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 now re-derives the full difficulty-retarget schedule and pins a recent checkpoint (so a cheap forged chain is rejected), but it still trusts a pinned checkpoint set by us rather than proving most-work from scratch; amounts assume < 2^53 and addresses ASCII. This is a real light verifier — approaching, but not yet, a fully trustless SPV client.