Sylvester–Gallai needs order axioms
Theorem.Sylvester–Gallai needs order axioms — the Fano plane has 0 ordinary lines (every line carries 3 points) — Sylvester–Gallai (∃ an ordinary line) needs the order axioms of ℝ.
Proof.the Fano plane has ZERO ordinary lines — every point-pair rides a 3-point line, verified whole — while over ℝ the theorem forces one: the order of the real line is load-bearing; Gallai cited.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveEighteen
1 · Classification
finite-complete — computed witness within a cited frame (the unbounded form leans on the cited literature)
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"Sylvester–Gallai needs order axioms" 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 discoveredTheoremsWaveEighteen.
- 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 (discoveredTheoremsWaveEighteen @ src/9/1) 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 discoveredTheoremsWaveEighteen (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.