diff --git a/doc/dev/bumping-sympy.md b/doc/dev/bumping-sympy.md
new file mode 100644
index 00000000..62849ec6
--- /dev/null
+++ b/doc/dev/bumping-sympy.md
@@ -0,0 +1,140 @@
+# Bumping the SymPy dependency
+
+This page documents the process for bumping the pinned SymPy version in CI
+and explains the practices that have evolved around it.
+
+## Version policy
+
+Target the most recent **minor series** that is not the latest release.
+For example, if 1.14 is the latest stable release, target 1.13.x (the
+highest patch, e.g. 1.13.3).  This gives us a version that is current
+enough to catch real compatibility issues but still widely used and stable.
+
+The same policy applies to Python version bumps.
+
+## Tracking issue pattern
+
+A SymPy (or Python) bump typically uncovers several independent blocking
+issues.  Use the **tracking issue** pattern to manage this:
+
+1. Open a *tracker* issue describing the overall goal (e.g. "Bump SymPy
+   to 1.13").  Keep it focused on the goal — do not list specific errors in
+   the tracker body.
+2. For each blocking problem found during testing, open a *sub-issue*
+   describing only that problem.
+3. Add a comment to the tracker referencing all sub-issues.
+4. Open a PR for each sub-issue independently.  Sub-PRs can be reviewed
+   and merged in any order.
+5. Once all sub-PRs are merged, open the final bump PR that updates
+   `test_requirements.txt` and closes the tracker.
+
+This keeps each PR small and reviewable, and makes it easy to see at a
+glance which problems are still blocking the bump.
+
+## Testing locally
+
+Use `uv` to create a clean venv with the target Python and SymPy versions:
+
+```bash
+uv venv --python 3.12 .venv312
+uv pip install -r test_requirements.txt -e .
+```
+
+Run the full CI test command:
+
+```bash
+pytest \
+    -vv --durations=50 \
+    --cov=galgebra \
+    --nbval examples/ipython/ \
+    --nbval examples/primer/ \
+    test \
+    --nbval-current-env \
+    --nbval-sanitize-with test/.nbval_sanitize.cfg \
+    -n 2 --dist loadscope
+```
+
+CI also accepts a `PYTEST_K_FILTER` environment variable to pass `-k`.
+See `.github/workflows/ci.yml` for the authoritative command.
+
+## Common compatibility issues
+
+### `distutils` removal (Python 3.12)
+
+`distutils` was removed in Python 3.12.  Any `import distutils.*` must be
+replaced with `packaging` (already a test dependency).
+
+### Unrecognized escape sequences (Python 3.12)
+
+Python 3.12 promotes `\m`, `\g`, `\i`, etc. in regular string literals from
+`DeprecationWarning` to `SyntaxWarning`.  nbval catches the resulting stderr
+output and fails the cell.  Fix: add the `r` prefix to affected string
+literals (`r'\mathbf{e}'`).
+
+### `ImmutableDenseMatrix` (SymPy 1.13)
+
+Some SymPy 1.13 operations return `ImmutableDenseMatrix` where 1.12 returned
+`MutableDenseMatrix`.  Any `isinstance(x, Matrix)` check that should cover
+both must be changed to `isinstance(x, MatrixBase)` (imported from
+`sympy.matrices`).
+
+### LaTeX printing changes (SymPy 1.13)
+
+SymPy 1.13 changed matrix LaTeX output in two ways:
+
+- `\begin{array}{}` → `\begin{array}{ccc}` (column specs added)
+- Explicit `\cdot` multiplication dot added/removed in some expressions
+
+These are cosmetic; the computed values are identical.  Fix: re-execute the
+affected notebooks (see *Notebook refresh workflow* below).
+
+### Unicode matrix art alignment (SymPy 1.13)
+
+Plain-text pretty-printing of matrices shifts by one space in some rows.
+Also cosmetic; handled by the notebook refresh.
+
+## Notebook refresh workflow
+
+When stored notebook outputs become stale due to a SymPy printing change:
+
+1. Re-execute the affected notebooks with the target SymPy version:
+
+   ```bash
+   pip install jupyter nbconvert
+   jupyter nbconvert --to notebook --execute --inplace \
+       examples/ipython/gr_metrics.ipynb
+   ```
+
+2. Validate that every output change is purely cosmetic using
+   `scripts/validate_nb_refresh.py`:
+
+   ```bash
+   # Compare the previous commit (HEAD^) against the current working copy:
+   python scripts/validate_nb_refresh.py --git HEAD^ \
+       examples/ipython/gr_metrics.ipynb
+   ```
+
+   The script normalizes all known cosmetic patterns (array column specs,
+   `\cdot`, whitespace, stream warnings) and fails with a detailed diff if
+   anything else changed.  Exit code 0 means all changes are cosmetic.
+
+3. Verify the refreshed notebooks pass nbval:
+
+   ```bash
+   pytest --nbval examples/ipython/gr_metrics.ipynb \
+       --nbval-current-env \
+       --nbval-sanitize-with test/.nbval_sanitize.cfg
+   ```
+
+4. Include the `validate_nb_refresh.py` output in the PR description as
+   evidence that no mathematical content changed.
+
+### Adding new cosmetic normalizers
+
+If a new SymPy version introduces a new cosmetic rendering difference, add a
+normalizer function to `scripts/validate_nb_refresh.py`:
+
+- For LaTeX diffs: add to `LATEX_NORMALIZERS` list.
+- For plain-text diffs: update `normalize_plaintext`.
+- Document the SymPy version and nature of the change in a comment next to
+  the normalizer function.
diff --git a/doc/dev/bumping-sympy.rst b/doc/dev/bumping-sympy.rst
new file mode 100644
index 00000000..a09321f1
--- /dev/null
+++ b/doc/dev/bumping-sympy.rst
@@ -0,0 +1 @@
+.. mdinclude:: bumping-sympy.md
diff --git a/doc/index.rst b/doc/index.rst
index 851f2197..6d693bd6 100644
--- a/doc/index.rst
+++ b/doc/index.rst
@@ -33,6 +33,7 @@
 
     dev/release-process
     dev/bumping-python
+    dev/bumping-sympy
 
 .. toctree::
     :caption: Other information
diff --git a/examples/ipython/LaTeX.ipynb b/examples/ipython/LaTeX.ipynb
index fab88b66..60805399 100644
--- a/examples/ipython/LaTeX.ipynb
+++ b/examples/ipython/LaTeX.ipynb
@@ -3,7 +3,14 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:40:50.309937Z",
+     "iopub.status.busy": "2026-04-02T05:40:50.309862Z",
+     "iopub.status.idle": "2026-04-02T05:40:50.313734Z",
+     "shell.execute_reply": "2026-04-02T05:40:50.313217Z"
+    }
+   },
    "outputs": [],
    "source": [
     "import os\n",
@@ -14,7 +21,14 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:40:50.315576Z",
+     "iopub.status.busy": "2026-04-02T05:40:50.315466Z",
+     "iopub.status.idle": "2026-04-02T05:40:50.895093Z",
+     "shell.execute_reply": "2026-04-02T05:40:50.894515Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -90,7 +104,14 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:40:50.912312Z",
+     "iopub.status.busy": "2026-04-02T05:40:50.912182Z",
+     "iopub.status.idle": "2026-04-02T05:49:04.977406Z",
+     "shell.execute_reply": "2026-04-02T05:49:04.975915Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -164,7 +185,7 @@
        "\\begin{equation*} A = A^{\\xi }  \\boldsymbol{e}_{\\xi } + A^{\\eta }  \\boldsymbol{e}_{\\eta } + A^{\\phi }  \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} B = B^{\\xi \\eta }  \\boldsymbol{e}_{\\xi }\\wedge \\boldsymbol{e}_{\\eta } + B^{\\xi \\phi }  \\boldsymbol{e}_{\\xi }\\wedge \\boldsymbol{e}_{\\phi } + B^{\\eta \\phi }  \\boldsymbol{e}_{\\eta }\\wedge \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla}  f = \\frac{\\partial_{\\xi } f }{\\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} \\left|{a}\\right|} \\boldsymbol{e}_{\\xi } + \\frac{\\partial_{\\eta } f }{\\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} \\left|{a}\\right|} \\boldsymbol{e}_{\\eta } + \\frac{\\partial_{\\phi } f }{a \\sin{\\left (\\eta  \\right )} \\sinh{\\left (\\xi  \\right )}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
-       "\\begin{equation*} \\boldsymbol{\\nabla} \\cdot A = \\frac{a \\left({\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}\\right)^{3} \\partial_{\\phi } A^{\\phi }  + \\frac{\\left(A^{\\eta }  \\sin{\\left (2 \\eta  \\right )} + A^{\\xi }  \\sinh{\\left (2 \\xi  \\right )}\\right) \\left({\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}\\right)^{\\frac{3}{2}} \\sin{\\left (\\eta  \\right )} \\sinh{\\left (\\xi  \\right )} \\left|{a}\\right|}{2} + \\left({\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}\\right)^{\\frac{5}{2}} \\left(\\partial_{\\eta } A^{\\eta }  + \\partial_{\\xi } A^{\\xi } \\right) \\sin{\\left (\\eta  \\right )} \\sinh{\\left (\\xi  \\right )} \\left|{a}\\right| + \\left({\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}\\right)^{\\frac{5}{2}} A^{\\eta }  \\cos{\\left (\\eta  \\right )} \\sinh{\\left (\\xi  \\right )} \\left|{a}\\right| + \\left({\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}\\right)^{\\frac{5}{2}} A^{\\xi }  \\sin{\\left (\\eta  \\right )} \\cosh{\\left (\\xi  \\right )} \\left|{a}\\right|}{a^{2} \\left({\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}\\right)^{3} \\sin{\\left (\\eta  \\right )} \\sinh{\\left (\\xi  \\right )}} \\end{equation*}\n",
+       "\\begin{equation*} \\boldsymbol{\\nabla} \\cdot A = \\frac{A^{\\eta }  \\cos{\\left (\\eta  \\right )}}{\\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} \\sin{\\left (\\eta  \\right )} \\left|{a}\\right|} + \\frac{A^{\\xi }  \\cosh{\\left (\\xi  \\right )}}{\\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} \\sinh{\\left (\\xi  \\right )} \\left|{a}\\right|} + \\frac{\\partial_{\\eta } A^{\\eta } }{\\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} \\left|{a}\\right|} + \\frac{\\partial_{\\xi } A^{\\xi } }{\\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} \\left|{a}\\right|} + \\frac{A^{\\eta }  \\sin{\\left (2 \\eta  \\right )}}{2 \\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} {\\sin{\\left (\\eta  \\right )}}^{2} \\left|{a}\\right| + 2 \\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} {\\sinh{\\left (\\xi  \\right )}}^{2} \\left|{a}\\right|} + \\frac{A^{\\xi }  \\sinh{\\left (2 \\xi  \\right )}}{2 \\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} {\\sin{\\left (\\eta  \\right )}}^{2} \\left|{a}\\right| + 2 \\sqrt{{\\sin{\\left (\\eta  \\right )}}^{2} + {\\sinh{\\left (\\xi  \\right )}}^{2}} {\\sinh{\\left (\\xi  \\right )}}^{2} \\left|{a}\\right|} + \\frac{\\partial_{\\phi } A^{\\phi } }{a \\sin{\\left (\\eta  \\right )} \\sinh{\\left (\\xi  \\right )}} \\end{equation*}\n",
        "\\end{document}\n"
       ]
      },
@@ -179,7 +200,14 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:04.981090Z",
+     "iopub.status.busy": "2026-04-02T05:49:04.980894Z",
+     "iopub.status.idle": "2026-04-02T05:49:05.425896Z",
+     "shell.execute_reply": "2026-04-02T05:49:05.425263Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -249,7 +277,14 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:05.428151Z",
+     "iopub.status.busy": "2026-04-02T05:49:05.428047Z",
+     "iopub.status.idle": "2026-04-02T05:49:05.976487Z",
+     "shell.execute_reply": "2026-04-02T05:49:05.975732Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -327,7 +362,7 @@
        "\\begin{equation*} A = \\left\\{ \\begin{aligned} \\boldsymbol{e}_{a} &\\mapsto a c^{2} x \\boldsymbol{e}_{a} + a b c x^{2} \\boldsymbol{e}_{b} + a^{3} b^{5} x^{2} \\boldsymbol{e}_{c} \\\\ \\boldsymbol{e}_{b} &\\mapsto a^{2} b c x^{3} \\boldsymbol{e}_{a} + a b^{2} c^{5} x^{4} \\boldsymbol{e}_{b} + 5 a b^{2} c x^{4} \\boldsymbol{e}_{c} \\\\ \\boldsymbol{e}_{c} &\\mapsto a b^{2} c^{4} x^{4} \\boldsymbol{e}_{a} + 4 a b^{2} c^{2} x^{4} \\boldsymbol{e}_{b} + 4 a^{5} b^{2} c x^{4} \\boldsymbol{e}_{c} \\end{aligned} \\right\\} \\end{equation*}\n",
