SharedFrameData
===============

.. py:class:: t3toolbox.backend.sharing.SharedFrameData

   The per-frame companion of the shared geometry: everything the tied projection,
   the shared retraction, and the group-spectrum diagnostics need, derived from an
   orthogonal frame by :py:func:`fv_shared_frame_data` (never stored inside a frame).

   All array fields carry the frame stack ``C`` leading; ``groups`` / ``row_splits`` are
   static structure (jax aux). One entry per NONTRIVIAL group (>= 2 modes), in canonical
   order; ``svd_*`` is the thin SVD of the stacked matrix
   ``M_g = concat_i(S_i^T)`` -- deliberately an SVD, never a Cholesky/Gram: the solve gets
   the intrinsic least-squares sensitivity, ``svd_s`` IS the group spectrum ``s_g`` at full
   (non-squared) accuracy, and the clipped pseudoinverse is well-defined at the
   rank-deficient points rank continuation visits.

   **What ``s_g`` is** (representation-independent -- a property of the represented tensor
   ``T`` and the partition alone): the singular values of the concatenated matricization
   ``[T_(i1) | ... | T_(ik)]`` over the group's modes; equivalently
   ``s_g^2 = eig(sum_i Gamma_i)`` (the summed mode Grams), equivalently the singular values
   of the Jacobian of ``T`` with respect to a gauged tied motion of the shared factor -- the
   exact analog of what a per-mode Tucker spectrum is to an unshared factor. Note the scale:
   every mode carries the full norm, so ``sum_j s_gj^2 = k * ||T||^2`` (a group of ``k``
   modes inflates the spectrum by ``sqrt(k)``; the factor cancels in every condition-number
   ratio). Cf. Peshekhonov, Arzhantsev & Rakhuba (2024, SF-Tucker) and Molozhavenko &
   Rakhuba (2026, SF-ETT), whose algorithms compute this same object.


   .. py:attribute:: groups
      :type:  tuple


   .. py:attribute:: row_splits
      :type:  tuple


   .. py:attribute:: centers
      :type:  tuple


   .. py:attribute:: svd_U
      :type:  tuple


   .. py:attribute:: svd_s
      :type:  tuple


   .. py:attribute:: svd_Vt
      :type:  tuple