tv_retract
==========

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

   .. code-block:: python

      def tv_retract(
              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, ...],  # tucker_cores (retracted T3, base-point ranks)
          typ.Tuple[NDArray, ...],  # tt_cores
      ]:


   Retract a frame-variations tangent vector onto the fixed-rank manifold.

   Forms the shifted doubled-rank embedding (base point + v) via :py:func:`tv_to_t3`
   (``include_shift=True``) and truncates it back to the **base point's own ranks** -- the Tucker
   ``up`` ranks and ``left`` TT ranks read off the frame cores -- with the implicit T3-SVD, yielding
   a point on the manifold of the base point's ranks.

   The truncation is the implicit T3-SVD (Algorithm 10) of Alger et al. (2026), "Tucker Tensor
   Train Taylor Series" (arXiv:2603.21141).