Remark. Isotropy pieces of the background parameter set [fdip-005P]

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For a fixed incidence presentation \(\Pi \), let \(K_\Pi \) be its finite group of signed endpoint automorphisms, acting on the parameter set \(\mathcal B_\Pi \). The isotropy group at \(b\) is \((K_\Pi )_b=\{g:g b=b\}\). Partition \(\mathcal B_\Pi \) by the conjugacy class in \(K_\Pi \) of this stabilizer. These are its isotropy pieces. If \(b'=g b\), then \((K_\Pi )_{b'}=g(K_\Pi )_b g^{-1}\), so an orbit stays inside one piece. Between different presentations, graph arrows still identify the automorphism groups, without placing all of them inside one common acting group.

Write \(b_{a,b}\) for the alternating four-cycle with positive weights \((a,b,a,b)\), fixed \(m>0\), and retained pair \(\{0,2\}\). The signed stabilizer calculation gives \[\operatorname {Aut}_{\mathsf {Bg}^{\pm }}(b_{a,b}) \cong \begin {cases} C_2&a\ne b,\\ C_2\times C_2&a=b. \end {cases}\] This is a change of isotropy at equal weights. In the directed groupoid the uniformly oriented unit cycle admits only rotations; preserving \(\{0,2\}\) leaves the identity and half-turn, so its stabilizer is \(C_2\). Reflections belong to the larger signed convention.

The orthogonal stabilizers of the same pair \((Q,P_R)\) are \((C_2)^2\) and \((C_2)^3\), as proved in Proposition [fdip-005I]. They describe a different comparison and do not define the graph isotropy pieces above. Here strata can mean only this set-level partition by isotropy type. No smooth stratification, analytic singularity or physical phase transition follows from it. Those further interpretations require additional structure and arguments.