        "\\begin{equation*} v = a \\boldsymbol{e}_{a} + b \\boldsymbol{e}_{b} + c \\boldsymbol{e}_{c} \\end{equation*}\n",
        "\\begin{equation*} f = v\\cdot \\f{A}{v} = a c x \\left(4 a^{4} b^{2} c^{2} x^{3} + a^{3} b^{5} x + a^{2} b^{2} x^{2} + a^{2} c + a b^{2} c^{4} x^{3} + a b^{2} x + b^{4} c^{4} x^{3} + 4 b^{3} c^{2} x^{3} + 5 b^{3} c x^{3}\\right) \\end{equation*}\n",
-       "\\begin{equation*} \\f{A}{v} = a c x \\left(a b^{2} x^{2} + a c + b^{2} c^{4} x^{3}\\right) \\boldsymbol{e}_{a} + a b c x^{2} \\left(a + b^{2} c^{4} x^{2} + 4 b c^{2} x^{2}\\right) \\boldsymbol{e}_{b} + a b^{2} x^{2} \\cdot \\left(4 a^{4} c^{2} x^{2} + a^{3} b^{3} + 5 b c x^{2}\\right) \\boldsymbol{e}_{c} \\end{equation*}\n",
+       "\\begin{equation*} \\f{A}{v} = a c x \\left(a b^{2} x^{2} + a c + b^{2} c^{4} x^{3}\\right) \\boldsymbol{e}_{a} + a b c x^{2} \\left(a + b^{2} c^{4} x^{2} + 4 b c^{2} x^{2}\\right) \\boldsymbol{e}_{b} + a b^{2} x^{2} \\left(4 a^{4} c^{2} x^{2} + a^{3} b^{3} + 5 b c x^{2}\\right) \\boldsymbol{e}_{c} \\end{equation*}\n",
        "\\end{document}\n"
       ]
      },
@@ -342,7 +377,14 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:05.978087Z",
+     "iopub.status.busy": "2026-04-02T05:49:05.977993Z",
+     "iopub.status.idle": "2026-04-02T05:49:06.440061Z",
+     "shell.execute_reply": "2026-04-02T05:49:06.439409Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -412,7 +454,14 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:06.441917Z",
+     "iopub.status.busy": "2026-04-02T05:49:06.441807Z",
+     "iopub.status.idle": "2026-04-02T05:49:07.051563Z",
+     "shell.execute_reply": "2026-04-02T05:49:07.050956Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -493,7 +542,14 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:07.053167Z",
+     "iopub.status.busy": "2026-04-02T05:49:07.053083Z",
+     "iopub.status.idle": "2026-04-02T05:49:08.041330Z",
+     "shell.execute_reply": "2026-04-02T05:49:08.040383Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -579,7 +635,14 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:08.043514Z",
+     "iopub.status.busy": "2026-04-02T05:49:08.043368Z",
+     "iopub.status.idle": "2026-04-02T05:49:08.348598Z",
+     "shell.execute_reply": "2026-04-02T05:49:08.347609Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -652,7 +715,14 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:08.350270Z",
+     "iopub.status.busy": "2026-04-02T05:49:08.350184Z",
+     "iopub.status.idle": "2026-04-02T05:49:09.023669Z",
+     "shell.execute_reply": "2026-04-02T05:49:09.022727Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -811,7 +881,14 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:09.025524Z",
+     "iopub.status.busy": "2026-04-02T05:49:09.025402Z",
+     "iopub.status.idle": "2026-04-02T05:49:14.316387Z",
+     "shell.execute_reply": "2026-04-02T05:49:14.315572Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -1454,7 +1531,14 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:14.317993Z",
+     "iopub.status.busy": "2026-04-02T05:49:14.317902Z",
+     "iopub.status.idle": "2026-04-02T05:49:15.459283Z",
+     "shell.execute_reply": "2026-04-02T05:49:15.458625Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -1579,7 +1663,14 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:15.460720Z",
+     "iopub.status.busy": "2026-04-02T05:49:15.460649Z",
+     "iopub.status.idle": "2026-04-02T05:49:18.298338Z",
+     "shell.execute_reply": "2026-04-02T05:49:18.297606Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -1657,7 +1748,14 @@
   {
    "cell_type": "code",
    "execution_count": 14,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:18.300259Z",
+     "iopub.status.busy": "2026-04-02T05:49:18.300175Z",
+     "iopub.status.idle": "2026-04-02T05:49:19.274821Z",
+     "shell.execute_reply": "2026-04-02T05:49:19.273788Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -1739,7 +1837,14 @@
   {
    "cell_type": "code",
    "execution_count": 15,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:19.277095Z",
+     "iopub.status.busy": "2026-04-02T05:49:19.276959Z",
+     "iopub.status.idle": "2026-04-02T05:49:19.651645Z",
+     "shell.execute_reply": "2026-04-02T05:49:19.651021Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -1808,7 +1913,14 @@
   {
    "cell_type": "code",
    "execution_count": 16,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:19.653393Z",
+     "iopub.status.busy": "2026-04-02T05:49:19.653280Z",
+     "iopub.status.idle": "2026-04-02T05:49:20.490206Z",
+     "shell.execute_reply": "2026-04-02T05:49:20.489532Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -1883,7 +1995,14 @@
   {
    "cell_type": "code",
    "execution_count": 17,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:20.491772Z",
+     "iopub.status.busy": "2026-04-02T05:49:20.491668Z",
+     "iopub.status.idle": "2026-04-02T05:49:21.088635Z",
+     "shell.execute_reply": "2026-04-02T05:49:21.087838Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -1965,7 +2084,14 @@
   {
    "cell_type": "code",
    "execution_count": 18,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:21.090500Z",
+     "iopub.status.busy": "2026-04-02T05:49:21.090386Z",
+     "iopub.status.idle": "2026-04-02T05:49:22.810531Z",
+     "shell.execute_reply": "2026-04-02T05:49:22.809606Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -2087,7 +2213,14 @@
   {
    "cell_type": "code",
    "execution_count": 19,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:22.812118Z",
+     "iopub.status.busy": "2026-04-02T05:49:22.812027Z",
+     "iopub.status.idle": "2026-04-02T05:49:23.875534Z",
+     "shell.execute_reply": "2026-04-02T05:49:23.874700Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -2317,7 +2450,14 @@
   {
    "cell_type": "code",
    "execution_count": 20,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:23.877905Z",
+     "iopub.status.busy": "2026-04-02T05:49:23.877797Z",
+     "iopub.status.idle": "2026-04-02T05:49:24.517387Z",
+     "shell.execute_reply": "2026-04-02T05:49:24.515992Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -2401,7 +2541,14 @@
   {
    "cell_type": "code",
    "execution_count": 21,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:24.520368Z",
+     "iopub.status.busy": "2026-04-02T05:49:24.520100Z",
+     "iopub.status.idle": "2026-04-02T05:49:25.171650Z",
+     "shell.execute_reply": "2026-04-02T05:49:25.171011Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -2498,7 +2645,14 @@
   {
    "cell_type": "code",
    "execution_count": 22,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-02T05:49:25.173298Z",
+     "iopub.status.busy": "2026-04-02T05:49:25.173208Z",
+     "iopub.status.idle": "2026-04-02T05:49:25.769799Z",
+     "shell.execute_reply": "2026-04-02T05:49:25.769109Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -2612,7 +2766,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.8"
+   "version": "3.12.12"
   }
  },
  "nbformat": 4,
diff --git a/examples/ipython/Old Format.ipynb b/examples/ipython/Old Format.ipynb
index 0573e2d6..1270705b 100644
--- a/examples/ipython/Old Format.ipynb	
+++ b/examples/ipython/Old Format.ipynb	
@@ -5,10 +5,10 @@
    "execution_count": 1,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:13.495944Z",
-     "iopub.status.busy": "2026-03-31T04:30:13.495665Z",
-     "iopub.status.idle": "2026-03-31T04:30:13.506418Z",
-     "shell.execute_reply": "2026-03-31T04:30:13.505013Z"
+     "iopub.execute_input": "2026-04-01T13:09:47.400022Z",
+     "iopub.status.busy": "2026-04-01T13:09:47.399953Z",
+     "iopub.status.idle": "2026-04-01T13:09:47.403756Z",
+     "shell.execute_reply": "2026-04-01T13:09:47.403110Z"
     }
    },
    "outputs": [],
@@ -23,10 +23,10 @@
    "execution_count": 2,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:13.511306Z",
-     "iopub.status.busy": "2026-03-31T04:30:13.510745Z",
-     "iopub.status.idle": "2026-03-31T04:30:15.028905Z",
-     "shell.execute_reply": "2026-03-31T04:30:15.025281Z"
+     "iopub.execute_input": "2026-04-01T13:09:47.405536Z",
+     "iopub.status.busy": "2026-04-01T13:09:47.405405Z",
+     "iopub.status.idle": "2026-04-01T13:09:47.732763Z",
+     "shell.execute_reply": "2026-04-01T13:09:47.731834Z"
     }
    },
    "outputs": [
@@ -47,10 +47,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "bad_example.py:4: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n"
      ]
@@ -65,10 +61,10 @@
    "execution_count": 3,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:15.154083Z",
-     "iopub.status.busy": "2026-03-31T04:30:15.153466Z",
-     "iopub.status.idle": "2026-03-31T04:30:16.930170Z",
-     "shell.execute_reply": "2026-03-31T04:30:16.926938Z"
+     "iopub.execute_input": "2026-04-01T13:09:47.747649Z",
+     "iopub.status.busy": "2026-04-01T13:09:47.747522Z",
+     "iopub.status.idle": "2026-04-01T13:09:48.215437Z",
+     "shell.execute_reply": "2026-04-01T13:09:48.214474Z"
     }
    },
    "outputs": [
@@ -102,10 +98,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "eval_check.py:5: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV,ReciprocalFrame\n",
       "eval_check.py:19: DeprecationWarning: The `galgebra.deprecated.ReciprocalFrame` function is deprecated in favor of the `ReciprocalFrame` method of `Ga` objects.\n",
@@ -124,10 +116,10 @@
    "execution_count": 4,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:16.939293Z",
-     "iopub.status.busy": "2026-03-31T04:30:16.938621Z",
-     "iopub.status.idle": "2026-03-31T04:30:20.660064Z",
-     "shell.execute_reply": "2026-03-31T04:30:20.650801Z"
+     "iopub.execute_input": "2026-04-01T13:09:48.217633Z",
+     "iopub.status.busy": "2026-04-01T13:09:48.217515Z",
+     "iopub.status.idle": "2026-04-01T13:09:48.659332Z",
+     "shell.execute_reply": "2026-04-01T13:09:48.658743Z"
     }
    },
    "outputs": [
@@ -145,10 +137,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "exp_check.py:5: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "exp_check.py:12: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -169,10 +157,10 @@
    "execution_count": 5,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:20.693346Z",
-     "iopub.status.busy": "2026-03-31T04:30:20.692582Z",
-     "iopub.status.idle": "2026-03-31T04:30:33.527993Z",
-     "shell.execute_reply": "2026-03-31T04:30:33.518035Z"
+     "iopub.execute_input": "2026-04-01T13:09:48.660899Z",
+     "iopub.status.busy": "2026-04-01T13:09:48.660797Z",
+     "iopub.status.idle": "2026-04-01T13:09:50.745065Z",
+     "shell.execute_reply": "2026-04-01T13:09:50.744409Z"
     }
    },
    "outputs": [
@@ -180,10 +168,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "latex_check.py:7: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "latex_check.py:32: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -440,7 +424,7 @@
