5. Selection Rules and Enumeration

5.1. Basic Predicates

valid_spin_state(two_j, two_m) tests whether m is an allowed projection of j. triangle_condition(two_j1, two_j2, two_j3) tests the SU(2) triangle inequalities and the integral-sum condition. Both functions are constexpr and noexcept.

static_assert(wigner::valid_spin_state(3, -1));
static_assert(wigner::triangle_condition(1, 1, 2));

5.2. Allowed Projections and Couplings

allowed_projections(two_j) returns 2m values in ascending order from -two_j to two_j. Invalid negative angular momentum produces an empty vector.

allowed_couplings(two_j1, two_j2) returns every admissible 2J in ascending order. The three specialized helpers also enforce coupling to a specified total angular momentum:

• allowed_couplings_12 enumerates intermediate j12;

• allowed_couplings_13 enumerates intermediate j13; and

• allowed_couplings_23 enumerates intermediate j23.

For example:

for (int two_j12 : wigner::allowed_couplings_12(
         two_j1, two_j2, two_j3, two_J))
{
    // Construct the ((j1 j2)j12, j3)J basis state.
}

This is the preferred way to construct a coupling basis. It avoids evaluating large rectangular collections that are known to vanish.

5.3. Canonical Symbol Arguments

The allowed_canonical_*_arguments functions enumerate one representative from each implemented symmetry orbit up to a maximum doubled angular momentum. They are useful for regression scans, compact reference tables, and future cache-generation experiments.

const auto representatives =
    wigner::allowed_canonical_6j_arguments(max_two_j);

The 3j canonicalization uses column permutations and simultaneous magnetic projection reversal. The 6j canonicalization uses tetrahedral column symmetries. The 9j canonicalization uses row permutations, column permutations, and transposition. Current 12j canonicalization uses cyclic ring rotations and interchange of the j and l rows.

5.4. Scaling Warning

Canonical enumeration is intentionally exhaustive and grows rapidly with max_two_j. The 9j and especially 12j helpers are designed for small controlled scans, not for constructing a production channel basis at large angular momentum.

In analysis code, construct the physically allowed coupling tree first with the triangle helpers, then evaluate only the coefficients required by that tree. This preserves the physical organization of the basis and scales much better than scanning all symbol arguments.

5.5. Ordering Guarantees

Projection and coupling vectors are returned in ascending numerical order. Canonical argument vectors use deterministic lexicographic representatives. These guarantees make tables and regression files reproducible, but they are not a persistent serialization format. If a future application stores these objects, it should write the named quantum numbers rather than relying on a vector index.