T3Tangent.corewise_inner
========================

.. py:method:: t3toolbox.manifold.T3Tangent.corewise_inner(other)

   .. code-block:: python

      def corewise_inner(self, other: 'T3Tangent'):


   The raw corewise (coordinate) dot of two tangents' variations -- **not** the HS inner product.

   Vectorized over the stack: returns an array of shape :py:attr:`stack_shape` (``K + C``), one dot
   per stacked tangent (a scalar when unstacked). The same-frame precondition is checked.

   This is the *coordinate* inner product; it equals Hilbert-Schmidt only on an orthonormal, gauged,
   (minimal-rank) frame. For that semantic -- with the orthogonal/gauge preconditions checked -- use
   :py:meth:`ManifoldGeometry.inner` (or :py:meth:`CorewiseGeometry.inner` for the Euclidean metric).
   The gauged identity ``<v, v'>_HS = sum_i <dU_i, dU_i'> + sum_i <dG_i, dG_i'>`` is Appendix A.3 of
   Alger et al. (2026), "Tucker Tensor Train Taylor Series" (arXiv:2603.21141).