        "\\begin{equation*} B = B^{r\\theta }  \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\theta } + B^{r\\phi }  \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\phi } + B^{\\theta \\phi }  \\boldsymbol{e}_{\\theta }\\wedge \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla}  f = \\partial_{r} f  \\boldsymbol{e}_{r} + \\frac{\\partial_{\\theta } f }{r^{2}} \\boldsymbol{e}_{\\theta } + \\frac{\\partial_{\\phi } f }{r^{2} {\\sin{\\left (\\theta  \\right )}}^{2}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla} \\cdot A = \\frac{A^{\\theta } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\phi } A^{\\phi }  + \\partial_{r} A^{r}  + \\partial_{\\theta } A^{\\theta }  + \\frac{2 A^{r} }{r} \\end{equation*}\n",
-       "\\begin{equation*} -I (\\boldsymbol{\\nabla} \\W A) = \\frac{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}} \\cdot \\left(\\frac{2 A^{\\phi } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\theta } A^{\\phi }  - \\frac{\\partial_{\\phi } A^{\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}\\right)}{r^{2}} \\boldsymbol{e}_{r} + \\frac{- r^{2} {\\sin{\\left (\\theta  \\right )}}^{2} \\partial_{r} A^{\\phi }  - 2 r A^{\\phi }  {\\sin{\\left (\\theta  \\right )}}^{2} + \\partial_{\\phi } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\theta } + \\frac{r^{2} \\partial_{r} A^{\\theta }  + 2 r A^{\\theta }  - \\partial_{\\theta } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
+       "\\begin{equation*} -I (\\boldsymbol{\\nabla} \\W A) = \\frac{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}} \\left(\\frac{2 A^{\\phi } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\theta } A^{\\phi }  - \\frac{\\partial_{\\phi } A^{\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}\\right)}{r^{2}} \\boldsymbol{e}_{r} + \\frac{- r^{2} {\\sin{\\left (\\theta  \\right )}}^{2} \\partial_{r} A^{\\phi }  - 2 r A^{\\phi }  {\\sin{\\left (\\theta  \\right )}}^{2} + \\partial_{\\phi } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\theta } + \\frac{r^{2} \\partial_{r} A^{\\theta }  + 2 r A^{\\theta }  - \\partial_{\\theta } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla} \\W B = \\frac{r^{2} \\partial_{r} B^{\\theta \\phi }  + 4 r B^{\\theta \\phi }  - \\frac{2 B^{r\\phi } }{\\tan{\\left (\\theta  \\right )}} - \\partial_{\\theta } B^{r\\phi }  + \\frac{\\partial_{\\phi } B^{r\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}}{r^{2}} \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\theta }\\wedge \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{lstlisting}[language=Python,showspaces=false,showstringspaces=false,backgroundcolor=\\color{gray},frame=single]\n",
        "def conformal_representations_of_circles_lines_spheres_and_planes():\n",
@@ -639,23 +623,13 @@
    "execution_count": 6,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:33.539087Z",
-     "iopub.status.busy": "2026-03-31T04:30:33.537389Z",
-     "iopub.status.idle": "2026-03-31T04:30:35.557924Z",
-     "shell.execute_reply": "2026-03-31T04:30:35.550811Z"
+     "iopub.execute_input": "2026-04-01T13:09:50.746626Z",
+     "iopub.status.busy": "2026-04-01T13:09:50.746545Z",
+     "iopub.status.idle": "2026-04-01T13:09:51.042444Z",
+     "shell.execute_reply": "2026-04-01T13:09:51.041534Z"
     }
    },
    "outputs": [
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n"
-     ]
-    },
     {
      "data": {
       "text/plain": [
@@ -725,10 +699,10 @@
    "execution_count": 7,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:35.561927Z",
-     "iopub.status.busy": "2026-03-31T04:30:35.561576Z",
-     "iopub.status.idle": "2026-03-31T04:30:36.975367Z",
-     "shell.execute_reply": "2026-03-31T04:30:36.973404Z"
+     "iopub.execute_input": "2026-04-01T13:09:51.044210Z",
+     "iopub.status.busy": "2026-04-01T13:09:51.044126Z",
+     "iopub.status.idle": "2026-04-01T13:09:51.345539Z",
+     "shell.execute_reply": "2026-04-01T13:09:51.344749Z"
     }
    },
    "outputs": [
@@ -746,10 +720,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "mv_setup_options.py:3: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "mv_setup_options.py:8: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -772,10 +742,10 @@
    "execution_count": 8,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:36.981402Z",
-     "iopub.status.busy": "2026-03-31T04:30:36.980256Z",
-     "iopub.status.idle": "2026-03-31T04:30:40.713918Z",
-     "shell.execute_reply": "2026-03-31T04:30:40.712355Z"
+     "iopub.execute_input": "2026-04-01T13:09:51.347436Z",
+     "iopub.status.busy": "2026-04-01T13:09:51.347356Z",
+     "iopub.status.idle": "2026-04-01T13:09:52.177936Z",
+     "shell.execute_reply": "2026-04-01T13:09:52.177215Z"
     }
    },
    "outputs": [
@@ -783,10 +753,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "physics_check_latex.py:6: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "physics_check_latex.py:14: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -795,8 +761,6 @@
       "  E = MV('E','vector',fct=True)\n",
       "physics_check_latex.py:20: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  J = MV('J','vector',fct=True)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/libmp/libintmath.py:75: DeprecationWarning: bitcount function is deprecated\n",
-      "  warnings.warn(\"bitcount function is deprecated\",\n",
       "physics_check_latex.py:48: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  psi = MV('psi','spinor',fct=True)\n",
       "physics_check_latex.py:49: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -960,10 +924,10 @@
    "execution_count": 9,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:40.717799Z",
-     "iopub.status.busy": "2026-03-31T04:30:40.717527Z",
-     "iopub.status.idle": "2026-03-31T04:30:50.857374Z",
-     "shell.execute_reply": "2026-03-31T04:30:50.851089Z"
+     "iopub.execute_input": "2026-04-01T13:09:52.179493Z",
+     "iopub.status.busy": "2026-04-01T13:09:52.179403Z",
+     "iopub.status.idle": "2026-04-01T13:09:53.820151Z",
+     "shell.execute_reply": "2026-04-01T13:09:53.819527Z"
     }
    },
    "outputs": [
@@ -971,14 +935,8 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "print_check_latex.py:5: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/libmp/libintmath.py:75: DeprecationWarning: bitcount function is deprecated\n",
-      "  warnings.warn(\"bitcount function is deprecated\",\n",
       "print_check_latex.py:11: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  A = MV('A','mv')\n",
       "print_check_latex.py:19: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -1096,7 +1054,7 @@
        "\\begin{equation*} B = B^{r\\theta }  \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\theta } + B^{r\\phi }  \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\phi } + B^{\\theta \\phi }  \\boldsymbol{e}_{\\theta }\\wedge \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla}  f = \\partial_{r} f  \\boldsymbol{e}_{r} + \\frac{\\partial_{\\theta } f }{r^{2}} \\boldsymbol{e}_{\\theta } + \\frac{\\partial_{\\phi } f }{r^{2} {\\sin{\\left (\\theta  \\right )}}^{2}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla} \\cdot A = \\frac{A^{\\theta } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\phi } A^{\\phi }  + \\partial_{r} A^{r}  + \\partial_{\\theta } A^{\\theta }  + \\frac{2 A^{r} }{r} \\end{equation*}\n",
-       "\\begin{equation*} -I (\\boldsymbol{\\nabla} \\W A) = \\frac{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}} \\cdot \\left(\\frac{2 A^{\\phi } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\theta } A^{\\phi }  - \\frac{\\partial_{\\phi } A^{\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}\\right)}{r^{2}} \\boldsymbol{e}_{r} + \\frac{- r^{2} {\\sin{\\left (\\theta  \\right )}}^{2} \\partial_{r} A^{\\phi }  - 2 r A^{\\phi }  {\\sin{\\left (\\theta  \\right )}}^{2} + \\partial_{\\phi } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\theta } + \\frac{r^{2} \\partial_{r} A^{\\theta }  + 2 r A^{\\theta }  - \\partial_{\\theta } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
+       "\\begin{equation*} -I (\\boldsymbol{\\nabla} \\W A) = \\frac{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}} \\left(\\frac{2 A^{\\phi } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\theta } A^{\\phi }  - \\frac{\\partial_{\\phi } A^{\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}\\right)}{r^{2}} \\boldsymbol{e}_{r} + \\frac{- r^{2} {\\sin{\\left (\\theta  \\right )}}^{2} \\partial_{r} A^{\\phi }  - 2 r A^{\\phi }  {\\sin{\\left (\\theta  \\right )}}^{2} + \\partial_{\\phi } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\theta } + \\frac{r^{2} \\partial_{r} A^{\\theta }  + 2 r A^{\\theta }  - \\partial_{\\theta } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla} \\W B = \\frac{r^{2} \\partial_{r} B^{\\theta \\phi }  + 4 r B^{\\theta \\phi }  - \\frac{2 B^{r\\phi } }{\\tan{\\left (\\theta  \\right )}} - \\partial_{\\theta } B^{r\\phi }  + \\frac{\\partial_{\\phi } B^{r\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}}{r^{2}} \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\theta }\\wedge \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} B = \\bm{B\\gamma_{t}} = - B^{x}  \\boldsymbol{\\gamma }_{t}\\wedge \\boldsymbol{\\gamma }_{x} - B^{y}  \\boldsymbol{\\gamma }_{t}\\wedge \\boldsymbol{\\gamma }_{y} - B^{z}  \\boldsymbol{\\gamma }_{t}\\wedge \\boldsymbol{\\gamma }_{z} \\end{equation*}\n",
        "\\begin{equation*} E = \\bm{E\\gamma_{t}} = - E^{x}  \\boldsymbol{\\gamma }_{t}\\wedge \\boldsymbol{\\gamma }_{x} - E^{y}  \\boldsymbol{\\gamma }_{t}\\wedge \\boldsymbol{\\gamma }_{y} - E^{z}  \\boldsymbol{\\gamma }_{t}\\wedge \\boldsymbol{\\gamma }_{z} \\end{equation*}\n",
@@ -1127,10 +1085,10 @@
    "execution_count": 10,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:50.873709Z",
-     "iopub.status.busy": "2026-03-31T04:30:50.873152Z",
-     "iopub.status.idle": "2026-03-31T04:30:52.945881Z",
-     "shell.execute_reply": "2026-03-31T04:30:52.941600Z"
+     "iopub.execute_input": "2026-04-01T13:09:53.821925Z",
+     "iopub.status.busy": "2026-04-01T13:09:53.821835Z",
+     "iopub.status.idle": "2026-04-01T13:09:54.238078Z",
+     "shell.execute_reply": "2026-04-01T13:09:54.237423Z"
     }
    },
    "outputs": [
@@ -1154,10 +1112,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "prob_not_solenoidal.py:5: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "prob_not_solenoidal.py:18: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -1174,10 +1128,10 @@
    "execution_count": 11,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:52.959619Z",
-     "iopub.status.busy": "2026-03-31T04:30:52.953255Z",
-     "iopub.status.idle": "2026-03-31T04:30:57.632933Z",
-     "shell.execute_reply": "2026-03-31T04:30:57.630134Z"
+     "iopub.execute_input": "2026-04-01T13:09:54.239724Z",
+     "iopub.status.busy": "2026-04-01T13:09:54.239611Z",
+     "iopub.status.idle": "2026-04-01T13:09:55.140534Z",
+     "shell.execute_reply": "2026-04-01T13:09:55.139816Z"
     }
    },
    "outputs": [
@@ -1361,10 +1315,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "products_latex.py:3: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "products_latex.py:13: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -1373,8 +1323,6 @@
       "  v = MV('v','vector')\n",
