Safety and Liveness of Cross-Domain State Preservation under Byzantine Faults: A Mechanized Proof in Isabelle/HOL

Abstract: Formally guaranteeing the safety and liveness of regulatory state transitions in cross-domain state synchronization systems is a problem of growing importance as tokenized assets are increasingly operated across heterogeneous blockchain networks and off-chain ledgers. This paper presents a mechanized proof of 2,348 lines in Isabelle/HOL establishing two complementary properties. First, cross-domain state preservation (safety): a regulatory state transition performed on one domain is faithfully reflected across all connected domains with structural preservation. This guarantee encompasses bidirectional roundtrip preservation, consistency across an arbitrary finite set of domains, and per-asset isolation. Second, liveness under Byzantine faults: in the presence of up to f < n/3 Byzantine nodes, we prove deterministic resolution of conflicting regulatory actions, deadlock freedom, and starvation freedom. In the combination of these two properties, the liveness proof discharges the honest-node assumption of the safety proof under Byzantine faults, promoting conditional safety to an unconditional guarantee. The seven generic locales derived in this process are domain-independent and reusable for arbitrary domains via Isabelle/HOL's interpretation mechanism. The application context is a regulatory state transition model based on the RCP framework (arXiv:2603.29278), which systematizes 31 requirements from 15 global financial regulatory authorities. All proof artifacts build in Isabelle/HOL without sorry or oops, have been submitted to the Archive of Formal Proofs (under review), and are publicly available on GitHub.

• The paper establishes a mechanized Isabelle/HOL proof ensuring deterministic state preservation and progress in regulatory transitions, even with Byzantine faults.
• It employs a modular locale hierarchy to capture state-machine structure and cross-domain synchronization, guaranteeing atomic integrity across distributed ledgers.
• The approach discharges the honest-node assumption by integrating fair leader rotation and strict lock timeouts, ensuring consistent and live asset state management.

Mechanized Verification of Cross-Domain State Preservation and Liveness under Byzantine Faults

Abstract and Motivations

The mechanized proof effort presented in "Safety and Liveness of Cross-Domain State Preservation under Byzantine Faults: A Mechanized Proof in Isabelle/HOL" (2604.03844) addresses a foundational challenge in cross-chain regulatory asset management: providing unconditional guarantees of both safety and liveness properties in the synchronization of regulatory state transitions across heterogeneous domains, even in the presence of Byzantine faults. The proof targets the context of tokenized assets subject to regulatory action where consistent and liveness-assured propagation of state transitions (e.g., asset freeze, confiscation) is paramount to prevent regulatory arbitrage and operational deadlocks. The framework is rooted in a regulatory state transition model derived from the Regulatory Compliance Protocol (RCP) (Kim et al., 31 Mar 2026), mapping 31 requirements from 15 financial authorities.

Formal System and Threat Model

The presented model abstracts a distributed system comprising a finite set of domains (blockchains, ledgers) with state machines per asset. State transitions are deterministic and subject to atomic, preemptive per-asset locking. The proof artifact is specified in Isabelle/HOL and parameterized to be domain-agnostic. Safety properties are initially proved under the honest participant assumption; liveness properties relax this to a Byzantine model permitting up to $f<n/3$ malicious actors, requiring $n \geq 3f+1$ nodes for resilience. Byzantine participants may withhold locks (denial-of-service) or manipulate leadership, but the model incorporates timeout-based forced releases and fair leader rotation to guarantee progress.

Regulatory State Transition Model

The model operationalizes a finite automaton with five regulatory states (ACTIVE, FROZEN, SEIZED, CONFISCATED, RESTRICTED) and seven regulatory actions (FREEZE, SEIZE, CONFISCATE, RESTRICT, UNFREEZE, UNRESTRICT, RELEASE). The transition function is deterministic and total; only 12 out of 35 possible transitions are admissible, reflecting concrete global regulatory constraints. The proof rigorously establishes three invariants: terminal absorptivity of CONFISCATED, universal reachability of CONFISCATE from any non-terminal state, and absence of state self-loops.

