siteswap average is the ball count
Theorem.siteswap average is the ball count — juggleable ⇔ (i ↦ i + s_i mod n) ∈ Sym(ℤ/n); b = (Σ_{i<n} s_i)/n ∈ ℕ.
Proof.a vanilla siteswap is JUGGLEABLE exactly when i ↦ (i + s_i) mod n permutes ℤ/n — no two balls arrive on one beat — and for every juggleable pattern the ball count is the plain average of its digits, the integrality of that average being part of the claim rather than an assumption. Exhausted over 5088 juggleable patterns with period ≤ 5 and heights ≤ 9 (a height is written as a digit, so the bound is the largest digit read off VORTEX_SEQUENCE; the period bound is a compute budget, and that is stated refutably rather than in prose: one period PAST the window, at period 6 with heights bounded by the period instead of the digit, the same identity is exhausted again and fails nowhere): every average an integer, every average equal to an independent simulation counting balls in flight, and 4/4 colliding patterns rejected. The simulation was wrong twice before it agreed — once counting only throws spanning beat 0, once off by one because the ball thrown AT the sampled instant is in hand and still a ball — so the agreement is between two routes, one of which had to be repaired to reach it. PRIOR ART: Buhler, Eisenbud, Graham & Wright, Juggling Drops and Descents (1994); NOT Shannon's juggling theorem, which relates dwell, flight and vacant times to balls and hands and is a different result.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/fire/physics/index.ts#siteswapAverageProven
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
"siteswap average is the ball count" 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 siteswapAverageProven.
- 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 (siteswapAverageProven @ src/fire/physics) 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 siteswapAverageProven (src/fire/physics/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.