tv_oblique_gauge_projection
===========================

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

   .. code-block:: python

      def tv_oblique_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 while preserving the tangent vector.

   Generalizes Holtz, Rohwedder & Schneider (2012), "On manifolds of tensors of fixed TT-rank".
   The Tucker perturbation is made perpendicular to U (compensating through the down/outer cores),
   then the TT variations are made left-perpendicular (compensating through the right cores).
   Stack-aware. Ragged path only (uniform deferred).

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