Lemma. Orientation checked cycle by cycle [frap-000U]
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    Let \(g=(\pi ,a)\) be a framed transport on finitely many positions \(I\), with a faithful local group \(K\) on nonempty \(\Omega \) and bijective local frames. On a cycle

    \[i_0\longmapsto i_1\longmapsto \cdots \longmapsto i_{r-1} \longmapsto i_0\]

    of \(\pi \), the return transport in the first fiber has framed map \(a_{i_{r-1}}\cdots a_{i_1}a_{i_0}\). Its class in \(K_{\mathrm {ab}}\) is that cycle's contribution to \(\chi (g)=\prod _{i\in I}[a_i]\). Consequently, identity return transport on every position cycle implies \(\chi (g)=1\). This is the cycle criterion.

    Proof. 

      Apply the coordinate action \(r\) times. Each local map is applied after the preceding one, giving the stated product with the first map on the right. In the abelianization its factors may be reordered; multiplying over disjoint cycles uses each factor exactly once. A fixed position is a cycle of length one and must also be checked.

      This is a sufficient test derived from the canonical character of [marchetti2025rubik, Section 3.1, p. 7]. A generator's finite order alone does not make its return map on each shorter position cycle the identity; the local transport on that cycle is the quantity to check.