E0 · SU(2) carrier
Selector, kernel, projector.
Three spin-2 qudits. The total Casimir is diagonal in the Clebsch–Gordan basis. K = (C−6I)(C−30I) kills exactly two sectors. Their dimensions are 25 and 22. That is the whole origin of 47.
Sector ledger
| J | C = J(J+1) | m_J | dim | κ | K² | P₄₇(C) | addr |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 180 | 32,400 | 0 | 0–0reject |
| 1 | 2 | 3 | 9 | 112 | 12,544 | 0 | 1–9reject |
| 2 | 6 | 5 | 25 | 0 | 0 | 1 | 10–34E47 |
| 3 | 12 | 4 | 28 | -108 | 11,664 | 0 | 35–62reject |
| 4 | 20 | 3 | 27 | -140 | 19,600 | 0 | 63–89reject |
| 5 | 30 | 2 | 22 | 0 | 0 | 1 | 90–111E47 |
| 6 | 42 | 1 | 13 | 432 | 186,624 | 0 | 112–124reject |
Σ dim = 125 = 125. ker K = 47.
Exact projector
P₄₇(C) = C(C−2I)(C−12I)(C−20I)(C−31I)(C−42I) / 1,814,400
The C−31I factor is intentional. On this finite spectrum the interpolant is 1 at C = 6 and C = 30 and 0 at every other Casimir eigenvalue. Defining tests: P² = P, P† = P, KP = 0, Tr P = 47.
- C = 0 → P = 0
- C = 2 → P = 0
- C = 6 → P = 1
- C = 12 → P = 0
- C = 20 → P = 0
- C = 30 → P = 1
- C = 42 → P = 0
Coupled-basis addresses
Canonical order: J ascending, then j₁₂, then M descending. Kernel occupies two contiguous blocks. Invalid 7-qubit codewords 125–127 sit outside the carrier.
J=0
J=1
J=2
J=3
J=4
J=5
J=6
inv
010–34 E₆90–111 E₃₀125–127
E₆ (J=2, M₂⊗R₂) · E₃₀ (J=5, M₅⊗R₅) · complement 78 · invalid 3