Let
M = B0 ⊕ R ⊕ B3
for the pair node
R = Z_3[eta]/(eta^2 - 3 eta).
Any endomorphism of M commuting with the vertex idempotents must be diagonal:
(lambda0, phi, lambda3)
with lambda0 ∈ Z_3, lambda3 ∈ Z_3, and phi ∈ End_R(R)=R.
Write phi as multiplication by (a,b) ∈ R ⊂ Z_3 × Z_3. Commuting with the branch maps i0,q0,i3,q3 forces
lambda0 = a,
lambda3 = b.
So the center is exactly
Z(End_R(B0 ⊕ R ⊕ B3)) ≅ R.
For the full singular Hecke order
O_sing ≅ Z_3(V7) × R,
the same argument gives
Z(End_{O_sing}(Z_3(V7) ⊕ B0 ⊕ R ⊕ B3)) ≅ O_sing.
So the Hecke node is not just represented by the CM generator algebra: it is recovered exactly as its center.
Brute-force enumeration in the 8-dimensional residue Auslander algebra shows that the center is 2-dimensional over F_3, with basis
1 = e0 + eR + e3,
t.
Since t^2 = 0, this residue center is
F_3[t]/(t^2),
which is exactly the dual-number shadow of the pair node.
This is the closest finite shadow so far: the residue quiver algebra remembers the full node through its center, and the center reduces to the same dual numbers that appeared in the syndrome algebra.