vortex doubling orbit is ⟨2⟩ in (ℤ/9ℤ)*
Theorem.vortex doubling orbit is ⟨2⟩ in (ℤ/9ℤ)* — 1→2→4→8→7→5 is the cyclic subgroup generated by 2 mod 9, order 6 = φ(9).
Proof.1→2→4→8→7→5 is the cyclic subgroup generated by 2 mod 9, order 6 = φ(9) — 2 is a primitive root of 9 = 3² exactly as wave 56 classifies, and dr(a·b)=dr(dr(a)·dr(b)) for a,b ≤ 200: the I Ching vortex is the unit group in motion.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveSixtyOne
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
"vortex doubling orbit is ⟨2⟩ in (ℤ/9ℤ)*" 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 discoveredTheoremsWaveSixtyOne.
- 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 (discoveredTheoremsWaveSixtyOne @ 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 discoveredTheoremsWaveSixtyOne (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.