Verifiability is the backbone of most ledger systems to realize credible authentication. However, existing permissioned blockchains and centralized ledger databases lack rigorous verifiability to authenticate all facts (i.e., what-when-who validation). Besides, they suffer from high verification cost to a continually growing immutable storage. In this paper, we introduce verification principles behind LedgerDB, a centralized ledger database that achieves both strong external auditability and fast verification. We coin a novel concept called Dasein Verification that composes of three validation factors what-when-who to formalize ledger auditing. Regarding what, LedgerDB devises fam (fractal accumulating model) to accelerate existence verification, and CM-Tree for efficient lineage verification. Veri-fiable data mutations are also supported. For when, we discuss attacks on existing time pegging protocols that compromise the authenticity of timestamps, and propose a time notary protocol to resolve those threats. Evaluations show that fam and CM- Tree significantly outperform traditional approaches. Compared to Hyperledger Fabric, LedgerDB achieves 23x higher verification throughput with 500 x lower latency in notarization applications, and 3 x higher throughput with 300 x lower latency in lineage tracking applications. As a public-cloud ledger service, the end-to-end verification latencies of LedgerDB are on average 50 x and 1000x lower than that of QLDB in the above applications, respectively.