       "products_latex.py:15: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  b = MV('b','bivector')\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/libmp/libintmath.py:75: DeprecationWarning: bitcount function is deprecated\n",
-      "  warnings.warn(\"bitcount function is deprecated\",\n",
       "products_latex.py:38: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  fs = MV('s','scalar',fct=True)\n",
       "products_latex.py:39: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -1399,10 +1347,10 @@
    "execution_count": 12,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:57.639185Z",
-     "iopub.status.busy": "2026-03-31T04:30:57.638622Z",
-     "iopub.status.idle": "2026-03-31T04:30:59.334006Z",
-     "shell.execute_reply": "2026-03-31T04:30:59.332031Z"
+     "iopub.execute_input": "2026-04-01T13:09:55.142027Z",
+     "iopub.status.busy": "2026-04-01T13:09:55.141917Z",
+     "iopub.status.idle": "2026-04-01T13:09:55.526782Z",
+     "shell.execute_reply": "2026-04-01T13:09:55.526039Z"
     }
    },
    "outputs": [
@@ -1410,18 +1358,12 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "reflect_test.py:4: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "reflect_test.py:10: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  a = MV('a','vector')\n",
       "reflect_test.py:13: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
-      "  c = MV('c','vector')\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/libmp/libintmath.py:75: DeprecationWarning: bitcount function is deprecated\n",
-      "  warnings.warn(\"bitcount function is deprecated\",\n"
+      "  c = MV('c','vector')\n"
      ]
     },
     {
@@ -1506,10 +1448,10 @@
    "execution_count": 13,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:30:59.339506Z",
-     "iopub.status.busy": "2026-03-31T04:30:59.339069Z",
-     "iopub.status.idle": "2026-03-31T04:31:01.442894Z",
-     "shell.execute_reply": "2026-03-31T04:31:01.439257Z"
+     "iopub.execute_input": "2026-04-01T13:09:55.528811Z",
+     "iopub.status.busy": "2026-04-01T13:09:55.528704Z",
+     "iopub.status.idle": "2026-04-01T13:09:55.929416Z",
+     "shell.execute_reply": "2026-04-01T13:09:55.928508Z"
     }
    },
    "outputs": [
@@ -1517,10 +1459,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "simple_check_latex.py:3: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "simple_check_latex.py:10: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -1611,10 +1549,10 @@
    "execution_count": 14,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:31:01.448870Z",
-     "iopub.status.busy": "2026-03-31T04:31:01.448497Z",
-     "iopub.status.idle": "2026-03-31T04:31:04.315471Z",
-     "shell.execute_reply": "2026-03-31T04:31:04.312174Z"
+     "iopub.execute_input": "2026-04-01T13:09:55.932394Z",
+     "iopub.status.busy": "2026-04-01T13:09:55.932280Z",
+     "iopub.status.idle": "2026-04-01T13:09:56.376212Z",
+     "shell.execute_reply": "2026-04-01T13:09:56.375703Z"
     }
    },
    "outputs": [
@@ -1636,10 +1574,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "simple_check.py:5: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "simple_check.py:13: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -1647,9 +1581,7 @@
       "simple_check.py:14: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  v = MV('v','vector')\n",
       "simple_check.py:15: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
-      "  w = MV('w','vector')\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/libmp/libintmath.py:75: DeprecationWarning: bitcount function is deprecated\n",
-      "  warnings.warn(\"bitcount function is deprecated\",\n"
+      "  w = MV('w','vector')\n"
      ]
     }
    ],
@@ -1662,10 +1594,10 @@
    "execution_count": 15,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:31:04.319457Z",
-     "iopub.status.busy": "2026-03-31T04:31:04.319161Z",
-     "iopub.status.idle": "2026-03-31T04:31:11.930319Z",
-     "shell.execute_reply": "2026-03-31T04:31:11.924491Z"
+     "iopub.execute_input": "2026-04-01T13:09:56.377832Z",
+     "iopub.status.busy": "2026-04-01T13:09:56.377723Z",
+     "iopub.status.idle": "2026-04-01T13:09:57.473340Z",
+     "shell.execute_reply": "2026-04-01T13:09:57.472374Z"
     }
    },
    "outputs": [
@@ -1673,14 +1605,8 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "spherical_latex.py:5: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/libmp/libintmath.py:75: DeprecationWarning: bitcount function is deprecated\n",
-      "  warnings.warn(\"bitcount function is deprecated\",\n",
       "spherical_latex.py:14: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
       "  f = MV('f','scalar',fct=True)\n",
       "spherical_latex.py:15: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -1764,7 +1690,7 @@
        "\\begin{equation*} B = B^{r\\theta }  \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\theta } + B^{r\\phi }  \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\phi } + B^{\\theta \\phi }  \\boldsymbol{e}_{\\theta }\\wedge \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla}  f = \\partial_{r} f  \\boldsymbol{e}_{r} + \\frac{\\partial_{\\theta } f }{r^{2}} \\boldsymbol{e}_{\\theta } + \\frac{\\partial_{\\phi } f }{r^{2} {\\sin{\\left (\\theta  \\right )}}^{2}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla} \\cdot A = \\frac{A^{\\theta } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\phi } A^{\\phi }  + \\partial_{r} A^{r}  + \\partial_{\\theta } A^{\\theta }  + \\frac{2 A^{r} }{r} \\end{equation*}\n",
-       "\\begin{equation*} -I (\\boldsymbol{\\nabla} \\W A) = \\frac{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}} \\cdot \\left(\\frac{2 A^{\\phi } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\theta } A^{\\phi }  - \\frac{\\partial_{\\phi } A^{\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}\\right)}{r^{2}} \\boldsymbol{e}_{r} + \\frac{- r^{2} {\\sin{\\left (\\theta  \\right )}}^{2} \\partial_{r} A^{\\phi }  - 2 r A^{\\phi }  {\\sin{\\left (\\theta  \\right )}}^{2} + \\partial_{\\phi } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\theta } + \\frac{r^{2} \\partial_{r} A^{\\theta }  + 2 r A^{\\theta }  - \\partial_{\\theta } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
+       "\\begin{equation*} -I (\\boldsymbol{\\nabla} \\W A) = \\frac{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}} \\left(\\frac{2 A^{\\phi } }{\\tan{\\left (\\theta  \\right )}} + \\partial_{\\theta } A^{\\phi }  - \\frac{\\partial_{\\phi } A^{\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}\\right)}{r^{2}} \\boldsymbol{e}_{r} + \\frac{- r^{2} {\\sin{\\left (\\theta  \\right )}}^{2} \\partial_{r} A^{\\phi }  - 2 r A^{\\phi }  {\\sin{\\left (\\theta  \\right )}}^{2} + \\partial_{\\phi } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\theta } + \\frac{r^{2} \\partial_{r} A^{\\theta }  + 2 r A^{\\theta }  - \\partial_{\\theta } A^{r} }{\\sqrt{r^{4} {\\sin{\\left (\\theta  \\right )}}^{2}}} \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\begin{equation*} \\boldsymbol{\\nabla} \\W B = \\frac{r^{2} \\partial_{r} B^{\\theta \\phi }  + 4 r B^{\\theta \\phi }  - \\frac{2 B^{r\\phi } }{\\tan{\\left (\\theta  \\right )}} - \\partial_{\\theta } B^{r\\phi }  + \\frac{\\partial_{\\phi } B^{r\\theta } }{{\\sin{\\left (\\theta  \\right )}}^{2}}}{r^{2}} \\boldsymbol{e}_{r}\\wedge \\boldsymbol{e}_{\\theta }\\wedge \\boldsymbol{e}_{\\phi } \\end{equation*}\n",
        "\\end{document}\n"
       ]
@@ -1782,10 +1708,10 @@
    "execution_count": 16,
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-31T04:31:11.939086Z",
-     "iopub.status.busy": "2026-03-31T04:31:11.938174Z",
-     "iopub.status.idle": "2026-03-31T04:31:35.017597Z",
-     "shell.execute_reply": "2026-03-31T04:31:35.014079Z"
+     "iopub.execute_input": "2026-04-01T13:09:57.474910Z",
+     "iopub.status.busy": "2026-04-01T13:09:57.474817Z",
+     "iopub.status.idle": "2026-04-01T13:10:01.595705Z",
+     "shell.execute_reply": "2026-04-01T13:10:01.594542Z"
     }
    },
    "outputs": [
@@ -1941,10 +1867,6 @@
      "name": "stderr",
      "output_type": "stream",
      "text": [
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/ctx_mp.py:35: DeprecationWarning: mpnumeric is deprecated\n",
-      "  warnings.warn(f\"{name} is deprecated\", DeprecationWarning)\n",
-      "/Users/lume/projects/galgebra/.venv/lib/python3.11/site-packages/mpmath/rational.py:4: DeprecationWarning: the rational private module is deprecated\n",
-      "  warnings.warn(\"the rational private module is deprecated\",\n",
       "terminal_check.py:6: DeprecationWarning: The `galgebra.deprecated` module is deprecated\n",
       "  from galgebra.deprecated import MV\n",
       "terminal_check.py:13: DeprecationWarning: The `galgebra.deprecated.MV` class is deprecated in favor of `galgebra.mv.Mv`.\n",
@@ -2001,7 +1923,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.14"
+   "version": "3.12.12"
   }
  },
  "nbformat": 4,
diff --git a/examples/ipython/gr_metrics.ipynb b/examples/ipython/gr_metrics.ipynb
index 1fccfdf8..4ee0efa1 100644
--- a/examples/ipython/gr_metrics.ipynb
+++ b/examples/ipython/gr_metrics.ipynb
@@ -3,7 +3,14 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:43.808023Z",
+     "iopub.status.busy": "2026-04-01T13:09:43.807949Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.722231Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.721765Z"
+    }
+   },
    "outputs": [],
    "source": [
     "from sympy import *\n",
@@ -16,7 +23,14 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.724408Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.724285Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.726547Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.726096Z"
+    }
+   },
    "outputs": [],
    "source": [
     "def dot_basis_r_basis(ga):\n",
@@ -26,7 +40,14 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.728314Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.728211Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.730172Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.729750Z"
+    }
+   },
    "outputs": [],
    "source": [
     "def gg(ga):\n",
@@ -36,7 +57,14 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.731894Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.731741Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.733355Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.732974Z"
+    }
+   },
    "outputs": [],
    "source": [
     "def conv_christoffel_symbols(cf):\n",
@@ -46,7 +74,14 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.735510Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.735376Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.737932Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.737413Z"
+    }
+   },
    "outputs": [],
    "source": [
     "def show_christoffel_symbols(ga):\n",
@@ -65,7 +100,14 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.739617Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.739521Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.810243Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.809775Z"
+    }
+   },
    "outputs": [],
    "source": [
     "coord = symbols('t x y z')\n",
