography, and formal verification domains.

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Table: Core Components of MMC in Different Domains

Domain	Certification Target	Mechanism
LLMs	Modal correctness of majority vote	Sequential e-process, SNR/entropy optimization
Distributed protocols	Security / fairness of outcomes	Martingale mixture bounds, gap analysis
Verification/Control	Satisfaction probability bound	Supermartingale certificate, stochastic invariants

The Martingale Majority Certificate serves as a rigorous, adaptive mechanism for certifying group decisions within the statistical structure provided by martingale theory, PAC-Bayes bounds, and quantitative supermartingale analysis.