Remark. What the corner character can detect [frap-0011]
Remark. What the corner character can detect [frap-0011]
For a \(d\)-simplex or \(d\)-cube with \(d\geq 3\), the local corner groups have rotation subgroup \(A_d\). Their contribution to the total-orientation character passes through \((A_d)_{\mathrm {ab}}\). The abelianization calculation therefore measures exactly what this detector can see, as in [marchetti2025rubik, Section 4.1, p. 11, and equation (19), p. 14].
In dimension three the target is \(C_3\), the familiar corner twist sum. In dimension four the local group has grown to the nonabelian \(A_4\), but the character still sees only a \(C_3\) quotient: every local change in the Klein four group disappears from it. In dimensions five and above the local group is perfect, so this corner character is identically trivial on the ambient framed transports.
A lost detector does not classify the legal moves. Other piece types, permutation signs and the subgroup actually generated by facet turns may still impose constraints. Even when the target is nontrivial, conservation requires a generator check; the local abelianization alone does not supply one.