Joint Lyapunov Certificates for K-Agent Generative AI Governance: Stochastic Stability, Emergent Ensemble Risk, and Zero-Knowledge Governance Attestation

The paper develops a rigorous mathematical framework for the governance of systems of K self-adapting generative AI models under Model Risk Management (MRM) principles. The central thesis is that when multiple models share a meta-learning coupling γ through an interaction matrix A, the per-agent Lyapunov analysis that underpins standard MRM is provably insufficient: individual agents can each satisfy their declared stability bounds while the joint system is in a regime of emergent ensemble-level drift. The paper formalizes this gap through the Joint Lyapunov Proof (JLP)—a cryptographic and stochastic protocol that attests, without revealing proprietary weights, that the aggregate dynamics Wt = Wt1,..., WtK satisfy MRM Ongoing Monitoring standard at every validation epoch τn.

The paper's main contributions are fourfold. First, it gives a complete characterization of the infinitesimal generator LV of the joint quadratic Lyapunov function V(W) = 1/2 ∥W∥2 under the coupled Itô SDE and proves that the naive self-decay bound αeff = 2αself fails as soon as γ > 0. Second, it derives the exact critical coupling threshold γ*(A) = αself / λmin(A), where λmin(A) is the most negative eigenvalue (real part) of A, above which the system loses mean-square stability. The destabilizing mode is governed by λmin(A)—not the Perron root λmax(A), which is the consensus mode, is the most stable direction, and equals 1 for every row-stochastic topology (so it cannot encode topology-dependent risk). The quadratic Lyapunov function yields a safe sufficient certificate γ < αself / λmin(Asym), Asym = 1/2(A + A⊤), which coincides with γ* for normal topologies and is strictly conservative for non-normal ones (the star), so it never certifies an unstable system. Third, it proves a Noise-Floor Theorem establishing that the stationary second moment satisfies E[V∞] = β/(2αself) with β = 1/2 Kdσ2 exactly in the decoupled case. Fourth, it identifies the correct target for zero-knowledge attestation: a per-epoch Succinct Non-Interactive Argument of Knowledge (SNARK) on the live weights proving the pointwise generator inequality LV(Wt) < −αV(Wt) + β is vacuous, because by the stability theorem the inequality holds at every W once the coefficients are in the stable regime, and it is in fact satisfied with greater slack as ∥Wt∥ → ∞, so a blown-up system passes most comfortably. The property that actually governs stability—γ < γ*(A)—is a function of the declared (A, αself, γ) and needs no weights; the paper therefore constructs a SNARK πJLP that proves the static spectral certificate αself I + γAsym ≻ 0 on a committed coupling matrix via a Cholesky factorization, established once per configuration rather than at every epoch.

The paper is organized as follows. Section 2 defines the multi-agent SDE and its standing assumptions. Section 3 develops the joint Lyapunov theory, culminating in the Stability Theorem and the Noise-Floor Theorem. Section 4 analyzes the stability threshold and proves the critical coupling formula. Section 6 constructs the ZK-SNARK. Section 7 gives a detailed treatment of the MRM mapping. Section 8 presents five numerical studies. Section 9 discusses implications and Section 10 concludes.

The dynamics are governed by the coupled system of Itô SDEs: dWtk = µk(Wt) dt + σ(Wtk) dBtk, k = 1,..., K, where the drift and diffusion are µk(W) = −∇Lk(Wk) + γ Σj Akj(Φ − Wj) and σ(Wk) = σ0 Id. Under Assumption 2.1 (self-decay), each agent's loss is αself-strongly convex and isotropic about the origin, with Lk(w) = 1/2 αself ∥w∥2, so the drift simplifies to µk(W) = −αself Wk + γ Σj Akj(Φ − Wj). Assumption 2.2 (admissible adjacency matrix) requires A ∈ RK×K with Akj ≥ 0, Akk = 0, and row-stochasticity (Σj Akj = 1 for all k). Assumption 2.3 bounds the meta-parameter: ∥Φ∥ ≤ Φmax < ∞.

The paper considers three canonical topologies: complete graph (Akj = 1/(K−1) for all j ≠ k, symmetric, λmin(A) = −1/(K−1) = −0.25 at K=5), ring graph (Ak,k±1 = 1/2, symmetric, λmin(A) = cos(4π/5) = −0.809 at K=5), and star graph (hub with A0j = 1/(K−1), spokes with Ak0 = 1, non-normal, λmin(A) = −1, λmin(Asym) = −1.25 at K=5). Ordered by the destabilizing eigenvalue λmin(A)—complete 0.25 < ring 0.809 < star 1.0 at K=5—they have decreasing exact critical coupling γ* = αself / λmin(A) (complete 0.40 > ring 0.124 > star 0.10), so the complete graph tolerates the most coupling and the star the least.

