diff --git a/.github/workflows/python-app.yml b/.github/workflows/python-app.yml
index 4bf2ff8..72e811a 100644
--- a/.github/workflows/python-app.yml
+++ b/.github/workflows/python-app.yml
@@ -27,7 +27,7 @@ jobs:
       run: |
         sudo apt-get install freeglut3-dev
         python -m pip install --upgrade pip
-        pip install point-cloud-registration
+        pip install -e . pykdtree
         pip install q3dviewer==1.1.6
         pip install pytest
     - name: Test with pytest
diff --git a/README.md b/README.md
index 57572b8..b1800f7 100644
--- a/README.md
+++ b/README.md
@@ -71,6 +71,7 @@ print("Estimated Transform matrix:\n", T_new)
 - [x] **Point-to-Plane ICP** – Improved accuracy using normal constraints  
 - [ ] **Generalized ICP (GICP)** – Handles anisotropic noise and improves robustness  
 - [x] **Normal Distributions Transform (NDT)** – Grid-based registration for high-noise environments  
+- [x] **Degeneracy detection + solution remapping** – Hold unobservable DOFs instead of drifting along them  
 - [ ] **Further optimizations** while staying pure Python  
 ### Demo
 
@@ -106,6 +107,38 @@ python3 demo_matching.py
 
 ![demo](imgs/demo.png)
 
+#### Degeneracy Detection & Solution Remapping
+
+Real-world geometry often under-constrains registration — a featureless
+seafloor, a straight corridor, a staircase. On such scenes the
+unconstrained Gauss-Newton solve slides the estimate along the
+unobservable directions, driven by nothing but noise. `align()` can
+instead eigendecompose its 6×6 Hessian, flag the degenerate directions,
+and hold them at the initial guess while the constrained ones converge
+(solution remapping: Zhang, Kaess & Singh, ICRA 2016; Hinduja, Ho &
+Kaess, IROS 2019):
+
+```python
+icp = PlaneICP(max_iter=50, max_dist=1.0)
+icp.set_target(target)
+T = icp.align(scan, use_solution_remapping=True, lm_damping=True)
+print(icp.last_hessian)  # inspect the observability yourself
+```
+
+```bash
+python3 demo_degeneracy.py --save   # writes imgs/degeneracy_*.png (needs matplotlib)
+```
+
+![Degeneracy demo](imgs/degeneracy_stair.png)
+
+On the synthetic staircase in `data/` only cross-step translation is
+unobservable: plain plane-ICP drifts centimetres along it, solution
+remapping holds it near the initial value while the other five DOFs
+still converge. On a flat plane three DOFs (tx, ty, yaw) are degenerate
+and the effect is an order of magnitude larger. matplotlib is needed
+only by this demo — the feature itself adds no dependencies to the
+library.
+
 ### Comparison of Registration Methods
 
 | Method                        | Objective Function*                                         | Data Representation   | Speed         | Precision    |
diff --git a/data/README.md b/data/README.md
index 7c7ae3d..7cd61fd 100644
--- a/data/README.md
+++ b/data/README.md
@@ -1,4 +1,6 @@
-# License Information for B-01.pcd
+# Data
+
+## B-01.pcd
 
 The file `B-01.pcd` is used to test our algorithms and is sourced from the following dataset:
 
@@ -21,4 +23,24 @@ Under the terms of the license, you must:
 2. **Provide a link to the license**: [CC BY 4.0](https://opendefinition.org/licenses/cc-by/).
 3. **Indicate changes**: If any modifications were made to the original data.
 
-For more details, visit the [license page](https://opendefinition.org/licenses/cc-by/).
\ No newline at end of file
+For more details, visit the [license page](https://opendefinition.org/licenses/cc-by/).
+## Synthetic degeneracy clouds
+
+`synthetic_staircase_{target,source}.pcd` and
+`synthetic_plane_{target,source}.pcd` are generated by
+[`generate_synthetic.py`](generate_synthetic.py) with fixed seeds —
+regeneration is byte-identical:
+
+```bash
+python3 data/generate_synthetic.py
+```
+
+They are used by `demo_degeneracy.py` and by the test suite
+(`tests/test_synthetic_data.py`). The staircase constrains every DOF
+except cross-step translation (one degenerate direction); the flat
+plane leaves tx, ty and yaw unobservable (three degenerate directions).
+Target and source are independent samplings of the same surface, so
+registration sees realistic correspondence noise.
+
+This is original synthetic data, licensed with the repository (MIT).
+No attribution requirements.
diff --git a/data/generate_synthetic.py b/data/generate_synthetic.py
new file mode 100644
index 0000000..c8a5bb8
--- /dev/null
+++ b/data/generate_synthetic.py
@@ -0,0 +1,158 @@
+#!/usr/bin/env python3
+"""
+Deterministic generators for the synthetic degenerate test clouds.
+
+Two scenes, each under-constraining registration in a known way:
+
+* staircase — treads (normals +z) and risers (normals +x) constrain
+  tx, tz and all three rotations, but nothing observes translation
+  across the stair width: exactly one degenerate DOF (ty).
