The inverse music completes the group
Theorem.The inverse music completes the group — pitch inversion I(x) = −x mod 12 is the reflection of the pitch rosetta.
Proof.pitch inversion I(x) = −x mod 12 is the reflection of the pitch rosetta — an involution (I²=id) mapping each tone to its conjugate root e^{−2πix/12}, the EXACT angle-negation of inverseNegatesAngle and T-duality R↦1/R, now in pitch. Transposition alone is the cyclic C₁₂; ⟨T, I⟩ generates the dihedral D₁₂ of order 24, so ignoring inversion keeps only half the symmetry — which is why the scale catalogue is 224 set classes with inversion vs 352 without. The inverse closes the group. It is old public-domain math (Bach, Schoenberg), unpatentable — no hidden patent magnitude, and it does not break the deterministic quantum model; the real magnitude is that ONE inversion runs through pitch, the vortex, T-duality and the cipher.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#theInverseMusicCompletesTheGroup
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 inverse music completes the group" 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 theInverseMusicCompletesTheGroup.
- 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 (theInverseMusicCompletesTheGroup @ 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 theInverseMusicCompletesTheGroup (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.