Theorem 3.3 (Generator of the joint Lyapunov function) gives the exact formula: LV(W) = −2αself V(W) + γ Σk,j Akj ⟨Wk, Φ − Wj⟩ + 1/2 Kdσ02. Lemma 3.4 (Coupling term bound) shows γ Σk,j Akj ⟨Wk, Φ − Wj⟩ ≤ γΦmax √K ∥W∥ − γ λmin(Asym) ∥W∥2. Corollary 3.6 (Generator upper bound) gives LV(W) ≤ −(2αself + 2γλmin(Asym) − ε)V(W) + Cε, where Cε = 1/2 Kdσ02 + γ2Φ2maxK/(4ε).

Theorem 3.7 (Joint Lyapunov stability — sufficient certificate) states that if γ 0 and β > 0 such that LV(W) ≤ −αeff V(W) + β for all W, with αeff = 2(αself − γλmin(Asym)) − ε0 > 0 and β = 1/2 Kdσ02 + γ2Φ2maxK/(4ε0). The process has a unique stationary distribution π, and Eπ[V(W)] ≤ β/αeff.

Theorem 3.8 (Noise floor) states that for γ = 0 and Φ = 0, the SDE decomposes into Kd independent scalar Ornstein–Uhlenbeck processes, with unique stationary distribution π = ⊗k=1K N(0, σ02/(2αself)Id), and Eπ[V(W)] = β0/(2αself) where β0:= 1/2 Kdσ02.

Theorem 4.2 (Critical coupling threshold) is the central result. With γ*(A):= αself / λmin(A), the family of processes parameterized by γ ≥ 0 satisfies: (i) Stable regime: if γ < γ*(A), the joint drift operator M = αself I + γ(A ⊗ Id) is positive-stable, Wt has a unique (Gaussian) stationary distribution, and E[V(Wt)] → Eπ[V(W)] exponentially fast. (ii) Critical point: if γ = γ*(A), the mode along the λmin(A) eigenvector is marginally stable and E[V(Wt)] grows linearly in t. (iii) Unstable regime: if γ > γ*(A), the mean along v = ζ ⊗ u diverges, ∥E[Wt]∥ ≥ c e(γλmin(A)−αself)t, and E[V(Wt)] ≥ 1/2 ∥E[Wt]∥2 → ∞.

Remark 4.1 explains why the spectral radius cannot govern topology-dependent risk: by Perron–Frobenius, every row-stochastic non-negative A has λmax(A) = 1, so a threshold of the form γ* = c αself/λmax(A) is c αself for every admissible topology—a topology-independent number. The resolution is that the Perron root is the most stabilized direction, not the destabilizing one; the genuine instability is controlled by λmin(A), the most negative eigenvalue of A.

Remark 4.5 distinguishes the exact threshold γ*(A) = αself / λmin(A) from the Lyapunov certificate γcert(A) = αself / λmin(Asym). Since λmin(Asym) ≥ λmin(A) always, γcert(A) ≤ γ*(A), with equality iff A is normal. For the symmetric complete and ring graphs the two coincide (0.40 and 0.124). For the non-normal star they separate: γ* = αself/1 = 0.10 but γcert = αself/1.25 = 0.08. The gap is the price of a quadratic certificate on non-normal coupling: V cannot certify the genuinely-stable band γ ∈ (0.08, 0.10) for the star. Crucially the certificate errs on the safe side—it never certifies an unstable system—so it remains the right object to attest.

Theorem 5.1 (Emergent ensemble risk) formalizes the central cautionary result. Consider the decoupled system (γ = 0, Φ = 0) with K agents and a hidden drift h ∈ Rd added to every agent's dynamics, so the true drift of agent k is µk(Wk) = −αself Wk + h. Define the honest noise floor V∞hon = Kdσ02/(4αself). If the drift magnitude satisfies K/(2αself2) ∥h∥2 = δ · V∞hon with 0 < δ < θind/(V∞hon/K) − 1, then: (i) No individual agent is flagged: for every k, Eπ[Vk] (1 + δ/2) · V∞hon. The proof shows that with hidden drift h, the stationary distribution of each agent is N(hi/αself, σ02/(2αself)), so Eπ[Vk] = dσ02/(4αself) + ∥h∥2/(2αself2). The individual threshold is θind = 2V∞hon/K, and the condition ensures ∥h∥2 < 2αself2 V∞hon/K, so no agent is individually flagged. The joint stationary value is Vjoint = K·Eπ[Vk] = K∥h∥2/(2αself2) + V∞hon = (1+δ)V∞hon, which exceeds the joint threshold (1+δ/2)V∞hon.