Mechanized Safety (Cross-Domain State Preservation)

Locale Abstraction

The proof engineer introduces a modular locale hierarchy capturing state-machine structure, preservation mappings, and cross-domain consistency in a reusable fashion. Each locale encapsulates a set of assumptions, extending via Isabelle/HOL’s {interpretation} mechanism. The core notions include:

• state_machine: formalizes deterministic partial transition systems with terminal states.
• state_preservation: encapsulates commutation of transitions under synchronization.
• symmetric_state_preservation: asserts information-preserving (injective) bidirectional synchronization.
• multi_domain_preservation: fortifies the reasoning to $N$-domain consistency with asset isolation.

Key Results

• Sequential Preservation: The transition mapping preserves finite action sequences, ensuring deterministic final states post-synchronization.
• Cross-Domain Consistency: After atomic synchronization, all chains holding an asset converge to the same regulatory state.
• Sync Isolation: Synchronization of a given asset does not affect others.

The regulatory domain instantiation proves that all protocol primitives (transition, lock, unlock) preserve the global consistency invariant and synthesize a protocol-agnostic design rationale.

Mechanized Liveness under Byzantine Faults

Locale Structure

Three domain-independent liveness locales formalize:

• priority_system: deterministic, injective mapping of regulatory requests to a total order for conflict resolution.
• deadlock_free_locking: strict lock timeout ensures no permanent resource locking.
• fair_leader_system: bounded leader schedule ensures all pending actions are eventually executed, preventing starvation via a fairness assumption over the scheduler (concrete in the BFT/VRF leader election context).

Combined Regulatory Instance

Regulatory-specific instantiations provide authority-based priorities (via a four-component key), and concretize the BFT system parameters reflecting real-world network deployments. Proofs of safety–liveness composition discharge the honest-node assumption, achieving the unconditional guarantee.

Starvation, Deadlock, and Determinism

• Determinism is enforced via the total order on regulatory requests.
• Deadlock is structurally prevented by bounded lock lifetimes, rendering attempts at lock withholding non-attack vectors.
• Guaranteed progress is achieved via fair scheduling bounds, with exponential decay in adversary success probability as the fairness window $k$ increases.

Combined Property: Assumption Discharge and Full Compositional Guarantee

The central technical contribution is an assume-guarantee structure: liveness proofs discharge the honest-node condition required for safety, thus extending conditional safety to unconditional safety plus liveness even in the Byzantine setting. The proof explicitly ensures that, starting from any valid global state, any legal regulatory action is guaranteed to succeed and preserve invariants if attempted when the asset is not locked, independent of adversarial behavior, provided threshold security parameters are met.

Comparison to Related Mechanized Verification

Compared to the Merkle Functor pattern of Lochbihler & Maric (FMBC 2020), which focuses on authenticated data structure integrity within a single domain and does not address multi-domain consistency or Byzantine settings, the present work subsumes prior approaches into a cross-domain, BFT-resilient synchronization paradigm. Prior formal verification efforts on BFT protocols (Velisarios, HotStuff) in Coq and Agda typically omit regulatory-specific properties and/or full liveness under Byzantine faults; their focus is on safety and generic consensus termination, never end-to-end regulatory operational semantics.

Implications, Practical Guidance, and Future Directions

This work delivers parameters directly relevant for protocol design in decentralized regulatory systems:

• Minimal node set: $n \geq 3f+1$
• Sync lock timeout: $T > 0$ (tuned for underlying network)
• Fairness bound: determined by VRF leader schedule security margin
• Priority function: injectivity over composite regulatory attributes

The compositional, domain-independent locales admit reapplication beyond regulatory compliance: orchestration of distributed database transactions, asset bridges, and more.

However, model limitations such as assumption of reliable communication, closed system operation, and non-modeled financial operations, delineate avenues for extension—e.g., formal refinement to implementation languages, dynamic domain reconfiguration, and adaptation to partial synchrony or permissionless open settings.

Conclusion

The mechanized proof herein establishes an Isabelle/HOL-checked foundation for cross-domain regulatory state integrity, liveness, and resilience to Byzantine threats. The developed locale artifacts are modular, reusable, and facilitate rigorous system design for next-generation regulated digital assets. All source artifacts are available on GitHub, with submission to the Archive of Formal Proofs.

The implications extend to both formal methods for blockchain protocols and the practical deployment of compliance-centric cross-chain infrastructures. Future research will focus on composing data integrity and state preservation proofs, bridging the gap towards end-to-end assurance across multi-layered, heterogeneous trust environments in the decentralized finance ecosystem.