What this proves — every hash runs on YOUR device (pure SHA-256). It checks the header chain descends from the ONE pinned genesis, meets its work, and that a second independent node reaches the same tip; then it binds a coin to a membership proof (and proves a fake coin absent) at the current tip. So a single node cannot feed you a forged chain or a swapped coin.
What it does NOT prove yet (honest) — it does not re-derive the full difficulty-retarget schedule, so it trusts that the honest chain has the most work rather than proving it; it trusts the ≥1 of 2 nodes shown (not zero-trust); amounts assume < 2^53 and addresses ASCII; and it runs over plain HTTP (a domain + HTTPS is the next step). This is a real light verifier — not yet a full independent SPV client.