Remark. Spin-frame compatibility is separate from internal covariance [fdip-009S]

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The internal gauge theorem uses \(I_S\otimes \rho (g(x))\) on field values. It does not cover an arbitrary unitary transformation on \(S\). Already in one coordinate, the chosen gamma matrix is \[X=\begin {pmatrix}0&1\\1&0\end {pmatrix}.\] For \(Z=\operatorname {diag}(1,-1)\), one has \(ZXZ=-X\). With trivial links and fixed lattice directions, \[ZD_0Z=-D_0\ne D_0.\] The inequality holds for every permitted period \(N\ge 3\): the Fourier momentum \(2\pi /N\) has nonzero sine. Thus even a constant unitary change on the spin factor fails the claimed fixed-gamma conjugacy unless the Clifford/frame data are transformed compatibly.

A spin action instead comes with its specified vector action and Clifford compatibility. For a constant orthogonal frame change \(R\), the relevant relation has the form \[\sigma (s)c(v)\sigma (s)^{-1}=c(Rv).\] An action on lattice differences must respect the same change of directions; a varying frame needs additional transport data. No such spin-frame transformation is part of the internal gauge statement above. The compact real Spin cover uses the positive quadratic form, and its action is on the spatial vector carrier. It is not, by its name alone, an action on an arbitrary internal fiber.

In dimension three, the quaternion map \(SU(2)\to SO(3)\) has kernel \(\{1,-1\}\). Its binary finite subgroups supply useful representations, but their roles remain distinct. For example the faithful two-dimensional fiber \(F=\mathbb C^2\) of \(2T\) can be used internally in \(S\otimes F\); then it commutes with the gamma matrices on \(S\). The corresponding three-dimensional rotation representation kills the center and is different data. Neither choice identifies the internal factor with the spatial Clifford module or provides a lattice frame action.