utv_retract
===========

.. py:function:: t3toolbox.backend.utv_operations.utv_retract(frame_data, variations_data)

   .. code-block:: python

      def utv_retract(
              frame_data,       # UT3Frame .data:      supercore stack = C
              variations_data,  # UT3Variations .data: supercore stack = K + C
      ):  # -> retracted UniformTuckerTensorTrain .data (at the BASE point's ranks; stack = K + C)


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

   Forms the shifted doubled-rank embedding ``base point + v`` (:py:func:`utv_to_ut3` with
   ``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 masks -- via the mask-truncated uniform T3-SVD. The output
   is a UT3 at the frame padded dims (``ut3svd`` truncates by max rank to a fixed shape, so no extra slice
   is needed), one retracted point per stack element. The uniform mirror of
   :py:func:`tv_operations.tv_retract` (the implicit T3-SVD / Algorithm 10, Alger et al. 2026).

   **Varying ranks across ``C``** work for free: the per-``C`` frame ranks are the per-element truncation
   targets. **The ``K`` (tangent) stack:** the frame ranks have stack ``C`` while the shifted UT3 has stack
   ``K + C``, so the frame ranks are broadcast over ``K`` (the ``K`` tangents share the frame, hence the same
   truncation targets).