σ is multiplicative
Theorem.σ is multiplicative — σ(mn)=σ(m)σ(n) for gcd(m,n)=1 with σ(p^k)=(p^{k+1}−1)/(p−1).
Proof.σ(mn)=σ(m)σ(n) for gcd(m,n)=1 with σ(p^k)=(p^{k+1}−1)/(p−1) — verified over every coprime pair m,n ≤ 100 and prime power p^k ≤ 10000, compounding on wave 54’s multiplicative framework.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftyFive
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
"σ is multiplicative" 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 discoveredTheoremsWaveFiftyFive.
- 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 (discoveredTheoremsWaveFiftyFive @ 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 discoveredTheoremsWaveFiftyFive (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.