triangular–square bridge 8T_k+1 = (2k+1)²
Theorem.triangular–square bridge 8T_k+1 = (2k+1)² — every triangular number T_k = k(k+1)/2 satisfies the exact identity.
Proof.every triangular number T_k = k(k+1)/2 satisfies the exact identity — the algebraic key turning a square into an odd square that carries sum-of-squares to the figurate numbers.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/thunder/waves/index.ts#discoveredTheoremsWaveFiftyTwo
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
"triangular–square bridge 8T_k+1 = (2k+1)²" 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 discoveredTheoremsWaveFiftyTwo.
- 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 (discoveredTheoremsWaveFiftyTwo @ src/thunder/waves) 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 discoveredTheoremsWaveFiftyTwo (src/thunder/waves/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.