Inversion negates the angle
Theorem.Inversion negates the angle — arg(1/z) = −arg(z) ∧ R(θ)⁻¹ = R(−θ) — one reflection law in four guises.
Proof.one reflection law in four guises, computed: arg(1/z)=−arg(z), R(θ)⁻¹=R(−θ), and on the vortex 6-cycle of (ℤ/9ℤ)* the inverse of 2^k sits at −k; geometric inversion v/|v|² keeps the coordinate (the gap is one conjugation) while intersection angles survive — 1/z is conformal.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#inverseNegatesAngle
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
"Inversion negates the angle" 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 inverseNegatesAngle.
- 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 (inverseNegatesAngle @ src/9/1) 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 inverseNegatesAngle (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.