tv_orthogonal_gauge_projection
==============================

.. py:function:: t3toolbox.backend.tv_operations.tv_orthogonal_gauge_projection(frame, variations)

   .. code-block:: python

      def tv_orthogonal_gauge_projection(
              frame:      typ.Tuple[
                  typ.Sequence[NDArray],  # up_tucker_cores
                  typ.Sequence[NDArray],  # down_tt_cores
                  typ.Sequence[NDArray],  # left_tt_cores
                  typ.Sequence[NDArray],  # right_tt_cores
              ],
              variations: typ.Tuple[
                  typ.Sequence[NDArray],  # tucker_variations
                  typ.Sequence[NDArray],  # tt_variations
              ],
      ) -> typ.Tuple[
          typ.Tuple[NDArray, ...],  # gauged_tucker_variations
          typ.Tuple[NDArray, ...],  # gauged_tt_variations
      ]:


   Project the variations onto the gauge-satisfying subspace (orthogonal projection).

   Changes the represented tangent vector. The result satisfies, for an orthogonal frame,
   ``U_i V_i^T = 0`` (all i) and ``einsum('...abi,...abj->...ij', L_i, H_i) = 0`` (i = 0..d-2).
   Stack-aware. Ragged path only (uniform deferred).

   Gauge conditions (48)-(49), Appendix A.3, of Alger et al. (2026), "Tucker Tensor Train
   Taylor Series" (arXiv:2603.21141).