The 48 row forms are the group order
Theorem.The 48 row forms are the group order — |rows| = 12 × 4 = 48 — transposition × {P, I, R, RI}, the serial group orbit.
Proof.a theorem replacing the axiom every twelve-tone textbook lists as a fact: transposition+inversion generate the dihedral D₁₂ (order 24) on the pitch axis, retrograde is a commuting C₂ on the orthogonal time axis, so the row-operation group is D₁₂ × C₂ of order 48 — the 48 forms are the group ORDER, derived from the generators, not memorised. It factors on the day's waves: 48 = 12 × 4 = |C₁₂| (the circle-of-fifths rosetta) × |V₄| (the four P/I/R/RI forms). Caveat (orbit-stabiliser): a symmetric row has fewer distinct forms.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#theFortyEightRowFormsAreTheGroupOrder
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 48 row forms are the group order" 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 theFortyEightRowFormsAreTheGroupOrder.
- 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 (theFortyEightRowFormsAreTheGroupOrder @ 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 theFortyEightRowFormsAreTheGroupOrder (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.