Proposition 6.1 (The per-epoch weight proof is vacuous) shows that proving the pointwise generator inequality LV(Wt) < −αV(Wt) + β at the live weights is the wrong target. (i) The inequality holds at every W ∈ RdK, so a proof that it holds at the particular live state Wt conveys no information beyond what the publicly-declared coefficients already imply. (ii) The check is satisfied with ever-greater slack the larger the weights grow: LV(W) + αV(W) − β → −∞ as ∥W∥ → ∞, so a system whose weights have blown up passes the stability test most comfortably of all.

The correct attestation target is the static spectral certificate. The paper constructs πJLP = ZK-SNARK (A, γ): αself IK + γAsym ≻ 0 ∧ Hash(A, γ, Key) = H, where Asym = 1/2(A + A⊤). The positive-definiteness αself I + γAsym ≻ 0 is equivalent to γ < γcert(A), the safe (conservative) certificate; it is the natural object to prove because it is decidable by a Cholesky factorization, which a circuit can verify without computing eigenvalues. The circuit verifies S = LL⊤ entrywise (O(K3) multiplications) and that each diagonal entry Lii is nonzero, replacing an eigenvalue computation by verification of an algebraic identity. The proof is established once per configuration rather than at every epoch.

Proposition 6.2 (Soundness and completeness of πJLP) states that a ZK-SNARK can be instantiated using the Groth16 construction with: (i) Completeness: if γ < γcert(A) the certifying Cholesky factor exists and the prover can always generate a valid proof πJLP. (ii) Soundness: with overwhelming probability, no polynomial-time prover can produce a valid πJLP when αself I + γAsym ̸≻ 0. (iii) Zero-knowledge: the proof reveals no information about the committed A beyond the membership statement.

The paper maps the JLP onto the three MRM pillars. For Conceptual Soundness, the firm must justify the choice of V as a proxy for ensemble-level model risk, demonstrate that A accurately reflects the actual model interaction topology, calibrate αself, σ0, γ, and Φ to observed weight dynamics, and compute γ*(A) with a documented safety margin. For Ongoing Monitoring, the JLP provides continuous attestation with daily cryptographic proofs stored on the governance ledger, early warning via the sequence V(Wτn), granular decomposition of the total risk into self-decay, coupling, and noise components, and noise floor calibration against the closed-form baseline. For Model Validation, the MV function can independently verify the topology declaration, check the historical sequence of proofs, and use the governance injection probe as a challenge procedure.

The paper formalizes a new MRM risk category—the coupling gap—defined as ∆coup:= γ̂ · λmin(Â) − αself. When ∆coup > 0 the operational coupling has crossed the exact threshold γ*(Â) and the joint second moment diverges; individual MRM is then completely inadequate. A more conservative trigger fires when γ̂ · λmin(Âsym) > αself, which is the quantity the spectral SNARK attests.

The numerical studies validate all theoretical claims. Study C1 confirms that αeff(γ) is strictly decreasing in γ (from 0.264 at γ=0 to −0.049 at γ=0.50), the residual LV + 2αself V grows rapidly with γ (from 0.018 to 4.671), and βeff remains at exactly βnaive = 0.02500 for all stable γ. Study C2 confirms the critical coupling by topology: the empirical critical coupling tracks γ* = αself / λmin(A) to within Monte Carlo error for all three topologies (complete 0.404 vs 0.400, ring 0.120 vs 0.124, star 0.100 vs 0.100). Study C3 verifies the noise floor formula to within Monte Carlo error across eight configurations, with relative errors from 0.17% to 5.33% and all discrepancies within two standard errors. Study C4 shows the Φt injection probe response: at γ=0.20, the probe is detected in just 13 steps; at γ=0.50, detection takes 183 steps due to higher baseline variance; at γ=0.05, detection requires >2,000 steps. Study C5 demonstrates emergent ensemble risk with K=10 agents: a hidden drift of hper dim = 0.00935 leaves all 10 individual agents below their thresholds (mean Vk = 0.04139 vs threshold 0.04977), while the joint V = 0.41393 exceeds the joint threshold 0.37327 by 11%.

The paper's limitations include the linear dynamics and homogeneity across agents (Assumption 2.1 takes each agent's loss to be exactly the isotropic quadratic Lk(w) = 1/2 αself ∥w∥2 with the same αself for every agent), non-quadratic Lyapunov functions, non-stationary meta-parameters, and adversarial meta-parameters. The paper reads its contribution as a tractable idealization that isolates one mechanism—how coupling topology governs the stability of an interacting ensemble—and makes it exactly computable, not as a description of the dynamics of production generative models.

