partition recurrence from the pentagonal theorem
Theorem.partition recurrence from the pentagonal theorem — p(n) = Σ_k (−1)^{k−1}(p(n−g_k)+p(n−g_−k)).
Proof.p(n) = Σ_k (−1)^{k−1}(p(n−g_k)+p(n−g_−k)) — because ∏(1−x^k) inverts Σp(n)x^n, giving an O(n√n) recurrence matching the brute partition DP for every n ≤ 60 (p(60)=966467).
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftyThree
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
"partition recurrence from the pentagonal theorem" 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 discoveredTheoremsWaveFiftyThree.
- 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 (discoveredTheoremsWaveFiftyThree @ 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 discoveredTheoremsWaveFiftyThree (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.