= Pure Neumann reflection transplantation preserves Euler characteristic
{title2=$\chi=V-E+F$}
Consider assemblies of congruent polygonal tiles with same-labelled faces paired by reflection and pure <Neumann boundary conditions> on unpaired faces. An invertible intertwiner for all face <permutation matrices> preserves the <Euler characteristic>. The number of tiles is the dimension of the representation. For each face label the number of edge classes is $(n+\operatorname{tr}M_s)/2$. For each vertex label the number of vertex classes is the number of orbits of the subgroup generated by incident-face involutions, hence the dimension of its invariant vectors. Intertwining preserves all these numbers. Summing the vertex and edge counts proves equal $V-E+F$, including boundary corners. This argument uses genuine permutation matrices; signed matrices encoding mixed <boundary conditions> do not give the same topological count.
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