Improvements to AI Systems Based on This Paper:

1. Ensemble-Level Stability Monitoring: AI systems managing multiple interacting generative models can now detect emergent drift that individual-model monitoring misses. The improved system computes the joint Lyapunov function V(W) = ½‖W‖2 across all agents and flags instability when the coupling parameter γ approaches the critical threshold γ*(A) = α self/λ min(A), rather than relying solely on per-agent stability bounds.

2. Topology-Aware Risk Assessment: The system can now quantify how its interaction topology (complete, ring, star, or arbitrary graph) affects stability. It uses λ min(A) (the most negative eigenvalue of the adjacency matrix) instead of the spectral radius, correctly identifying that star topologies are most fragile (γ* = 0.10) while complete graphs are most robust (γ* = 0.40) for K=5 agents.

3. Zero-Knowledge Governance Attestation: The system can generate cryptographic proofs (ZK-SNARKs) that certify its coupling matrix A satisfies the stability condition α self·I + γ·A sym ≻ 0 without revealing proprietary model weights. This enables regulatory compliance (e.g., Model Risk Management) while preserving intellectual property, with proofs established once per configuration rather than per epoch.

4. Early Warning via Noise-Floor Calibration: The system can compare its observed stationary second moment E[V∞] against the closed-form baseline β0/(2α self) = Kdσ02/(4α self). Deviations beyond Monte Carlo error (validated at 0.17%–5.33% relative error) trigger early warnings of hidden drift or coupling anomalies before individual agents exceed their thresholds.

5. Hidden Drift Detection: The system can detect a common hidden drift h applied to all agents that evades individual monitoring. By computing the joint metric V joint = Σk E[Vk], it flags when V joint exceeds (1+δ/2)·V∞ hon even when every individual agent's Vk remains below its threshold θ ind, as demonstrated with K=10 agents (joint V = 0.41393 vs threshold 0.37327, while all individuals stay below 0.04977).

6. Avoidance of Vacuous Stability Proofs: The system now targets the correct attestation object—the static spectral certificate γ < γ cert(A)—rather than proving the pointwise generator inequality at live weights, which is vacuous and actually passes with greater slack as weights blow up. This prevents false confidence in unstable systems.

7. Conservative Safety Margins: For non-normal topologies (e.g., star graphs), the system uses the conservative certificate γ cert(A) = α self/λ min(A sym), which is strictly smaller than the exact threshold γ*(A). This ensures it never certifies an unstable system, even though it may under-certify stable bands (e.g., γ ∈ (0.08, 0.10) for the star).

8. Probe-Based Stress Testing: The system can inject a controlled perturbation Φ t into the coupling dynamics and measure detection latency as a function of γ. At γ=0.20, detection occurs in 13 steps; at γ=0.50, it takes 183 steps; at γ=0.05, over 2,000 steps. This enables real-time assessment of monitoring sensitivity and early-warning capability.

9. Granular Risk Decomposition: The system can decompose total ensemble risk into self-decay (2α self·V), coupling (γ·ΣA kj⟨W k, Φ−W j⟩), and noise (½Kdσ02) components, enabling precise attribution of instability sources and targeted mitigation.

10. Coupling Gap Trigger: The system can compute the coupling gap Δ coup = γ̂·λ min(Â) − α self. When Δ coup > 0, it automatically escalates to emergency governance protocols, recognizing that individual model risk management is completely inadequate in this regime.

We develop a rigorous mathematical framework for the governance of systems of K self-adapting generative AI models under the principles of Model Risk Management (MRM). When multiple models share a meta-learning coupling through an interaction matrix, the per-agent Lyapunov analysis that underpins standard MRM is provably insufficient: individual agents can each satisfy their declared stability bounds while the joint system is in a regime of emergent ensemble-level drift. We formalize this gap through the Joint Lyapunov Proof (JLP)---a cryptographic and stochastic protocol that attests, without revealing proprietary weights, that the aggregate dynamics satisfy MRM Ongoing Monitoring standard at every validation epoch. Our main contributions are the following. We give a complete characterization of the infinitesimal generator of the joint quadratic Lyapunov function. We derive the exact critical coupling threshold above which the system loses mean-square stability. We prove a Noise-Floor Theorem and identify the correct target for zero-knowledge attestation. A per-epoch Succinct Non-Interactive Argument of Knowledge (SNARK) on the live weights is derived. All theoretical claims are validated against five numerical studies using a multi-agent softmax system.