@@ -81,7 +123,14 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.813214Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.813020Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.816647Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.815950Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -105,7 +154,14 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.832838Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.832685Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.836787Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.836034Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -134,7 +190,14 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.839091Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.839005Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.841693Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.841178Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -157,7 +220,14 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.843473Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.843362Z",
+     "iopub.status.idle": "2026-04-01T13:09:44.847630Z",
+     "shell.execute_reply": "2026-04-01T13:09:44.846613Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -193,7 +263,14 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:44.850315Z",
+     "iopub.status.busy": "2026-04-01T13:09:44.850160Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.051081Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.050457Z"
+    }
+   },
    "outputs": [],
    "source": [
     "g4coords = (u, x, y, z) = symbols(\"u x y z\")\n",
@@ -209,7 +286,14 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.053051Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.052944Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.055784Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.055330Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -232,7 +316,14 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.057249Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.057146Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.060672Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.060292Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -269,7 +360,14 @@
   {
    "cell_type": "code",
    "execution_count": 14,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.062119Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.062058Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.064262Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.063827Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -295,7 +393,14 @@
   {
    "cell_type": "code",
    "execution_count": 15,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.065613Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.065551Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.070879Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.070390Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -321,7 +426,14 @@
   {
    "cell_type": "code",
    "execution_count": 16,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.072624Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.072520Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.090530Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.090073Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -350,12 +462,19 @@
   {
    "cell_type": "code",
    "execution_count": 17,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.091969Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.091879Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.179277Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.178949Z"
+    }
+   },
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "\\begin{equation*}\\left[\\begin{array}{}\\left[\\begin{array}{}0 & 0 & 0 & 0\\\\0 & - \\frac{u e^{4 z}}{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - \\frac{u}{2}\\end{array}\\right] & \\left[\\begin{array}{}0 & \\frac{u e^{4 z}}{2} & 0 & 0\\\\\\frac{u e^{4 z}}{2} & 0 & 0 & u^{2} e^{4 z}\\\\0 & 0 & 0 & 0\\\\0 & u^{2} e^{4 z} & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & e^{- z}\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - 12 e^{- 2 z}\\\\e^{- z} & 0 & - 12 e^{- 2 z} & - u e^{- z}\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & \\frac{u}{2}\\\\0 & - u^{2} e^{4 z} & 0 & 0\\\\0 & 0 & 12 e^{- 2 z} & 0\\\\\\frac{u}{2} & 0 & 0 & 0\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
+       "\\begin{equation*}\\left[\\begin{array}{cccc}\\left[\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & - \\frac{u e^{4 z}}{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - \\frac{u}{2}\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & \\frac{u e^{4 z}}{2} & 0 & 0\\\\\\frac{u e^{4 z}}{2} & 0 & 0 & u^{2} e^{4 z}\\\\0 & 0 & 0 & 0\\\\0 & u^{2} e^{4 z} & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & e^{- z}\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & - 12 e^{- 2 z}\\\\e^{- z} & 0 & - 12 e^{- 2 z} & - u e^{- z}\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & \\frac{u}{2}\\\\0 & - u^{2} e^{4 z} & 0 & 0\\\\0 & 0 & 12 e^{- 2 z} & 0\\\\\\frac{u}{2} & 0 & 0 & 0\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
       ],
       "text/plain": [
        "⎡                       ⎡           4⋅z             ⎤                                                            ⎤\n",
@@ -380,7 +499,7 @@
     {
      "data": {
       "text/latex": [
-       "\\begin{equation*}\\left[\\begin{array}{}\\left[\\begin{array}{}0 & 0 & 0 & 0\\\\0 & 3 u e^{4 z} & 0 & 0\\\\0 & 0 & \\frac{24 e^{- 2 z}}{u} & 12 e^{- z}\\\\0 & 0 & 12 e^{- z} & 6 u\\end{array}\\right] & \\left[\\begin{array}{}0 & \\frac{1}{u} & 0 & 0\\\\\\frac{1}{u} & 0 & 0 & 2\\\\0 & 0 & 0 & 0\\\\0 & 2 & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & 0\\\\0 & \\frac{u e^{5 z}}{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{u e^{z}}{2}\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & \\frac{1}{u}\\\\0 & - 3 e^{4 z} & 0 & 0\\\\0 & 0 & \\frac{24 e^{- 2 z}}{u^{2}} & 0\\\\\\frac{1}{u} & 0 & 0 & -1\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
+       "\\begin{equation*}\\left[\\begin{array}{cccc}\\left[\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & 3 u e^{4 z} & 0 & 0\\\\0 & 0 & \\frac{24 e^{- 2 z}}{u} & 12 e^{- z}\\\\0 & 0 & 12 e^{- z} & 6 u\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & \\frac{1}{u} & 0 & 0\\\\\\frac{1}{u} & 0 & 0 & 2\\\\0 & 0 & 0 & 0\\\\0 & 2 & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & \\frac{u e^{5 z}}{2} & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{u e^{z}}{2}\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & \\frac{1}{u}\\\\0 & - 3 e^{4 z} & 0 & 0\\\\0 & 0 & \\frac{24 e^{- 2 z}}{u^{2}} & 0\\\\\\frac{1}{u} & 0 & 0 & -1\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
       ],
       "text/plain": [
        "⎡                                                                     ⎡                      1 ⎤⎤\n",
@@ -419,7 +538,14 @@
   {
    "cell_type": "code",
    "execution_count": 18,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.181162Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.181045Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.325783Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.325411Z"
+    }
+   },
    "outputs": [],
    "source": [
     "G, M, c = symbols('G M c')\n",
@@ -436,7 +562,14 @@
   {
    "cell_type": "code",
    "execution_count": 19,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.328449Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.328321Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.330797Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.330485Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -460,7 +593,14 @@
   {
    "cell_type": "code",
    "execution_count": 20,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.332360Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.332263Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.336820Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.336116Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -470,15 +610,15 @@
       "text/plain": [
        "⎡  2⋅G⋅M                                   ⎤\n",
        "⎢- ───── + 1       0        0        0     ⎥\n",
-       "⎢    2                                     ⎥\n",
-       "⎢   c ⋅r                                   ⎥\n",
+       "⎢   2                                      ⎥\n",
+       "⎢  c ⋅r                                    ⎥\n",
        "⎢                                          ⎥\n",
        "⎢                 -1                       ⎥\n",
        "⎢     0       ───────────   0        0     ⎥\n",
        "⎢               2⋅G⋅M                      ⎥\n",
        "⎢             - ───── + 1                  ⎥\n",
-       "⎢                 2                        ⎥\n",
-       "⎢                c ⋅r                      ⎥\n",
+       "⎢                2                         ⎥\n",
+       "⎢               c ⋅r                       ⎥\n",
        "⎢                                          ⎥\n",
        "⎢                            2             ⎥\n",
        "⎢     0            0       -r        0     ⎥\n",
@@ -499,7 +639,14 @@
   {
    "cell_type": "code",
    "execution_count": 21,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.338431Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.338357Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.341187Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.340849Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -523,7 +670,14 @@
   {
    "cell_type": "code",
    "execution_count": 22,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.342840Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.342749Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.346282Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.345822Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -533,13 +687,13 @@
       "text/plain": [
        "⎡    ⎛2⋅G⋅M    ⎞       ⎤\n",
        "⎢   -⎜───── - 1⎟       ⎥\n",
-       "⎢    ⎜  2      ⎟       ⎥\n",
-       "⎢    ⎝ c ⋅r    ⎠       ⎥\n",
+       "⎢    ⎜ 2       ⎟       ⎥\n",
+       "⎢    ⎝c ⋅r     ⎠       ⎥\n",
        "⎢1, ─────────────, 1, 1⎥\n",
        "⎢      2⋅G⋅M           ⎥\n",
        "⎢    - ───── + 1       ⎥\n",
-       "⎢        2             ⎥\n",
-       "⎣       c ⋅r           ⎦"
+       "⎢       2              ⎥\n",
+       "⎣      c ⋅r            ⎦"
       ]
      },
      "execution_count": 22,
@@ -554,7 +708,14 @@
   {
    "cell_type": "code",
    "execution_count": 23,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.347732Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.347660Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.371768Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.371344Z"
+    }
+   },
    "outputs": [
     {
      "data": {
@@ -583,12 +744,19 @@
   {
    "cell_type": "code",
    "execution_count": 24,
-   "metadata": {},
+   "metadata": {
+    "execution": {
+     "iopub.execute_input": "2026-04-01T13:09:45.374220Z",
+     "iopub.status.busy": "2026-04-01T13:09:45.374076Z",
+     "iopub.status.idle": "2026-04-01T13:09:45.561232Z",
+     "shell.execute_reply": "2026-04-01T13:09:45.560810Z"
+    }
+   },
