utv_orthogonal_gauge_projection
===============================

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

   .. code-block:: python

      def utv_orthogonal_gauge_projection(
              frame_data,       # UT3Frame .data
              variations_data,  # UT3Variations .data
      ):  # -> gauged variations .data (same masks; the tangent VECTOR changes)


   Orthogonally project the variations onto the gauge-satisfying subspace (the uniform mirror of
   :py:func:`tv_operations.tv_orthogonal_gauge_projection`). Removes the component of each Tucker
   variation parallel to its up-core ``U`` and of each left-interior TT variation parallel to its
   left-core ``P`` -- the gauge conditions (48)-(49), Appendix A.3 of Alger et al. (2026). The
   represented tangent vector CHANGES (orthogonal, not oblique).

   Mask-once up front (the frame and variation padding zeroed), then mask-agnostic einsums vectorized over
   the leading mode axis ``d`` (a per-core *map*); the last TT variation is left unchanged (the ``[:-1]``
   boundary). The output carries the variation's masks unchanged (gauge preserves the rank structure).