Catalan parity = Mersenne
Theorem.Catalan parity = Mersenne — {n ≤ 32 : C_n odd} = {0,1,3,7,15,31} = {2^k − 1}, exact BigInt on the sealed convolution.
Proof.{n ≤ 32 : C_n odd} = {0,1,3,7,15,31} = {2^k − 1}, exact BigInt on the sealed convolution — bounded witness; the all-n carry argument is Kummer, cited.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/thunder/waves/index.ts#emergenceContinuesWave
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
"Catalan parity = Mersenne" 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 emergenceContinuesWave.
- 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 (emergenceContinuesWave @ 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 emergenceContinuesWave (src/thunder/waves/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.