Exact finite results from Passes 4587–4591, built on the protected O+(8,2) quotient reconstructed in Pass 4579.
The W33-derived plus-type eight-space has 135 nonzero singular points and 270 maximal totally singular 4-spaces. The latter split into two families of 135, giving the three outer-node types of finite D4 triality.
Each cross-incidence matrix M satisfies MMᵀ=15I+3A, where A=SRG(135,70,37,35). After centering by D=9M−J, the cyclic multiplication law is D12 D23=54 D13.
The two opposite line-pairs of every W33 apartment produce two singular quotient vectors. Together with the apartment's singular fiber label they form one totally singular projective line. The resulting map is 1620 → 270, exactly six apartments per selected line.
The selected lines form a regular 135₆–270₃ incidence geometry. There are 810 point–line flags, and every flag has exactly two apartment lifts.
Join two singular points when they lie on a selected line. The graph has 135 vertices, degree 12, and 810 edges. Its exact spectrum is 12¹, 6¹⁵, 3²⁰, 0⁶⁰, (−3)²⁴, (−6)¹⁵.
Every adjacent pair has exactly one common neighbor. Therefore every edge lies in a unique triangle, and the graph has exactly 270 triangles. Those triangles are precisely the 270 apartment-selected singular lines.
The point–selected-line incidence matrix has rational rank 120 and binary rank 119.
A selected singular line has six W33 apartment lifts and lies in six maximal singular 4-spaces. The stabilizer has order 96. On the apartment lifts it acts transitively; on the maximal generators the orbit sizes are 3+2+1. Both actions have image order 12 and the same kernel of order 8, yet their permutation characters differ. Hence there is no stabilizer-equivariant bijection between the two six-sets.
The positive replacement is global: one 135-family stays transitive under PSp(4,3), while the other decomposes as 27+36+36+36. Each degree-36 orbit is genuinely the W33 spread G-set: its representative stabilizer has order 720 and fixes exactly one of the 36 spreads.
The rational rowspace of the 135×270 selected-line incidence matrix has dimension 120, and the protected quotient separately has 120 anisotropic points. The actual PSp(4,3) characters rule out an identification. All five deterministic transvection generators have the same trace 3 on both modules, but the short order-6 word g4*g0 has character −1 on the incidence rowspace and 3 on the anisotropic permutation module. The rational modules are therefore non-isomorphic.
Executable witnesses: analysis/w33_pass4587_w33_derived_d4_triality.py, analysis/w33_pass4588_apartment_triality_obstruction_spread_bridge.py, analysis/w33_pass4589_apartment_selected_singular_graph.py, and analysis/w33_pass4591_rank120_anisotropic_module_no_go.py.