ManifoldGeometry.retract
========================

.. py:method:: t3toolbox.manifold.ManifoldGeometry.retract(p)

   .. code-block:: python

      def retract(
              self,
              p:  T3Tangent,  # step (a tangent at the current point's frame)
      ) -> t3.TuckerTensorTrain:  # retracted point on M, at p's frame ranks


   Retract the step ``p`` to the manifold: shifted doubled-rank embedding, truncated to frame ranks.

   Forms ``base point + p`` and truncates back to ``p``'s frame ranks via the implicit T3-SVD
   (Algorithm 10). The current point is carried by ``p.frame`` (its orthonormal frame), so no
   separate point argument is needed.

   **Safe mode** requires ``p``'s frame to be **orthogonal** (raises otherwise); skipped under
   ``safety.unsafe()`` / a jax trace. ORTH only -- retract is gauge-invariant. Minimal rank is a
   documented *caveat*, not a precondition: on a non-minimal frame retract stays a valid
   first-order retraction but drops the numerically-redundant rank rather than preserving it
   strictly (``docs/numerical_contracts.md``).