multiplicative order divides φ(n)
Theorem.multiplicative order divides φ(n) — ord_n(a) | φ(n) — Lagrange in ⟨a⟩ ⊆ (ℤ/nℤ)*.
Proof.ord_n(a) | φ(n) — Lagrange in the unit group ⟨a⟩ ⊆ (ℤ/nℤ)*, verified for every coprime a, n ≤ 100 (the group-theoretic root of Euler’s theorem).
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftySix
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
"multiplicative order divides φ(n)" 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 discoveredTheoremsWaveFiftySix.
- 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 (discoveredTheoremsWaveFiftySix @ 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 discoveredTheoremsWaveFiftySix (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.