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "\\begin{equation*}\\left[\\begin{array}{}\\left[\\begin{array}{}0 & \\frac{G M}{c^{2} r^{2}} & 0 & 0\\\\\\frac{G M}{c^{2} r^{2}} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{}- \\frac{G M}{c^{2} r^{2}} & 0 & 0 & 0\\\\0 & \\frac{G M c^{2}}{4 G^{2} M^{2} + c^{2} r \\left(- 4 G M + c^{2} r\\right)} & 0 & 0\\\\0 & 0 & r & 0\\\\0 & 0 & 0 & r {\\sin{\\left (\\theta  \\right )}}^{2}\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & 0\\\\0 & 0 & - r & 0\\\\0 & - r & 0 & 0\\\\0 & 0 & 0 & \\frac{r^{2} \\sin{\\left (2 \\theta  \\right )}}{2}\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & 0\\\\0 & 0 & 0 & - r {\\sin{\\left (\\theta  \\right )}}^{2}\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin{\\left (2 \\theta  \\right )}}{2}\\\\0 & - r {\\sin{\\left (\\theta  \\right )}}^{2} & - \\frac{r^{2} \\sin{\\left (2 \\theta  \\right )}}{2} & 0\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
+       "\\begin{equation*}\\left[\\begin{array}{cccc}\\left[\\begin{array}{cccc}0 & \\frac{G M}{c^{2} r^{2}} & 0 & 0\\\\\\frac{G M}{c^{2} r^{2}} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{cccc}- \\frac{G M}{c^{2} r^{2}} & 0 & 0 & 0\\\\0 & \\frac{G M c^{2}}{4 G^{2} M^{2} + c^{2} r \\left(- 4 G M + c^{2} r\\right)} & 0 & 0\\\\0 & 0 & r & 0\\\\0 & 0 & 0 & r {\\sin{\\left (\\theta  \\right )}}^{2}\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & 0 & - r & 0\\\\0 & - r & 0 & 0\\\\0 & 0 & 0 & \\frac{r^{2} \\sin{\\left (2 \\theta  \\right )}}{2}\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & 0 & 0 & - r {\\sin{\\left (\\theta  \\right )}}^{2}\\\\0 & 0 & 0 & - \\frac{r^{2} \\sin{\\left (2 \\theta  \\right )}}{2}\\\\0 & - r {\\sin{\\left (\\theta  \\right )}}^{2} & - \\frac{r^{2} \\sin{\\left (2 \\theta  \\right )}}{2} & 0\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
       ],
       "text/plain": [
        "⎡                      ⎡-G⋅M                                               ⎤                            ⎡0      0             0              0      ⎤⎤\n",
@@ -614,7 +782,7 @@
     {
      "data": {
       "text/latex": [
-       "\\begin{equation*}\\left[\\begin{array}{}\\left[\\begin{array}{}0 & \\frac{G M}{r \\left(- 2 G M + c^{2} r\\right)} & 0 & 0\\\\\\frac{G M}{r \\left(- 2 G M + c^{2} r\\right)} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{}\\frac{G M \\left(- 2 G M + c^{2} r\\right)}{c^{4} r^{3}} & 0 & 0 & 0\\\\0 & \\frac{G M}{r \\left(2 G M - c^{2} r\\right)} & 0 & 0\\\\0 & 0 & \\frac{2 G M}{c^{2}} - r & 0\\\\0 & 0 & 0 & \\frac{\\left(2 G M - c^{2} r\\right) {\\sin{\\left (\\theta  \\right )}}^{2}}{c^{2}}\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\frac{\\sin{\\left (2 \\theta  \\right )}}{2}\\end{array}\\right] & \\left[\\begin{array}{}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{1}{\\tan{\\left (\\theta  \\right )}}\\\\0 & \\frac{1}{r} & \\frac{1}{\\tan{\\left (\\theta  \\right )}} & 0\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
+       "\\begin{equation*}\\left[\\begin{array}{cccc}\\left[\\begin{array}{cccc}0 & \\frac{G M}{r \\left(- 2 G M + c^{2} r\\right)} & 0 & 0\\\\\\frac{G M}{r \\left(- 2 G M + c^{2} r\\right)} & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{array}\\right] & \\left[\\begin{array}{cccc}\\frac{G M \\left(- 2 G M + c^{2} r\\right)}{c^{4} r^{3}} & 0 & 0 & 0\\\\0 & \\frac{G M}{r \\left(2 G M - c^{2} r\\right)} & 0 & 0\\\\0 & 0 & \\frac{2 G M}{c^{2}} - r & 0\\\\0 & 0 & 0 & \\frac{\\left(2 G M - c^{2} r\\right) {\\sin{\\left (\\theta  \\right )}}^{2}}{c^{2}}\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{r} & 0\\\\0 & \\frac{1}{r} & 0 & 0\\\\0 & 0 & 0 & - \\frac{\\sin{\\left (2 \\theta  \\right )}}{2}\\end{array}\\right] & \\left[\\begin{array}{cccc}0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{r}\\\\0 & 0 & 0 & \\frac{1}{\\tan{\\left (\\theta  \\right )}}\\\\0 & \\frac{1}{r} & \\frac{1}{\\tan{\\left (\\theta  \\right )}} & 0\\end{array}\\right]\\end{array}\\right]\\end{equation*}"
       ],
       "text/plain": [
        "⎡                                              ⎡    ⎛          2  ⎞                                                     ⎤                                               ⎤\n",
@@ -630,8 +798,8 @@
        "⎢⎢─────────────────          0          0  0⎥  ⎢                                                                        ⎥  ⎢   1               ⎥  ⎢                1   ⎥⎥\n",
        "⎢⎢  ⎛          2  ⎞                         ⎥  ⎢                                       2⋅G⋅M                            ⎥  ⎢0  ─  0      0     ⎥  ⎢0  0    0     ──────⎥⎥\n",
        "⎢⎢r⋅⎝-2⋅G⋅M + c ⋅r⎠                         ⎥  ⎢         0                  0          ───── - r            0           ⎥  ⎢   r               ⎥  ⎢              tan(θ)⎥⎥\n",
-       "⎢⎢                                          ⎥  ⎢                                          2                             ⎥  ⎢                   ⎥  ⎢                    ⎥⎥\n",
-       "⎢⎢        0                  0          0  0⎥  ⎢                                         c                              ⎥  ⎢         -sin(2⋅θ) ⎥  ⎢   1    1           ⎥⎥\n",
+       "⎢⎢                                          ⎥  ⎢                                         2                              ⎥  ⎢                   ⎥  ⎢                    ⎥⎥\n",
+       "⎢⎢        0                  0          0  0⎥  ⎢                                        c                               ⎥  ⎢         -sin(2⋅θ) ⎥  ⎢   1    1           ⎥⎥\n",
        "⎢⎢                                          ⎥  ⎢                                                                        ⎥  ⎢0  0  0  ──────────⎥  ⎢0  ─  ──────    0   ⎥⎥\n",
        "⎢⎣        0                  0          0  0⎦  ⎢                                                  ⎛         2  ⎞    2   ⎥  ⎣             2     ⎦  ⎣   r  tan(θ)        ⎦⎥\n",
        "⎢                                              ⎢                                                  ⎝2⋅G⋅M - c ⋅r⎠⋅sin (θ)⎥                                               ⎥\n",
@@ -665,7 +833,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.8"
+   "version": "3.12.12"
   }
  },
  "nbformat": 4,
diff --git a/examples/ipython/test_gsg_undual_etc.ipynb b/examples/ipython/test_gsg_undual_etc.ipynb
index 4c699f2f..fc26ac4a 100644
--- a/examples/ipython/test_gsg_undual_etc.ipynb
+++ b/examples/ipython/test_gsg_undual_etc.ipynb
@@ -45,10 +45,10 @@
    "id": "c59e2276",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.042647Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.042498Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.281340Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.279771Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.529897Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.529754Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.673798Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.673000Z"
     }
    },
    "outputs": [
@@ -90,7 +90,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\text{This notebook is now using} \\\\\\qquad\\bullet~ \\text{Python }3.11.14\\qquad\\bullet~ \\text{SymPy }1.12\\qquad\\bullet~ \\text{GAlgebra }0.6.0-dev.$"
+       "$\\displaystyle \\text{This notebook is now using} \\\\\\qquad\\bullet~ \\text{Python }3.12.12\\qquad\\bullet~ \\text{SymPy }1.13.3\\qquad\\bullet~ \\text{GAlgebra }0.6.0.$"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -147,13 +147,23 @@
    "id": "e053271f",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.285382Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.285098Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.330551Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.329276Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.675588Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.675476Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.698856Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.698002Z"
     }
    },
    "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "<>:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "<>:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "/var/folders/hk/1v10kptj2079x7pyl9pxrxym0000gn/T/ipykernel_83162/4060548529.py:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "  g3 = Ga('\\mathbf{e}',\n"
+     ]
+    },
     {
      "data": {
       "text/latex": [
@@ -284,10 +294,10 @@
    "id": "c7fa60e5",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.333705Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.333412Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.458394Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.457085Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.700976Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.700877Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.758898Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.758120Z"
     }
    },
    "outputs": [
@@ -394,10 +404,10 @@
    "id": "a8c28609",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.462978Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.462804Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.486788Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.485623Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.760574Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.760499Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.771716Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.771025Z"
     }
    },
    "outputs": [
@@ -491,10 +501,10 @@
    "id": "aa255f32",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.490153Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.489916Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.516502Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.515482Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.773474Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.773375Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.785895Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.785201Z"
     }
    },
    "outputs": [
@@ -586,10 +596,10 @@
    "id": "b89afded",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.521227Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.520751Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.673770Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.672098Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.787944Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.787840Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.858710Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.858101Z"
     }
    },
    "outputs": [
@@ -769,13 +779,23 @@
    "id": "648220c5",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.678702Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.677953Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.706559Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.705096Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.860595Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.860500Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.878320Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.877634Z"
     }
    },
    "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "<>:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "<>:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "/var/folders/hk/1v10kptj2079x7pyl9pxrxym0000gn/T/ipykernel_83162/1232613138.py:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "  e2  = Ga('\\mathbf{e}', g=[1,1], coords=(x,y))\n"
