Equivariant Face-Trace Evenness No-Go and the Relative Interior Q Counit
Symmetry forces even ordinary traces
The degree-two obstruction of entry269 cannot be repaired by changing the fillers of the two collapsed pure-sheet faces or by rescaling the six mixed face fillers.
Reflection pairs the pure faces with coefficients a and -a. Their coherently oriented top contribution is 2a. It pairs the three one-negative faces with the three two-negative faces with coefficients b and -b, contributing -6b. Every D3/reflection-equivariant ordinary face trace is therefore
2a - 6b,
which is even. The row [2,-6] has Smith factor 2 and cokernel Z/2; it never reaches the normalized primitive value one.
Thus normalization saturation, ramification, and the fundamental-sphere ambiguity do not supply the missing odd Q column while preserving literal support and symmetry.
Minimal symmetry-compatible enlargement
Entries266, 341--343 construct the finite literal images of the six cross-sheet edges and eight triangular faces of the conductor cross-polytope. The smallest remaining class is its reflection-odd relative interior class.
If an extraordinary proper/log-excess counit maps that class to literal Q with primitive coefficient +1, the top row becomes
[2, -6, 1].
Its Smith factor is one, and the relative interior generator alone reaches the normalized value one. This is the minimal symmetry-compatible repair.
The finite boundary maps do not construct the relative interior counit. The missing datum is a support-typed map of the cross-polytope relative fundamental class into the literal generic Q top, compatible with all six edge maps, all eight facet BC cells, and endpoint framing.
Evidence
- research/voevodsky/check_equivariant_face_trace_evenness_no_go.rs
- entries 143, 266, 268--269, 341--343.
{
"status": "falsified_scoped_equivariant_face_trace_oddness",
"ordinary_equivariant_trace": "2a-6b",
"ordinary_row": [2, -6],
"ordinary_smith_factors": [2],
"ordinary_cokernel": "Z/2",
"primitive_value_one_reachable": false,
"minimal_enlarged_row": [2, -6, 1],
"enlarged_smith_factors": [1],
"required_new_class": "reflection-odd relative interior Q counit",
"cross_polytope_edges_constructed_finitely": 6,
"cross_polytope_faces_constructed_finitely": 8,
"relative_interior_to_literal_Q_constructed": false,
"mapping_fiber_instantiated": false,
"physical_p_defined": false,
"physical_bockstein_defined": false,
"D8_Jordan_tested": false,
"graph_admission_claimed": false
}