t3_orthogonality_residual
=========================

.. py:function:: t3toolbox.backend.t3_orthogonalization.t3_orthogonality_residual(x, side)

   .. code-block:: python

      def t3_orthogonality_residual(
              x: typ.Tuple[typ.Sequence[NDArray], typ.Sequence[NDArray]],  # (tucker_cores, tt_cores)
              side: str,  # 'left' or 'right'
      ) -> NDArray:  # shape = stack_shape; max abs deviation from orthonormality per stack element (0 == exact)


   Non-enforcing check of T3 *left/right-orthogonal form*: Tucker cores down-orthogonal AND TT
   cores left- (``side='left'``) or right- (``side='right'``) orthogonal, **per stack element**.

   Returns the max-abs deviation from the identity over (each stacked block of), reduced over the
   **non-stack** axes (shape ``stack_shape``):

   - Tucker B_i, all i:                 ``einsum('...io,...jo->...ij', B, B) = I``
   - TT G_i, left  (i = 0..d-2):        ``einsum('...aib,...aic->...bc', G, G) = I``
   - TT G_i, right (i = 1..d-1):        ``einsum('...aib,...cib->...ac', G, G) = I``

   The boundary TT core (last for left, first for right) is the center remainder and is not checked.
   This is the form :py:func:`t3_left_orthogonalize` / :py:func:`t3_right_orthogonalize` produce and
   the form ``t3svd(..., assume_orthogonal=...)`` may assume.