power-sum closed forms
Theorem.power-sum closed forms — Σk = n(n+1)/2, Σ(2k−1) = n², Σk² = n(n+1)(2n+1)/6 each exact.
Proof.Σk = n(n+1)/2, Σ(2k−1) = n², Σk² = n(n+1)(2n+1)/6 each exact for all n ≤ 1000 — the triangular, square and pyramidal formulas.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveForty
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
"power-sum closed forms" 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 discoveredTheoremsWaveForty.
- 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 (discoveredTheoremsWaveForty @ 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 discoveredTheoremsWaveForty (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.