Authentication in Web 3.0 faces a structural conflict. Systems that offer full anonymity leave no lawful way to identify a malicious actor. Systems built for accountability expose a persistent wallet address to blockchain-graph analysis, or fall back on centralized key recovery. Existing designs solve one side of this conflict at the cost of the other. This paper presents PEUAP-W3, a Privacy-Enhanced and User-centric Authentication Protocol. Its contribution is the integration of five established components into a single deployed and formally analyzed system. A Circom 2 circuit of 1579 Groth16 constraints proves four facts in a single 192-byte on-chain proof: knowledge of an opening of the session credential commitment, an SpO₂ value inside an 85–100% band, single-use nonce binding, and HMAC integrity. Shamir (k = 2, n = 3) sharing distributes the identity payload across three independent relays. The coordinator reconstructs an identity only after a threshold vote has been recorded on chain. Credentials are issued as W3C Verifiable Credentials 2.0 in did:key form. Four Solidity contracts verify the proof on Ethereum Sepolia. Verification costs about 241,000 gas and takes roughly 3 ms. ProVerif and Scyther find no attack under the Dolev–Yao model. A concurrency sweep to 500 simultaneous requests completes 1191 requests with zero failures at about 15.4 requests per second. A behavioral gate screens commodity abuse as a supplementary control; it is not treated as a security boundary. Against a nine-property framework, PEUAP-W3 satisfies six properties. Three remain conditional and are not verified in the current deployment: biological-origin assurance and digital replay prevention, both of which need an attested sensor; and GDPR erasure equivalence. Here, formally verified refers to the protocol models and theorems, not to the complete deployed software.