+* plane — a featureless flat seafloor. Only tz, roll and pitch are
+  observable: three degenerate DOFs (tx, ty, yaw).
+
+Regeneration is byte-identical (fixed seeds, fixed format string), so
+the committed .pcd files can always be reproduced with
+
+    python3 data/generate_synthetic.py
+
+Design notes that carry the demo's story:
+
+* Faces keep a margin (~ the k-NN neighborhood radius) away from the
+  concave tread/riser folds. Without it, PCA normal estimation blends
+  the two faces at the fold, the blended normals pick up spurious
+  y-components, and the ty direction stops being flagged as degenerate.
+* Centering is deterministic (never the empirical mean), so clouds
+  sampled with different seeds lie on the *same* surface — required for
+  target/source pairs that emulate two scans of one scene.
+
+NumPy only; no external point-cloud dependencies.
+"""
+import argparse
+import os
+
+import numpy as np
+
+
+def make_staircase(n_steps=6, tread=1.0, riser=0.5, width=6.0,
+                   pts_per_face=220, margin=0.10, noise=0.01, seed=42):
+    """
+    Point cloud of a staircase marching along +x and rising in +z.
+
+    Step k contributes a tread (horizontal, normal +z) at height
+    (k+1)*riser and a riser (vertical, normal +x) at x = k*tread.
+    Returns an (N, 3) float64 array centered deterministically.
+    """
+    rng = np.random.default_rng(seed)
+    faces = []
+    for k in range(n_steps):
+        # Tread: z = (k+1)*riser, x in [k*tread+margin, (k+1)*tread-margin].
+        u = rng.uniform(k * tread + margin, (k + 1) * tread - margin,
+                        pts_per_face)
+        v = rng.uniform(0.0, width, pts_per_face)
+        tread_face = np.column_stack([
+            u, v, np.full(pts_per_face, (k + 1) * riser)])
+        # Riser: x = k*tread, z in [k*riser+margin, (k+1)*riser-margin].
+        w = rng.uniform(k * riser + margin, (k + 1) * riser - margin,
+                        pts_per_face)
+        v2 = rng.uniform(0.0, width, pts_per_face)
+        riser_face = np.column_stack([
+            np.full(pts_per_face, k * tread), v2, w])
+        faces.extend([tread_face, riser_face])
+    points = np.vstack(faces)
+    points += rng.normal(0.0, noise, points.shape)
+    center = np.array([n_steps * tread / 2.0, width / 2.0,
+                       n_steps * riser / 2.0])
+    return points - center
+
+
+def make_plane(half_extent=6.0, num_points=2500, noise=0.01, seed=7):
+    """
+    A featureless flat seafloor: uniform samples of z = 0 over
+    [-half_extent, half_extent]^2 with Gaussian surface noise.
+    """
+    rng = np.random.default_rng(seed)
+    xy = rng.uniform(-half_extent, half_extent, (num_points, 2))
+    z = rng.normal(0.0, noise, num_points)
+    return np.column_stack([xy, z])
+
+
+def save_pcd(path, points):
+    """
+    Write an Nx3 array as an ASCII PCD v0.7 file (x y z).
+
+    Coordinates are stored at float32 / six-decimal precision — the PCD
+    FIELDS declare SIZE 4 — so a save/load roundtrip is exact to ~1e-6,
+    not bit-exact against float64 input. Regeneration from the fixed
+    seeds is byte-identical because the quantization itself is
+    deterministic.
+    """
+    points = np.asarray(points, dtype=np.float32)
+    n = points.shape[0]
+    header = "\n".join([
+        "# .PCD v0.7 - Point Cloud Data file format",
+        "VERSION 0.7",
+        "FIELDS x y z",
+        "SIZE 4 4 4",
+        "TYPE F F F",
+        "COUNT 1 1 1",
+        f"WIDTH {n}",
+        "HEIGHT 1",
+        "VIEWPOINT 0 0 0 1 0 0 0",
+        f"POINTS {n}",
+        "DATA ascii",
+    ])
+    with open(path, "w") as f:
+        f.write(header + "\n")
+        for x, y, z in points:
+            f.write(f"{x:.6f} {y:.6f} {z:.6f}\n")
+
+
+def load_pcd(path):
+    """Read an ASCII PCD file with x y z fields into an (N, 3) array."""
+    points = []
+    with open(path) as f:
+        in_data = False
+        for line in f:
+            if in_data:
+                parts = line.split()
+                if len(parts) >= 3:
+                    points.append([float(parts[0]), float(parts[1]),
+                                   float(parts[2])])
+            elif line.startswith("DATA"):
+                if line.split()[1] != "ascii":
+                    raise ValueError(f"{path}: only ASCII PCD is supported")
+                in_data = True
+    return np.array(points)
+
+
+# (filename, generator, kwargs) — target/source pairs are independent
+# samplings of the same surface, so registration sees realistic
+# correspondence noise instead of a permutation of identical points.
