**Kaprekar Routine**
T(n) = sort_desc(n) - sort_asc(n)
Domain: {1000,...,9999} repdigits (|D| = 8991)
**Theorem 1**: dr(T(n)) = dr(n) mod 9 ✓
sort_desc(n) ≡ sort_asc(n) ≡ Σd_i (mod 9)
**Theorem 2**: T = ⊕_{r=1}^9 T_r
dim(T_r) ≈ 1111×1111
**Theorem 3**: ρ(T_r) ≤ 1/7 = 0.142857 (r∈{1,3,6,9})
**Theorem 4**: RSU Operator
A = T_r + T_r^T
H = diag(deg_i^1.2 × |v₂(i)|^0.4)
 = H^{-1}A → λ₂ = 1 - μ₂ = 6/7 = 0.857143 ✓
**Courant-Fischer**: |μ₂^eig - μ₂^RQ| < 10^{-14}