tv_to_t3
========

.. py:function:: t3toolbox.backend.tv_operations.tv_to_t3(frame, variations, include_shift = False)

   .. code-block:: python

      def tv_to_t3(
              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
              ],
              include_shift:  bool = False,  # False: tangent vector v. True: base point + v.
      ) -> typ.Tuple[
          typ.Tuple[NDArray, ...],  # tucker_cores (doubled Tucker ranks)
          typ.Tuple[NDArray, ...],  # tt_cores     (doubled TT ranks)
      ]:


   Doubled-rank Tucker tensor train representing a frame-variations tangent vector.

   The Tucker cores become ``[U_i; V_i]`` (stacked along the Tucker-rank axis); the TT cores form
   the standard block-bidiagonal embedding. With ``include_shift=True`` the base point is folded
   into the last TT core so the result represents ``base point + v``. Stack-aware.

   Equations (50)-(53) and Figure 20, Appendix A.3.1, of Alger et al. (2026),
   "Tucker Tensor Train Taylor Series" (arXiv:2603.21141).