+CLOUDS = [
+    ("synthetic_staircase_target.pcd", make_staircase,
+     dict(seed=42, pts_per_face=220)),
+    ("synthetic_staircase_source.pcd", make_staircase,
+     dict(seed=2024, pts_per_face=220)),
+    ("synthetic_plane_target.pcd", make_plane,
+     dict(seed=7, num_points=2500)),
+    ("synthetic_plane_source.pcd", make_plane,
+     dict(seed=99, num_points=2000)),
+]
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__.splitlines()[1])
+    parser.add_argument("--out-dir",
+                        default=os.path.dirname(os.path.abspath(__file__)),
+                        help="Directory to write the .pcd files into "
+                             "(default: this file's directory).")
+    args = parser.parse_args()
+    for name, generator, kwargs in CLOUDS:
+        path = os.path.join(args.out_dir, name)
+        points = generator(**kwargs)
+        save_pcd(path, points)
+        print(f"wrote {path} ({points.shape[0]} points)")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/data/synthetic_plane_source.pcd b/data/synthetic_plane_source.pcd
new file mode 100644
index 0000000..faab0f7
--- /dev/null
+++ b/data/synthetic_plane_source.pcd
@@ -0,0 +1,2011 @@
+# .PCD v0.7 - Point Cloud Data file format
+VERSION 0.7
+FIELDS x y z
+SIZE 4 4 4
+TYPE F F F
+COUNT 1 1 1
+WIDTH 2000
+HEIGHT 1
+VIEWPOINT 0 0 0 1 0 0 0
+POINTS 2000
+DATA ascii
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diff --git a/data/synthetic_plane_target.pcd b/data/synthetic_plane_target.pcd
new file mode 100644
index 0000000..545a563
--- /dev/null
+++ b/data/synthetic_plane_target.pcd
@@ -0,0 +1,2511 @@
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diff --git a/data/synthetic_staircase_target.pcd b/data/synthetic_staircase_target.pcd
new file mode 100644
index 0000000..1f383d7
--- /dev/null
+++ b/data/synthetic_staircase_target.pcd
@@ -0,0 +1,2651 @@
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diff --git a/demo_degeneracy.py b/demo_degeneracy.py
new file mode 100644
index 0000000..168de50
--- /dev/null
+++ b/demo_degeneracy.py
@@ -0,0 +1,310 @@
+#!/usr/bin/env python3
+"""
+Demo: degeneracy detection + solution remapping on under-constrained scenes.
+
+Two synthetic scenes from data/ (regenerated in memory if the .pcd files
+are absent):
+
+* staircase — treads and risers constrain everything except cross-step
+  translation (ty): exactly one degenerate DOF.
+* plane — a featureless flat seafloor: tx, ty and yaw are unobservable,
+  three degenerate DOFs.
+
+For each scene the source scan (an independent sampling of the same
+surface) is displaced by a known ground-truth transform and aligned back
+with PlaneICP twice: once plain (lm_damping only) and once with
+use_solution_remapping=True. The demo prints the DegeneracyResult and a
+per-DOF error table, and renders a 2x2 figure per case:
+
+    A  the scans before alignment (3D)
+    B  top-down view after alignment, with a zoom showing the drift
+    C  the Hessian eigenvalue spectrum vs the SR threshold
+    D  per-DOF |error| vs ground truth, plain vs SR
+
+Without solution remapping the optimizer slides along the unobservable
+directions, driven by nothing but sampling noise; with it, those
+directions hold at the initial guess while the observable ones converge.
+
+Usage:
+    python3 demo_degeneracy.py                     # interactive windows
+    python3 demo_degeneracy.py --save              # write imgs/degeneracy_*.png
+    python3 demo_degeneracy.py --case stair --save
+
+matplotlib is imported by this demo alone (pip install matplotlib); the
+degeneracy feature adds no dependencies to the library.
+"""
+import argparse
+import os
+import sys
+
+import numpy as np
+
+from point_cloud_registration import PlaneICP, analyse_hessian
+from point_cloud_registration.math_tools import expSO3, makeT, transform_points
+
+REPO_DIR = os.path.dirname(os.path.abspath(__file__))
+DEFAULT_DATA_DIR = os.path.join(REPO_DIR, "data")
+
+DOF_LABELS = ["tx", "ty", "tz", "roll", "pitch", "yaw"]
+
+# Okabe-Ito colorblind-safe colors (validated for CVD separation).
+COLORS = {
+    "target": "#999999",    # context layer
+    "initial": "#E69F00",
+    "baseline": "#D55E00",  # no solution remapping
+    "sr": "#0072B2",        # with solution remapping
+}
+
+CASES = {
+    "stair": {
+        "title": "Staircase — 1 degenerate DOF (ty, cross-step)",
+        "target_pcd": "synthetic_staircase_target.pcd",
+        "source_pcd": "synthetic_staircase_source.pcd",
+        "generator": ("make_staircase",
+                      dict(seed=42, pts_per_face=220),
+                      dict(seed=2024, pts_per_face=220)),
+        # Ground truth deliberately has zero component along the degenerate
+        # ty, so the SR-held value is also the correct one.
