Σ_{d|n} φ(d) = n to 1000
Theorem.Σ_{d|n} φ(d) = n to 1000 — Σ_{d|n} φ(d) = n complete.
Proof.Σ_{d|n} φ(d) = n complete for every n ≤ 1000 — the cyclic group partitioned by element order; Gauss cited for all n.
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
src/thunder/waves/index.ts#discoveredTheoremsWaveFive
1 · Classification
bounded-witness — 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
"Σ_{d|n} φ(d) = n to 1000" 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 discoveredTheoremsWaveFive.
- 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 (discoveredTheoremsWaveFive @ 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 discoveredTheoremsWaveFive (src/thunder/waves/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.