Towards Trustworthy Digital Twins: Verifiable Simulation via Recursive Zero-Knowledge Proofs

Industrial Digital Twins (DTs) require cryptographic verification for external audits without exposing proprietary models—a challenge for continuous-time simulations with thousands of ODE integration steps. Existing methods fail: monolithic SNARKs exceed memory limits beyond 3000 steps, while naive recursive approaches impose > 200× overhead. We present ZKP-DS, combining sensitivity-driven precision allocation (35% circuit reduction), hierarchical proof batching (26-39× speedup via parallelization), and probabilistic epoch sampling (78% verification reduction with > 99.99% fraud detection). Fixed-step Runge-Kutta integration in adaptive fixed-point arithmetic ensures both cryptographic soundness and IEEE-compliant numerical accuracy. Experiments on full-scale power systems (2000 states, 10,000 steps) achieve 5.6−8.5× computational overhead with sub-60ms constant-time verification on university GPU clusters. Results demonstrate that zero-knowledge proofs can provide mathematical integrity guarantees for industrial cyber-physical systems at practical costs, enabling trustworthy multi-stakeholder monitoring and regulatory compliance.