fv_base_point_tangent
=====================

.. py:function:: t3toolbox.backend.geometry.fv_base_point_tangent(frame)

   .. code-block:: python

      def fv_base_point_tangent(
              frame:  typ.Tuple,    # (U, O, P, Q) -- an orthogonal frame
      ) -> typ.Tuple:               # v_X: the attachment point X as a gauged tangent (tucker_var, tt_var)


   The attachment point ``X = (U, P)`` as a gauged tangent ``v_X`` -- **the DIRECT construction**: all
   variations zero except the last TT variation, set to the frame's last left core ``P_last``. This is
   already gauged (the last TT variation is the one slot with no gauge condition), so it needs neither an
   ambient projection nor a gauge projection: ``dense(v_X) = X`` and ``‖v_X‖_coord = ‖X‖_HS`` exactly.
   (It equals ``tv_project_t3_onto_tangent_space(frame, (U, P))`` -- verified -- but avoids that roundabout
   computation, whose environment contractions all collapse since the projected T3 IS the frame's own
   cores. See ``docs/contributor/fitting_internals.md`` §Regularization.)