ℤ[ω] is a Euclidean domain
Theorem.ℤ[ω] is a Euclidean domain — for z,w≠0 there is q with N(z−qw)<N(w).
Proof.for z,w≠0 there is q with N(z−qw)<N(w) — the hexagonal lattice gives N(r)≤N(w)/3, verified for every z and w≠0 on the grid; hence a UFD.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftyNine
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"ℤ[ω] is a Euclidean domain" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation discoveredTheoremsWaveFiftyNine.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (discoveredTheoremsWaveFiftyNine @ src/4/6) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes discoveredTheoremsWaveFiftyNine (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.