+        "rvec": np.array([0.02, -0.015, 0.08]),
+        "tvec": np.array([0.30, 0.0, -0.20]),
+        "png": "degeneracy_stair.png",
+    },
+    "plane": {
+        "title": "Flat plane — 3 degenerate DOFs (tx, ty, yaw)",
+        "target_pcd": "synthetic_plane_target.pcd",
+        "source_pcd": "synthetic_plane_source.pcd",
+        "generator": ("make_plane",
+                      dict(seed=7, num_points=2500),
+                      dict(seed=99, num_points=2000)),
+        "rvec": np.array([0.03, -0.02, 0.0]),
+        "tvec": np.array([0.0, 0.0, 0.4]),
+        "png": "degeneracy_plane.png",
+    },
+}
+
+
+def _generator_module(data_dir):
+    sys.path.insert(0, data_dir)
+    try:
+        import generate_synthetic
+    finally:
+        sys.path.pop(0)
+    return generate_synthetic
+
+
+def load_case(name, data_dir):
+    """Return (target, source_surface), from disk or regenerated in memory."""
+    case = CASES[name]
+    gen = _generator_module(data_dir)
+    clouds = []
+    fn_name, target_kwargs, source_kwargs = case["generator"]
+    for pcd, kwargs in ((case["target_pcd"], target_kwargs),
+                        (case["source_pcd"], source_kwargs)):
+        path = os.path.join(data_dir, pcd)
+        if os.path.exists(path):
+            clouds.append(gen.load_pcd(path))
+        else:
+            print(f"[demo] {path} not found - regenerating in memory")
+            clouds.append(getattr(gen, fn_name)(**kwargs))
+    return clouds[0], clouds[1]
+
+
+def pose_error(T_est, T_true):
+    """Per-DOF error [tx, ty, tz, roll, pitch, yaw] of T_est vs T_true."""
+    D = np.linalg.inv(T_true) @ T_est
+    R, t = D[:3, :3], D[:3, 3]
+    rot = 0.5 * np.array([R[2, 1] - R[1, 2],
+                          R[0, 2] - R[2, 0],
+                          R[1, 0] - R[0, 1]])
+    return np.concatenate([t, rot])
+
+
+def dominant_dof(eigenvector):
+    return DOF_LABELS[int(np.argmax(np.abs(eigenvector)))]
+
+
+def run_case(name, data_dir):
+    case = CASES[name]
+    target, source_surface = load_case(name, data_dir)
+    T_true = makeT(expSO3(case["rvec"]), case["tvec"])
+    scan = transform_points(np.linalg.inv(T_true), source_surface)
+    scan32 = scan.astype(np.float32)
+
+    results = {}
+    for mode, kwargs in (
+            ("baseline", dict(lm_damping=True)),
+            ("sr", dict(use_solution_remapping=True, lm_damping=True))):
+        engine = PlaneICP(max_iter=50, max_dist=1.0, tol=1e-6, k=10)
+        engine.set_target(target)
+        T = engine.align(scan32, **kwargs)
+        deg = analyse_hessian(engine.last_hessian.astype(float))
+        results[mode] = {
+            "T": T,
+            "deg": deg,
+            "errors": np.abs(pose_error(T, T_true)),
+            "aligned": transform_points(T, scan),
+        }
+    return {
+        "name": name,
+        "case": case,
+        "target": target,
+        "scan": scan,
+        "T_true": T_true,
+        "results": results,
+    }
+
+
+def print_report(out):
+    case = out["case"]
+    print(f"\n=== {case['title']} ===")
+    deg = out["results"]["sr"]["deg"]
+    print(f"eigenvalues      : {np.array2string(deg.eigenvalues, precision=2)}")
+    print(f"condition number : {deg.condition_number:.1f}")
+    print(f"SR threshold     : {deg.lambda_threshold:.1f}")
+    print(f"constrained DOFs : {deg.num_constrained_dof}/6")
+    flags = ", ".join(
+        f"{dominant_dof(deg.eigenvectors[:, i])} (lam={deg.eigenvalues[i]:.1f})"
+        for i in range(6) if deg.degenerate_mask[i])
+    print(f"degenerate       : {flags or 'none'}")
+
+    print(f"\n{'DOF':<6} {'no SR':>12} {'with SR':>12}")
+    base = out["results"]["baseline"]["errors"]
+    sr = out["results"]["sr"]["errors"]
+    for i, label in enumerate(DOF_LABELS):
+        unit = "m" if i < 3 else "rad"
+        print(f"{label:<6} {base[i]:>10.4f} {unit} {sr[i]:>10.4f} {unit}")
+
+
+def make_figure(out, plt):
+    case = out["case"]
+    target = out["target"]
+    scan = out["scan"]
+    base = out["results"]["baseline"]
+    sr = out["results"]["sr"]
+    deg = sr["deg"]
+
+    fig = plt.figure(figsize=(12.5, 9.5), constrained_layout=True)
+    fig.suptitle(case["title"], fontsize=14)
+
+    # A: before alignment (3D).
