Euler pentagonal recurrence to 50
Theorem.Euler pentagonal recurrence to 50 — p(n) = Σ_{k≥1} (−1)^{k+1} [ p(n − k(3k−1)/2) + p(n − k(3k+1)/2) ].
Proof.p(n) by generalized pentagonal numbers equals the raw partition DP for all n ≤ 50, p(50) = 204226 — the η-identity in exact integers.
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
src/thunder/verify/index.ts#discoveredTheoremsWaveTwelve
1 · Classification
bounded-witness — 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
"Euler pentagonal recurrence to 50" 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 discoveredTheoremsWaveTwelve.
- 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 (discoveredTheoremsWaveTwelve @ src/thunder/verify) 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 discoveredTheoremsWaveTwelve (src/thunder/verify/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.