Inverse is not reverse
Theorem.Inverse is not reverse — the precise correction: in the twelve-tone group INVERSION reflects PITCH (I(x)=−x mod 12, the value axis, order kept) while RETROGRADE reverses TIME (the order axis, pitches kept)..
Proof.the precise correction: in the twelve-tone group INVERSION reflects PITCH (I(x)=−x mod 12, the value axis, order kept) while RETROGRADE reverses TIME (the order axis, pitches kept). They are different operations on orthogonal axes — I(row) ≠ R(row) — each an involution, and because pitch and time are independent they COMMUTE (I∘R = R∘I). Conflating inverse with reverse collapses a two-axis structure to one and loses half of it, the same error as ignoring inversion in the necklace count.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#inverseIsNotReverse
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
"Inverse is not reverse" 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 inverseIsNotReverse.
- 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 (inverseIsNotReverse @ 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 inverseIsNotReverse (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.