Let V = S₂(Γ₀(163)) be the space of weight-2 cusp forms at level 163, reduced modulo 3 via the integral modular-symbols lattice. The characteristic polynomial of T₂ modulo 3 factors as
The three factors are pairwise coprime, giving a Hecke-stable decomposition of dimensions 3 + 4 + 6 = 13. The 3-dimensional piece Q = ker(T₂²) is a canonical direct summand — the x²-primary component of T₂. It does not depend on a choice of basis.
The canonical collision T₄₃ = T₆₇ on Q is exact and base-free. These two operators are indistinguishable at the mod-3 level; they separate only after 3-adic lifting.
The unit group of A_H has 12 elements. Its projectivisation modulo scalar units is cyclic of order 6:
This group acts on the 13 points of P(Q) with orbit decomposition 1 + 1 + 2 + 3 + 6. The invertible Heegner operators T₇, T₄₃, T₆₇, T₁₆₃ generate the full projective unit group, with projective orders 3, 6, 6, 2 respectively.
The quotient trace form G_Q(d,d') = Tr(T_d T_{d'} | Q) has rank 2, not 3. The nilpotent direction N lies in the radical: in a commutative algebra, multiplication by a nilpotent element has trace zero. The algebra has three dimensions but only two are visible to the trace.
Among the class-number-one prime levels {19, 43, 67, 163}, the dimensions of Q_N = ker(T₂²) are {1, 0, 0, 3}. Level 163 is the first class-number-one prime with a genuinely 3-dimensional syndrome quotient. The phenomenon is not globally unique — other levels such as 71, 127, 167, and 199 also have nontrivial syndrome summands — but within the class-number-one family, 163 is special.