The real challenge of multi-domain collaboration is not whether cross-chain is possible, but who pays for trust and how to audit it. Lagrange upgrades 'I trust him' to 'I have verified': data does not leave the domain, only verifiable conclusions are transmitted; proofs can be recursively aggregated, allowing one proof to be used multiple times, and the source/target domains can verify the same proof to avoid redundant calculations; proof settings include expiration and revocation/re-signing to prevent old rules from being abused under new regulations.
Implementation method: establish a 'proof budget table' (target volume, maximum verification latency, cache hit rate, and reuse window), front-load expensive computations to the edge, and only perform light verification on-chain; output a 'sandwich evidence chain': human-readable summary (source, strategy, audit number) + machine-readable credentials (parameters, state commitments, cost breakdown) + mathematical proof (recursive aggregation). Incident handling can be reviewed by aligning the summary and credentials, without everyone needing to read the circuit.
When trust can be orchestrated and priced like bandwidth and storage, the marginal costs of cross-domain collaboration will continue to decline. Lagrange's position is not a 'replacement for bridges', but rather a trust middleware for multi-domain collaboration—less exposure, more collaboration, strong reproducibility.