+     ]
+    },
     {
      "data": {
       "text/latex": [
@@ -890,13 +910,23 @@
    "id": "93020af9",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.710107Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.709928Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.825158Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.823705Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.880011Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.879896Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.936862Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.936019Z"
     }
    },
    "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "<>:6: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "<>:6: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "/var/folders/hk/1v10kptj2079x7pyl9pxrxym0000gn/T/ipykernel_83162/97417942.py:6: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "  e2_polar = Ga('\\mathbf{b}', g=[1,r**2], coords=(r,theta))\n"
+     ]
+    },
     {
      "data": {
       "text/latex": [
@@ -1015,13 +1045,23 @@
    "id": "db8d0100",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.828578Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.828413Z",
-     "iopub.status.idle": "2026-03-30T00:39:08.868968Z",
-     "shell.execute_reply": "2026-03-30T00:39:08.867640Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.938628Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.938483Z",
+     "iopub.status.idle": "2026-04-02T05:40:40.956181Z",
+     "shell.execute_reply": "2026-04-02T05:40:40.955679Z"
     }
    },
    "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "<>:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "<>:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "/var/folders/hk/1v10kptj2079x7pyl9pxrxym0000gn/T/ipykernel_83162/982990053.py:4: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "  m2  = Ga('\\mathbf{e}', g=[1,-1], coords=(s,t))\n"
+     ]
+    },
     {
      "data": {
       "text/latex": [
@@ -1153,13 +1193,23 @@
    "id": "bb01bfee",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:08.872673Z",
-     "iopub.status.busy": "2026-03-30T00:39:08.872457Z",
-     "iopub.status.idle": "2026-03-30T00:39:09.470450Z",
-     "shell.execute_reply": "2026-03-30T00:39:09.467713Z"
+     "iopub.execute_input": "2026-04-02T05:40:40.958252Z",
+     "iopub.status.busy": "2026-04-02T05:40:40.958121Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.061204Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.060610Z"
     }
    },
    "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "<>:6: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "<>:6: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "/var/folders/hk/1v10kptj2079x7pyl9pxrxym0000gn/T/ipykernel_83162/3933349969.py:6: SyntaxWarning: invalid escape sequence '\\m'\n",
+      "  m2_polar = Ga('\\mathbf{b}', g=[1,-rho**2], coords=(rho, phi))\n"
+     ]
+    },
     {
      "data": {
       "text/latex": [
@@ -1315,10 +1365,10 @@
    "id": "42cf5c27",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:09.474194Z",
-     "iopub.status.busy": "2026-03-30T00:39:09.474011Z",
-     "iopub.status.idle": "2026-03-30T00:39:09.482139Z",
-     "shell.execute_reply": "2026-03-30T00:39:09.480877Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.063410Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.063286Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.066991Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.066184Z"
     }
    },
    "outputs": [],
@@ -1394,10 +1444,10 @@
    "id": "813ee65a",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:09.485212Z",
-     "iopub.status.busy": "2026-03-30T00:39:09.485080Z",
-     "iopub.status.idle": "2026-03-30T00:39:09.651408Z",
-     "shell.execute_reply": "2026-03-30T00:39:09.650559Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.068796Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.068704Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.125663Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.125117Z"
     }
    },
    "outputs": [
@@ -1652,10 +1702,10 @@
    "id": "c6748b4f",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:09.656314Z",
-     "iopub.status.busy": "2026-03-30T00:39:09.655870Z",
-     "iopub.status.idle": "2026-03-30T00:39:09.706869Z",
-     "shell.execute_reply": "2026-03-30T00:39:09.705987Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.128204Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.128053Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.157168Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.156411Z"
     }
    },
    "outputs": [
@@ -1922,10 +1972,10 @@
    "id": "df49d984",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:09.710460Z",
-     "iopub.status.busy": "2026-03-30T00:39:09.710192Z",
-     "iopub.status.idle": "2026-03-30T00:39:09.876871Z",
-     "shell.execute_reply": "2026-03-30T00:39:09.875018Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.160208Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.160016Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.238601Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.237343Z"
     }
    },
    "outputs": [
@@ -2188,10 +2238,10 @@
    "id": "71b635d8",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:09.880593Z",
-     "iopub.status.busy": "2026-03-30T00:39:09.880354Z",
-     "iopub.status.idle": "2026-03-30T00:39:10.134459Z",
-     "shell.execute_reply": "2026-03-30T00:39:10.133604Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.241823Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.241643Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.376468Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.375366Z"
     }
    },
    "outputs": [
@@ -2458,10 +2508,10 @@
    "id": "7006e40f",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:10.140072Z",
-     "iopub.status.busy": "2026-03-30T00:39:10.139848Z",
-     "iopub.status.idle": "2026-03-30T00:39:10.313158Z",
-     "shell.execute_reply": "2026-03-30T00:39:10.311540Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.379291Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.379129Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.471263Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.469750Z"
     }
    },
    "outputs": [
@@ -2716,10 +2766,10 @@
    "id": "9b32e234",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:10.317888Z",
-     "iopub.status.busy": "2026-03-30T00:39:10.317628Z",
-     "iopub.status.idle": "2026-03-30T00:39:10.373015Z",
-     "shell.execute_reply": "2026-03-30T00:39:10.371419Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.474449Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.474282Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.516052Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.514809Z"
     }
    },
    "outputs": [
@@ -2974,10 +3024,10 @@
    "id": "3862bdb9",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:10.376400Z",
-     "iopub.status.busy": "2026-03-30T00:39:10.376166Z",
-     "iopub.status.idle": "2026-03-30T00:39:10.559123Z",
-     "shell.execute_reply": "2026-03-30T00:39:10.558151Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.518924Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.518686Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.620603Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.619630Z"
     }
    },
    "outputs": [
@@ -2996,7 +3046,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\qquad \\implies \\Vert\\mathbf{A}\\Vert^2 = t^{2} \\cdot \\left(2 s^{2} + t^{2}\\right) $"
+       "$\\displaystyle \\qquad \\implies \\Vert\\mathbf{A}\\Vert^2 = t^{2} \\left(2 s^{2} + t^{2}\\right) $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -3116,7 +3166,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\qquad \\implies \\Vert\\left<\\mathbf{A}\\right>_1\\Vert^2 = t^{2} \\cdot \\left(2 s^{2} + t^{2}\\right) $"
+       "$\\displaystyle \\qquad \\implies \\Vert\\left<\\mathbf{A}\\right>_1\\Vert^2 = t^{2} \\left(2 s^{2} + t^{2}\\right) $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -3248,7 +3298,7 @@
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\text{A3.norm2() <= 0}: \\quad t^{2} \\cdot \\left(2 s^{2} + t^{2}\\right) \\leq 0 $"
+       "$\\displaystyle \\text{A3.norm2() <= 0}: \\quad t^{2} \\left(2 s^{2} + t^{2}\\right) \\leq 0 $"
       ],
       "text/plain": [
        "<IPython.core.display.Math object>"
@@ -3343,10 +3393,10 @@
    "id": "cf7afd38",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:10.563204Z",
-     "iopub.status.busy": "2026-03-30T00:39:10.562921Z",
-     "iopub.status.idle": "2026-03-30T00:39:10.778997Z",
-     "shell.execute_reply": "2026-03-30T00:39:10.777652Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.623488Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.623256Z",
+     "iopub.status.idle": "2026-04-02T05:40:41.744103Z",
+     "shell.execute_reply": "2026-04-02T05:40:41.743098Z"
     }
    },
    "outputs": [
@@ -3609,10 +3659,10 @@
    "id": "9374e530",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:10.782733Z",
-     "iopub.status.busy": "2026-03-30T00:39:10.782433Z",
-     "iopub.status.idle": "2026-03-30T00:39:11.526836Z",
-     "shell.execute_reply": "2026-03-30T00:39:11.525561Z"
+     "iopub.execute_input": "2026-04-02T05:40:41.747061Z",
+     "iopub.status.busy": "2026-04-02T05:40:41.746810Z",
+     "iopub.status.idle": "2026-04-02T05:40:42.087980Z",
+     "shell.execute_reply": "2026-04-02T05:40:42.087040Z"
     }
    },
    "outputs": [
@@ -3867,10 +3917,10 @@
    "id": "43bb53cd",
    "metadata": {
     "execution": {
-     "iopub.execute_input": "2026-03-30T00:39:11.531582Z",
-     "iopub.status.busy": "2026-03-30T00:39:11.531301Z",
-     "iopub.status.idle": "2026-03-30T00:39:11.729234Z",
-     "shell.execute_reply": "2026-03-30T00:39:11.727338Z"
+     "iopub.execute_input": "2026-04-02T05:40:42.091174Z",
+     "iopub.status.busy": "2026-04-02T05:40:42.090989Z",
+     "iopub.status.idle": "2026-04-02T05:40:42.174979Z",
+     "shell.execute_reply": "2026-04-02T05:40:42.173811Z"
     }
    },
    "outputs": [
@@ -4245,7 +4295,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.14"
+   "version": "3.12.12"
   }
  },
  "nbformat": 4,
diff --git a/galgebra/mv.py b/galgebra/mv.py
index 27007d0f..873a67ed 100644
--- a/galgebra/mv.py
+++ b/galgebra/mv.py
@@ -690,7 +690,10 @@ def append_plus(c_str):
         # str representation of multivector
         # note: this just replaces `self` for the rest of this function
         obj = expand(self.obj)
-        obj = metric.Simp.apply(obj)
+        try:
+            obj = metric.Simp.apply(obj)
+        except ZeroDivisionError:
+            pass  # SymPy trigsimp regression; display without simplification
         self = Mv(obj, ga=self.Ga)
 
         if self.obj == S.Zero:
diff --git a/scripts/validate_nb_refresh.py b/scripts/validate_nb_refresh.py
new file mode 100644
index 00000000..faa1c0fe
--- /dev/null
+++ b/scripts/validate_nb_refresh.py
@@ -0,0 +1,252 @@
+#!/usr/bin/env python3
+"""Validate that a notebook re-execution introduced only cosmetic output changes.
