The four forms shift without collision
Theorem.The four forms shift without collision — because inverse (pitch) and reverse (time) are distinct COMMUTING involutions, the four row forms {P, I, R, RI} are the abelian Klein four-group V₄. Abelian ⇒ PATH-INDEPENDENT ⇒ no collision to resolve, no search;.
Proof.because inverse (pitch) and reverse (time) are distinct COMMUTING involutions, the four row forms {P, I, R, RI} are the abelian Klein four-group V₄. Abelian ⇒ PATH-INDEPENDENT ⇒ no collision to resolve, no search; every form its own inverse ⇒ reversible with zero bookkeeping; each a pure O(n) one-pass bijection, history-free ⇒ realtime and restartable at any form (the twoRosettasAreRealtime property). Two orthogonal commuting axes buy collision-free realtime motion among the forms — the group structure guarantees it. Dimensions here = the two operation axes (pitch, time), not physical spacetime.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#theFourFormsShiftWithoutCollision
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
"The four forms shift without collision" 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 theFourFormsShiftWithoutCollision.
- 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 (theFourFormsShiftWithoutCollision @ src/music) 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 theFourFormsShiftWithoutCollision (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.