T3Frame.has_minimal_ranks
=========================

.. py:method:: t3toolbox.frame_variations_format.T3Frame.has_minimal_ranks()

   .. code-block:: python

      def has_minimal_ranks(self) -> bool:


   True if the frame has minimal ranks.

   Minimal-rank frame means:
       - ``left_ranks == right_ranks``,
       - ``up_ranks == down_ranks``, and
       - those ranks (tucker_ranks=up_ranks, tt_ranks=left_ranks) are minimal for a regular
         Tucker tensor train of this shape (see :py:meth:`TuckerTensorTrain.get_minimal_ranks`).

   This is the **structural** minimal-rank check (cheap integer arithmetic on the ranks); for the
   **numerical** one (no stored rank numerically redundant) see :py:meth:`has_numerically_minimal_ranks`.
   Empirically (``docs/numerical_contracts.md``) minimal rank is **not** a correctness
   precondition for any verified operation -- ``inner``/``norm``-as-HS and ``manifold_dim`` are exact
   on a non-minimal orthonormal frame, and ``retract`` only loses *strict* rank preservation (it
   drops the redundant rank, staying a valid retraction). So this is a non-enforcing checker;
   ``T3Frame`` does not require minimal ranks at construction.

   Minimal (non-degenerate) ranks and their connection to matricizations and matrix unfoldings
   are discussed in Appendix A.2 of Alger et al. (2026), "Tucker Tensor Train Taylor Series"
   (arXiv:2603.21141).

   .. rubric:: Examples

   >>> import numpy as np
   >>> import t3toolbox.tucker_tensor_train as t3
   >>> import t3toolbox.frame_variations_format as bvf
   >>> np.random.seed(0)
   >>> x = t3.TuckerTensorTrain.randn((6, 7, 5), (2, 2, 2), (1, 2, 2, 1))  # minimal ranks
   >>> frame, _ = bvf.t3_orthogonal_representations(x)
   >>> print(frame.has_minimal_ranks)
   True
   >>> x2 = t3.TuckerTensorTrain.randn((14, 15, 16), (4, 5, 6), (1, 3, 2, 1))  # Tucker rank 4 > 1*3
   >>> frame2, _ = bvf.t3_orthogonal_representations(x2)
   >>> print(frame2.has_minimal_ranks)
   False