+    ax_a = fig.add_subplot(2, 2, 1, projection="3d")
+    step = max(1, target.shape[0] // 1500)
+    ax_a.scatter(*target[::step].T, s=2, c=COLORS["target"], alpha=0.35,
+                 label="target scan")
+    ax_a.scatter(*scan[::step].T, s=2, c=COLORS["initial"], alpha=0.6,
+                 label="source scan, initial pose")
+    ax_a.set_title("A - before alignment")
+    ax_a.set_xlabel("x [m]"); ax_a.set_ylabel("y [m]"); ax_a.set_zlabel("z [m]")
+    ax_a.view_init(elev=10, azim=-88)  # look down the stair width: step profile
+    ax_a.legend(loc="upper left", fontsize=8, markerscale=4)
+
+    # B: after alignment, top-down, with a zoom inset on the drift.
+    ax_b = fig.add_subplot(2, 2, 2)
+    ax_b.scatter(target[:, 0], target[:, 1], s=2, c=COLORS["target"],
+                 alpha=0.3, label="target")
+    ax_b.scatter(base["aligned"][:, 0], base["aligned"][:, 1], s=2,
+                 c=COLORS["baseline"], alpha=0.4, label="aligned, no SR")
+    ax_b.scatter(sr["aligned"][:, 0], sr["aligned"][:, 1], s=2,
+                 c=COLORS["sr"], alpha=0.4, label="aligned, with SR")
+    ax_b.set_title("B - after alignment (top-down)")
+    ax_b.set_xlabel("x [m]"); ax_b.set_ylabel("y [m]")
+    ax_b.set_aspect("equal")
+    ax_b.legend(loc="upper left", fontsize=8, markerscale=4)
+
+    # Zoom anchored to the +x/+y extreme of the target so the lateral
+    # drift of the vermillion cloud is visible at true scale.
+    anchor = target[np.argmax(target[:, 0] + target[:, 1])][:2]
+    half = 0.45
+    axins = ax_b.inset_axes([0.58, 0.05, 0.4, 0.4])
+    for pts, color in ((target, COLORS["target"]),
+                       (base["aligned"], COLORS["baseline"]),
+                       (sr["aligned"], COLORS["sr"])):
+        m = (np.abs(pts[:, 0] - anchor[0]) < half) & \
+            (np.abs(pts[:, 1] - anchor[1]) < half)
+        axins.scatter(pts[m, 0], pts[m, 1], s=4, c=color, alpha=0.5)
+    axins.set_xlim(anchor[0] - half, anchor[0] + half)
+    axins.set_ylim(anchor[1] - half, anchor[1] + half)
+    axins.set_xticks([]); axins.set_yticks([])
+    ax_b.indicate_inset_zoom(axins, edgecolor="black")
+
+    # C: eigenvalue spectrum vs SR threshold.
+    ax_c = fig.add_subplot(2, 2, 3)
+    colors = [COLORS["baseline"] if deg.degenerate_mask[i] else COLORS["sr"]
+              for i in range(6)]
+    eig_floor = max(deg.eigenvalues.min(), 1e-3)
+    ax_c.bar(range(6), np.maximum(deg.eigenvalues, eig_floor), color=colors,
+             width=0.6)
+    ax_c.axhline(deg.lambda_threshold, color="black", linestyle="--",
+                 linewidth=1.2)
+    ax_c.text(0.02, deg.lambda_threshold * 1.15,
+              "SR threshold  sqrt(lam_max/lam_min)", fontsize=8)
+    ax_c.set_yscale("log")
+    ax_c.set_xticks(range(6))
+    ax_c.set_xticklabels(
+        [dominant_dof(deg.eigenvectors[:, i]) for i in range(6)], fontsize=9)
+    ax_c.set_xlabel("eigen-direction (dominant DOF)")
+    ax_c.set_ylabel("eigenvalue")
+    ax_c.set_title(
+        f"C - Hessian spectrum - {deg.num_constrained_dof}/6 constrained "
+        f"(degenerate in vermillion)")
+
+    # D: per-DOF error vs ground truth.
+    ax_d = fig.add_subplot(2, 2, 4)
+    x = np.arange(6)
+    err_floor = 1e-6
+    ax_d.bar(x - 0.18, np.maximum(base["errors"], err_floor), width=0.36,
+             color=COLORS["baseline"], label="no SR")
+    ax_d.bar(x + 0.18, np.maximum(sr["errors"], err_floor), width=0.36,
+             color=COLORS["sr"], label="with SR")
+    ax_d.set_yscale("log")
+    ax_d.set_xticks(x)
+    ax_d.set_xticklabels(DOF_LABELS, fontsize=9)
+    ax_d.set_ylabel("|error| vs ground truth  [m or rad]")
+    ax_d.set_title("D - final pose error per DOF")
+    ax_d.legend(fontsize=8)
+
+    return fig
+
+
+def main(argv=None):
+    parser = argparse.ArgumentParser(
+        description="Degeneracy detection + solution remapping demo.")
