Sixty degrees decodes pi
Theorem.Sixty degrees decodes pi — the vortex step is τ/6 = π/3 with cos 60° = ½ exact;.
Proof.the vortex step is τ/6 = π/3 with cos 60° = ½ exact; three steps realize Euler as 2³ ≡ −1 (mod 9); Archimedes' hexagon-seeded perimeter doubling brackets π to 3.1410 < π < 3.1427 at the 96-gon; and ⟨x↦2x, x↦1−x⟩ closes to AGL(1,ℤ/9) of order 54 — the ring and the void generate everything (cross-verified with erpax the day it was found).
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#sixtyDegreesDecodesPi
Figure — the proof, computed and plotted
aₙ₊₁ = 2aₙbₙ/(aₙ+bₙ), bₙ₊₁ = √(aₙ₊₁·bₙ) (Archimedes, radius 1)
Computed by sixtyDegreesDecodesPi().rungs @ src/9/1 — recomputed on every build, zero tokens.
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
"Sixty degrees decodes pi" 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 sixtyDegreesDecodesPi.
- 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 (sixtyDegreesDecodesPi @ 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 sixtyDegreesDecodesPi (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.