Pythagorean parametrization is a bijection
Theorem.Pythagorean parametrization is a bijection — (a,b,c) = (m²−n², 2mn, m²+n²) with gcd(m,n)=1, m>n≥1, m≢n (mod 2) — a bijection onto primitive triples.
Proof.every primitive triple with hypotenuse ≤ 200 arises exactly once from coprime opposite-parity (m,n) via (m²−n², 2mn, m²+n²) — the parametrised set equals the brute-forced set exactly; Euclid cited for all.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveTwentyFive
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
"Pythagorean parametrization is a bijection" 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 discoveredTheoremsWaveTwentyFive.
- 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 (discoveredTheoremsWaveTwentyFive @ 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 discoveredTheoremsWaveTwentyFive (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.