+    parser.add_argument("--case", choices=["stair", "plane", "both"],
+                        default="both")
+    parser.add_argument("--save", action="store_true",
+                        help="write imgs/degeneracy_*.png instead of showing")
+    parser.add_argument("--show", action="store_true",
+                        help="open interactive windows (default)")
+    parser.add_argument("--data-dir", default=DEFAULT_DATA_DIR)
+    args = parser.parse_args(argv)
+
+    names = ["stair", "plane"] if args.case == "both" else [args.case]
+    outputs = [run_case(name, args.data_dir) for name in names]
+    for out in outputs:
+        print_report(out)
+
+    try:
+        if args.save and not args.show:
+            # Select a headless backend; must happen before pyplot loads.
+            import matplotlib
+            matplotlib.use("Agg")
+        import matplotlib.pyplot as plt
+    except ImportError:
+        print("\n[demo] matplotlib is not installed - skipping the figures "
+              "(pip install matplotlib)")
+        return 0
+
+    for out in outputs:
+        fig = make_figure(out, plt)
+        if args.save:
+            os.makedirs(os.path.join(REPO_DIR, "imgs"), exist_ok=True)
+            path = os.path.join(REPO_DIR, "imgs", out["case"]["png"])
+            fig.savefig(path, dpi=110)
+            print(f"[demo] wrote {path}")
+    if args.show or not args.save:
+        plt.show()
+    return 0
+
+
+if __name__ == "__main__":
+    raise SystemExit(main())
diff --git a/imgs/degeneracy_plane.png b/imgs/degeneracy_plane.png
new file mode 100644
index 0000000..265a9e3
Binary files /dev/null and b/imgs/degeneracy_plane.png differ
diff --git a/imgs/degeneracy_stair.png b/imgs/degeneracy_stair.png
new file mode 100644
index 0000000..b3b836e
Binary files /dev/null and b/imgs/degeneracy_stair.png differ
diff --git a/point_cloud_registration/__init__.py b/point_cloud_registration/__init__.py
index a7571e5..3cf0209 100644
--- a/point_cloud_registration/__init__.py
+++ b/point_cloud_registration/__init__.py
@@ -8,3 +8,4 @@
 from point_cloud_registration.voxel import VoxelGrid, voxel_filter, color_by_voxel
 from point_cloud_registration.estimate_normals import estimate_normals, get_norm_lines, estimate_norm_with_tree
 from point_cloud_registration.caratheodory import fast_caratheodory, create_gn_set
+from point_cloud_registration.degeneracy import DegeneracyResult, analyse_hessian, apply_sr_solve
diff --git a/point_cloud_registration/degeneracy.py b/point_cloud_registration/degeneracy.py
new file mode 100644
index 0000000..ec4e4da
--- /dev/null
+++ b/point_cloud_registration/degeneracy.py
@@ -0,0 +1,273 @@
+"""
+Degeneracy detection and solution remapping for point cloud registration.
+
+Least-squares registration (ICP and its variants) takes a Gauss-Newton step
+``dx = -H⁻¹ g`` at every iteration, where ``H`` is the 6×6 approximate Hessian
+(``JᵀJ``) accumulated over the correspondences and ``g`` is the gradient.  When
+the scene is geometrically under-determined — a flat plane, a straight
+corridor, a surface of revolution — ``H`` is rank-deficient or ill-conditioned
+along one or more directions, and the unconstrained solve slides the estimate
+along those directions driven by nothing but noise.
+
+This module implements *solution remapping*: eigendecompose ``H``, classify each
+eigen-direction as constrained or degenerate, and project the update so that its
+component along every degenerate direction is zero.  The estimate is then only
+updated where the data actually supports it, and stays put elsewhere.
+
+Scope: :func:`analyse_hessian` *classifies* directions and copes with an
+exactly rank-deficient ``H``; :func:`apply_sr_solve` still inverts ``H`` to form
+the step, so it requires an invertible one.  Damping a singular Hessian is the
+caller's responsibility — see :func:`apply_sr_solve` for the contract.
+
+References:
+    A. Hinduja, B.-J. Ho and M. Kaess, "Degeneracy-Aware Factors with
+    Applications to Underwater SLAM", IEEE/RSJ International Conference on
+    Intelligent Robots and Systems (IROS), 2019, pp. 1293-1299,
+    doi: 10.1109/IROS40897.2019.8968577 — introduced solution remapping
+    inside the ICP iteration itself (its Algorithm 1 is the procedure
+    implemented by :func:`analyse_hessian` + :func:`apply_sr_solve`,
+    including the ``sqrt(lambda_max / lambda_min)`` threshold).
+
+    J. Zhang, M. Kaess and S. Singh, "On Degeneracy of Optimization-based
+    State Estimation Problems", IEEE International Conference on Robotics
+    and Automation (ICRA), 2016 — the origin of the solution-remapping
+    update for optimization-based state estimation.
