MC-E47-CANONICAL-47FACE-CRYSTAL-20260930-001 · E0/E1
E47 Canonical
47-Face Dual Crystal
The 47-dimensional E47 kernel is resolved by canonical coupled angular-momentum labels |J,j12,m>. Each state is assigned one deterministic spherical channel; polar dualization then produces one facet per channel.
47canonical channels / facets
25 + 22J=2 and J=5 states
90 / 135 / 47V / E / F topology
21/21Python checks PASS
25 J=2 / C=6 channels · 22 J=5 / C=30 channels
One facet ↔ one exact coupled state.
Canonical basis
E47 = span{|J,j12,m>}
J=2: j12=0,1,2,3,4 and m=-2,…,2 → 25 states
J=5: j12=3,4 and m=-5,…,5 → 22 states
25 + 22 = 47Geometry
|J,j12,m> → n_a ∈ S²
C47 = {x ∈ R³ : n_a · x ≤ 1, a=1,…,47}
facet F_a ↔ |J_a,j12_a,m_a>
V=90, E=135, F=47
χ = V−E+F = 2Validation
| Quantity | Result |
|---|---|
| Basis orthonormality residual | 9.024×10⁻16 |
| C eigenlabel residual | 1.526×10⁻14 |
| J12² eigenlabel residual | 1.184×10⁻14 |
| K annihilation residual | 2.036×10⁻13 |
| Facet-plane residual | 3.331×10⁻16 |
| Face sizes | 3×4, 14×5, 23×6, 6×7, 1×8 |
Certificate PASS · 21/21 checks.
Evidence boundary
The coupled basis is canonical after fixing the coupling tree, Condon–Shortley Clebsch–Gordan phases, and ordering. The 3D spherical embedding is an explicit deterministic label-derived canonicalization rule; it is not claimed to be the unique Euclidean embedding forced by SU(2). No hardware or material-crystal claim is made.