t3_probe_corewise_transpose
===========================

.. py:function:: t3toolbox.backend.probing.t3_probe_corewise_transpose(ztildes, ww, core_pair, sum_over_probes = False)

   .. code-block:: python

      def t3_probe_corewise_transpose(
              ztildes:    typ.Sequence[NDArray],  # probe residuals, len=d, elm_shape=W+C+(Ni,)
              ww:         typ.Sequence[NDArray],  # probe vectors,   len=d, elm_shape=W+(Ni,)
              core_pair:  typ.Tuple[
                  typ.Sequence[NDArray],          # tucker_cores, len=d, elm_shape=C+(ni,Ni)
                  typ.Sequence[NDArray],          # tt_cores,     len=d, elm_shape=C+(ri,ni,r(i+1))
              ],
              sum_over_probes: bool = False,      # True: sum the probe stack W (the gradient J^T r)
      ) -> typ.Tuple[
          typ.Tuple[NDArray, ...],  # tucker-core gradients, same shapes as tucker_cores
          typ.Tuple[NDArray, ...],  # tt-core gradients,     same shapes as tt_cores
      ]:


   Corewise (non-manifold) transpose of :py:func:`t3_probe`: gradient of the probes w.r.t. the
   cores of the frame ``core_pair``, treated as independent variables.

   The probe analog of :py:func:`t3_apply_corewise_transpose` -- the Section 6.3 substitution
   ``P, Q, O -> G_i`` (``U`` non-orthogonal) into :py:func:`tv_probe_transpose`, i.e. that
   transpose at frame ``(U, G, G, G)``. Returns gradients shaped exactly like ``(tucker_cores,
   tt_cores)`` (a gradient, not a tensor; no ``|W|`` blow-up). For non-manifold optimizers (Adam,
   L-BFGS) fitting from probes. ``sum_over_probes=True`` sums the probe stack ``W`` (the gradient
   ``J^T r``); ``False`` keeps it. Math reference: Section 6.3, Alger et al. (2026) (arXiv:2603.21141).