+
+DOF order
+---------
+All 6-vectors and 6×6 matrices here use this library's tangent-space ordering::
+
+    [tx, ty, tz, ωx, ωy, ωz]
+
+i.e. translation first, rotation second — the ordering consumed by
+``math_tools.plus()`` and produced by the ``calc_H_g_e2()`` methods of the
+registration classes.  These are right-tangent (body-frame) increments: the
+step is applied as ``T @ [expSO3(dx[3:]) | dx[:3]]``, so the eigenvectors of
+``H`` — and therefore the directions this module classifies and zeroes — are
+body-frame directions of the current estimate.  Callers that work in a
+different convention (for example a factor-graph library that orders rotation
+first, or reasons in world-frame axes) must permute or rotate ``H`` and ``g``
+before calling and map the returned step back afterwards.
+
+This module depends on NumPy only.
+"""
+from __future__ import annotations
+
+from dataclasses import dataclass
+from typing import Optional
+
+import numpy as np
+
+
+@dataclass
+class DegeneracyResult:
+    """
+    Output of analyse_hessian().
+
+    Attributes
+    ----------
+    eigenvalues:
+        Ascending eigenvalues of H.
+    eigenvectors:
+        Corresponding eigenvectors as *columns*, so ``eigenvectors[:, i]`` is the
+        unit direction whose curvature is ``eigenvalues[i]``.
+    condition_number:
+        ``sqrt(λ_max / λ_min)``; also the adaptive degeneracy threshold when no
+        explicit threshold was supplied.  ``inf`` for a rank-zero Hessian.
+    lambda_threshold:
+        The eigenvalue threshold actually applied.
+    degenerate_mask:
+        ``True`` where the direction is degenerate (data does not constrain it).
+    num_constrained_dof:
+        Number of directions the data does constrain, 0-6.
+    is_degenerate:
+        ``True`` if at least one direction is degenerate.
+    V_constrained:
+        The constrained eigenvectors as unit-norm *rows*.
+    """
+    eigenvalues: np.ndarray       # (6,) ascending, in PCR order
+    eigenvectors: np.ndarray      # (6,6) columns are eigenvectors, in PCR order
+    condition_number: float       # sqrt(λ_max / λ_min)
+    lambda_threshold: float       # eigenvalue threshold used
+    degenerate_mask: np.ndarray   # (6,) bool, True = degenerate direction
+    num_constrained_dof: int      # count of non-degenerate directions
+    is_degenerate: bool           # True if any DOF is degenerate
+    # Non-degenerate eigenvector rows in PCR order, shape (num_constrained_dof, 6).
+    # Empty (0, 6) array when all directions are degenerate.
+    V_constrained: np.ndarray
+
+
+def analyse_hessian(
+    H: np.ndarray,
+    lambda_threshold: Optional[float] = None,
+) -> DegeneracyResult:
+    """
+    Eigendecompose a 6×6 registration Hessian (ATA) and identify degenerate
+    directions.
+
+    H must be in this library's DOF order: [tx, ty, tz, ωx, ωy, ωz].
+
+    Criterion
+    ---------
+    A direction is degenerate when its eigenvalue falls below the threshold
+
+        lambda_cn = sqrt(lambda_max / lambda_min)
+
+    the square root of the Hessian's condition number.  Note what that compares:
+    an eigenvalue, which carries the scale of the problem (correspondence count,
+    units of the residual), against a *unitless* ratio.  Classification therefore
+    depends on the absolute magnitude of H and not on its conditioning alone —
+    scaling H uniformly moves the eigenvalues but leaves lambda_cn untouched.
+    ``H = 2*I`` and ``H = 0.5*I`` are both perfectly conditioned (lambda_cn = 1),
+    yet the first has all six directions constrained and the second has all six
+    degenerate.  Concretely, the best-observed direction survives only when
+    ``lambda_max >= sqrt(lambda_max / lambda_min)``, i.e. when
+    ``lambda_max * lambda_min >= 1``.
+
+    This is the criterion as published and as implemented in the reference
+    solution-remapping code, and it is kept here unchanged for fidelity to it.
+    Callers who have a calibrated absolute curvature floor for their sensor and
+    geometry should use the lambda_threshold override instead of relying on it.
+
+    Structural zeros are handled in two phases so that exact rank deficiency does
+    not swamp the test:
+
+    1. Eigenvalues below ``1e-10 * lambda_max`` are treated as structural zeros
+       (a genuinely unobserved direction: too few correspondences, or a scene
+       with an exact continuous symmetry).  They are always degenerate.
+    2. lambda_cn is then formed from the smallest *non*-structural-zero
+       eigenvalue.  Without this, a single exact-zero eigenvalue would send the
+       condition number to infinity and mark every direction — including the
+       well-observed ones — degenerate.
+
+    A Hessian whose largest eigenvalue is itself below ``1e-10`` carries no
+    information at all (e.g. zero correspondences); every direction is reported
+    degenerate and ``V_constrained`` is an empty ``(0, 6)`` array.
+
+    This classification is well defined for a singular H, but classifying is not
+    solving: :func:`apply_sr_solve` inverts H and will raise on an exactly
+    singular one.  See its docstring for how to pair the two.