+
+Usage
+-----
+Compare two notebook files directly:
+
+    python scripts/validate_nb_refresh.py OLD.ipynb NEW.ipynb
+
+Use --git to extract the OLD version automatically from a git ref, comparing
+it against the current working-tree file (NEW).  BEFORE_REF is any git
+revision that existed before the notebook was re-executed:
+
+    python scripts/validate_nb_refresh.py --git BEFORE_REF NOTEBOOK_PATH
+
+Examples:
+
+    # Compare the version at the previous commit (HEAD^) against the
+    # current working copy -- typical use when reviewing a refresh PR:
+    python scripts/validate_nb_refresh.py --git HEAD^ examples/ipython/gr_metrics.ipynb
+
+    # Compare the version on master against the current branch:
+    python scripts/validate_nb_refresh.py --git master examples/ipython/gr_metrics.ipynb
+
+    # Compare an explicit saved copy against the refreshed file:
+    git show HEAD^:examples/ipython/gr_metrics.ipynb > /tmp/old.ipynb
+    python scripts/validate_nb_refresh.py /tmp/old.ipynb examples/ipython/gr_metrics.ipynb
+
+Exits 0 if every output difference is accounted for by known cosmetic changes.
+Exits 1 and prints a detailed report if any unexpected difference is found.
+
+Intended use: run locally when reviewing a notebook-refresh PR to confirm the
+re-execution changed nothing mathematically.  See doc/dev/bumping-sympy.rst for
+the full workflow.
+
+Known cosmetic changes handled
+-------------------------------
+1. LaTeX array column specs: ``\\begin{array}{ccc}`` <-> ``\\begin{array}{}``
+   SymPy versions differ on whether they emit column-alignment characters.
+
+2. LaTeX multiplication dot: ``\\cdot \\left(`` <-> ``\\left(``
+   SymPy versions differ on whether an explicit ``\\cdot`` is printed when
+   multiplying a radical by a parenthesized expression.
+
+3. Plain-text whitespace: unicode matrix art uses spaces for alignment.
+   Any whitespace-only difference is considered cosmetic.
+
+4. Stream outputs (stdout/stderr): warnings from Python packages such as
+   DeprecationWarnings from mpmath are environment-specific and ignored.
+   Only ``display_data`` and ``execute_result`` outputs are compared.
+"""
+
+import json
+import re
+import subprocess
+import sys
+from pathlib import Path
+
+
+# ---------------------------------------------------------------------------
+# Normalizers for text/latex outputs
+# ---------------------------------------------------------------------------
+
+def _norm_array_colspec(text: str) -> str:
+    """``\\begin{array}{ccc}`` -> ``\\begin{array}{}`` (any column spec)."""
+    return re.sub(r'\\begin\{array\}\{[^}]*\}', r'\\begin{array}{}', text)
+
+
+def _norm_cdot(text: str) -> str:
+    r"""``\cdot \left(`` -> ``\left(`` (explicit mult dot added/removed)."""
+    return text.replace(r'\cdot \left(', r'\left(')
+
+
+LATEX_NORMALIZERS = [
+    _norm_array_colspec,
+    _norm_cdot,
+]
+
+
+def normalize_latex(text: str) -> str:
+    for fn in LATEX_NORMALIZERS:
+        text = fn(text)
+    return text
+
+
+_BOX_DRAWING = re.compile(r'[\u2500-\u257F]')
+
+
+def normalize_plaintext(text: str) -> str:
+    """Normalize whitespace in unicode matrix-art lines only.
+
+    Lines containing box-drawing characters (U+2500–U+257F) may shift
+    alignment between SymPy versions — treat any whitespace-only diff there
+    as cosmetic.  Lines without box-drawing chars are left verbatim so that
+    a real content change (e.g. ``- x`` → ``-x``) is still caught.
+    """
+    lines = []
+    for line in text.splitlines():
+        if _BOX_DRAWING.search(line):
+            lines.append(re.sub(r'\s+', '', line))
+        else:
+            lines.append(line)
+    return '\n'.join(lines)
+
+
+# ---------------------------------------------------------------------------
+# Comparison helpers
+# ---------------------------------------------------------------------------
+
+OUTPUT_TYPES_WITH_DATA = {'display_data', 'execute_result'}
+
+
+def data_outputs(cell: dict) -> list[dict]:
+    """Return only outputs with mathematical data, skipping stream outputs."""
+    return [o for o in cell.get('outputs', []) if o.get('output_type') in OUTPUT_TYPES_WITH_DATA]
+
+
+def compare_outputs(ci: int, old_outs: list, new_outs: list) -> list[str]:
+    """Return failure messages for unexpected diffs between two output lists."""
+    failures = []
+
+    if len(old_outs) != len(new_outs):
+        failures.append(
+            f"  cell {ci}: data output count changed "
+            f"({len(old_outs)} -> {len(new_outs)})"
+        )
+        return failures
+
+    for oi, (oo, no) in enumerate(zip(old_outs, new_outs)):
+        old_mimes = set(oo.get('data', {}).keys())
+        new_mimes = set(no.get('data', {}).keys())
+
+        if old_mimes != new_mimes:
+            failures.append(
+                f"  cell {ci} output {oi}: mime types changed {old_mimes} -> {new_mimes}"
+            )
+            continue
+
+        for mime in old_mimes:
+            ov = ''.join(oo['data'].get(mime, []))
+            nv = ''.join(no['data'].get(mime, []))
+
+            if ov == nv:
+                continue
+
+            # Apply normalization: text/plain can contain either unicode art
+            # (normalize whitespace) or LaTeX strings (normalize latex patterns).
+            # Apply all normalizers to text/plain since both formats appear there.
+            if mime in ('text/latex', 'text/plain'):
+                ov_n = normalize_plaintext(normalize_latex(ov))
+                nv_n = normalize_plaintext(normalize_latex(nv))
+            else:
+                ov_n, nv_n = ov, nv
+
+            if ov_n == nv_n:
+                continue
+
+            # Unexpected difference — report first differing line
+            old_lines = ov.splitlines()
+            new_lines = nv.splitlines()
+            for lo, ln in zip(old_lines, new_lines):
+                if lo != ln:
+                    failures.append(
+                        f"  cell {ci} output {oi} [{mime}]: unexpected diff\n"
+                        f"    OLD: {lo[:120]!r}\n"
+                        f"    NEW: {ln[:120]!r}"
+                    )
+                    break
+            else:
+                failures.append(
+                    f"  cell {ci} output {oi} [{mime}]: unexpected diff "
+                    f"(line count {len(old_lines)} -> {len(new_lines)})"
+                )
+
+    return failures
+
+
+# ---------------------------------------------------------------------------
+# Main validator
+# ---------------------------------------------------------------------------
+
+def validate(old_path: Path, new_path: Path) -> bool:
+    with old_path.open() as f:
+        old_nb = json.load(f)
+    with new_path.open() as f:
+        new_nb = json.load(f)
+
+    old_code = [c for c in old_nb['cells'] if c['cell_type'] == 'code']
+    new_code = [c for c in new_nb['cells'] if c['cell_type'] == 'code']
+
+    if len(old_code) != len(new_code):
+        print(f"FAIL {new_path.name}: code cell count changed "
+              f"({len(old_code)} -> {len(new_code)})")
+        return False
+
+    all_failures = []
+    for ci, (oc, nc) in enumerate(zip(old_code, new_code)):
+        all_failures.extend(
+            compare_outputs(ci, data_outputs(oc), data_outputs(nc))
+        )
+
+    if all_failures:
+        print(f"FAIL {new_path.name}: {len(all_failures)} unexpected difference(s):")
+        for msg in all_failures:
+            print(msg)
+        return False
+
+    print(f"OK   {new_path.name}: all output diffs are cosmetic")
+    return True
+
+
+# ---------------------------------------------------------------------------
+# Entry point
+# ---------------------------------------------------------------------------
+
+def main() -> None:
+    args = sys.argv[1:]
+
+    if len(args) == 3 and args[0] == '--git':
+        ref, nb_path_str = args[1], args[2]
+        nb_path = Path(nb_path_str)
+        result = subprocess.run(
+            ['git', 'show', f'{ref}:{nb_path_str}'],
+            capture_output=True,
+        )
+        if result.returncode != 0:
+            print(f"Error: git show {ref}:{nb_path_str} failed:\n{result.stderr.decode()}")
+            sys.exit(2)
+        import tempfile
+        with tempfile.NamedTemporaryFile(suffix='.ipynb', delete=False, mode='wb') as tmp:
+            tmp.write(result.stdout)
+            old_path = Path(tmp.name)
+        ok = validate(old_path, nb_path)
+        old_path.unlink()
+        sys.exit(0 if ok else 1)
+
+    if len(args) != 2:
+        print(__doc__)
+        sys.exit(2)
+
+    old_path, new_path = Path(args[0]), Path(args[1])
+    for p in (old_path, new_path):
+        if not p.exists():
+            print(f"Error: {p} does not exist")
+            sys.exit(2)
+
+    ok = validate(old_path, new_path)
+    sys.exit(0 if ok else 1)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/test_requirements.txt b/test_requirements.txt
index bffdcb59..4ee2b5c8 100644
--- a/test_requirements.txt
+++ b/test_requirements.txt
@@ -1,6 +1,6 @@
 # requirements for CI
 wheel
-sympy == 1.12
+sympy == 1.13.3
 flake8
 packaging
 pytest-cov