Reducing the exact local presentation modulo 3 gives
A = O_sing / 3 O_sing ≅ F_3 × F_3[eps]/(eps^2).In the residue basis
[ e7bar , epairbar , nbar ],the regular module splits exactly as
A_reg ≅ chi_v7 ⊕ P_pair,where:
chi_v7 = <e7bar> is the split semisimple branch;P_pair = <epairbar, nbar> is the unique non-split self-extension of chi_pair.So the residue node is not a single length-3 indecomposable. Its only non-semisimplicity is internal to the glued pair block.
On the basis [epairbar, nbar], the pair block matrices are
E = [[0,0],[0,0]],
N = [[0,0],[2,0]],
I = [[1,0],[0,1]].So N kills the top and maps the top line onto the socle; this is exactly the dual-number Jordan block.
The Ext table is
Ext^1(chi_pair, chi_pair) ≅ F_3,
Ext^1(chi_v7, chi_pair) = 0,
Ext^1(chi_pair, chi_v7) = 0,
Ext^1(chi_v7, chi_v7) = 0.So the split V7 branch is completely disconnected from the extension theory. All residue-level deformation lives inside the V1/V5 pair.