+
+    If lambda_threshold is not None it overrides the adaptive threshold with a
+    fixed absolute eigenvalue floor (useful when a caller has a calibrated
+    minimum curvature for its sensor and geometry).  The reported condition
+    number is still computed, with lambda_min floored at ``1e-12`` to keep it
+    finite.
+    """
+    H = np.asarray(H, dtype=float)
+    assert H.shape == (6, 6), f"Expected 6×6 Hessian, got {H.shape}"
+
+    eigenvalues, eigenvectors = np.linalg.eigh(H)  # ascending order
+    lam_max = float(eigenvalues[-1])
+
+    # Zero or near-zero Hessian → no correspondences → all directions degenerate.
+    if lam_max < 1e-10:
+        return DegeneracyResult(
+            eigenvalues=eigenvalues,
+            eigenvectors=eigenvectors,
+            condition_number=float('inf'),
+            lambda_threshold=float('inf'),
+            degenerate_mask=np.ones(6, dtype=bool),
+            num_constrained_dof=0,
+            is_degenerate=True,
+            V_constrained=np.empty((0, 6)),
+        )
+
+    if lambda_threshold is not None:
+        # User-supplied override (e.g. from SR solve fixed threshold).
+        lam_min = float(max(eigenvalues[0], 1e-12))
+        condition_number = float(np.sqrt(lam_max / lam_min))
+        threshold = float(lambda_threshold)
+        degenerate_mask = eigenvalues < threshold
+    else:
+        # Two-phase: structural zeros first, then the condition-number test
+        # among the non-zeros.  Rank-deficient Hessians (flat surface,
+        # single-point) have exact-zero eigenvalues that must not inflate the
+        # condition number to infinity.
+        abs_zero_thresh = 1e-10 * lam_max
+        struct_zero_mask = eigenvalues < abs_zero_thresh
+        if struct_zero_mask.all():
+            condition_number = float('inf')
+            threshold = float('inf')
+            degenerate_mask = np.ones(6, dtype=bool)
+        else:
+            lam_min_pos = float(eigenvalues[~struct_zero_mask][0])
+            condition_number = float(np.sqrt(lam_max / lam_min_pos))
+            threshold = condition_number
+            # Structural zeros are degenerate no matter where the adaptive
+            # threshold lands: with a tight non-zero spectrum the threshold
+            # can fall below the structural-zero cutoff (e.g. five equal
+            # large eigenvalues give threshold 1), and the phase-1 verdict
+            # must not be overwritten by the phase-2 test.
+            degenerate_mask = (eigenvalues < threshold) | struct_zero_mask
+
+    constrained_indices = np.where(~degenerate_mask)[0]
+    num_constrained = int(len(constrained_indices))
+
+    # V_constrained: rows are non-degenerate eigenvectors in PCR order
+    V_constrained = eigenvectors[:, constrained_indices].T  # (num_constrained, 6)
+
+    return DegeneracyResult(
+        eigenvalues=eigenvalues,
+        eigenvectors=eigenvectors,
+        condition_number=condition_number,
+        lambda_threshold=threshold,
+        degenerate_mask=degenerate_mask,
+        num_constrained_dof=num_constrained,
+        is_degenerate=bool(np.any(degenerate_mask)),
+        V_constrained=V_constrained,
+    )
+
+
+def apply_sr_solve(
+    H: np.ndarray,
+    g: np.ndarray,
+    deg: DegeneracyResult,
+) -> np.ndarray:
+    """
+    Apply solution-remapping to the registration linear solve.
+
+    Standard solve:  dx = -H⁻¹ g
+    SR solve:        dx = -Vf_inv @ Vu_filtered @ H⁻¹ @ g
+
+    ``Vu_filtered`` is the matrix of eigenvector rows with the degenerate rows
+    replaced by zero, and ``Vf_inv`` maps back from the eigenbasis to the DOF
+    basis.  The composition projects the Gauss-Newton step onto the span of the
+    constrained eigen-directions: the optimizer still takes the full step where
+    the data supports it, and no step at all along the ill-conditioned DOFs.
+
+    When nothing is degenerate this reduces exactly to the standard solve.
+
+    Contract: H must be invertible.  Detecting degeneracy does not remove the
+    inversion — ``np.linalg.solve(H, g)`` runs on the H you pass, before any
+    projection, so an exactly singular H raises ``np.linalg.LinAlgError`` no
+    matter what ``deg`` says about it.  A caller whose raw Hessian can be exactly
+    rank-deficient (a plane fitted to coplanar points, fewer correspondences than
+    DOFs) should pass a regularized ``H + lambda*I`` here while taking ``deg``
+    from :func:`analyse_hessian` on the **raw** H.  An isotropic shift adds
+    lambda to every eigenvalue and leaves the eigenvectors untouched, so the
+    projection this function applies is unchanged; only the step length along the
+    constrained directions is damped, and analysing the raw H keeps the
+    degeneracy classification honest (damping before analysis raises lambda_min
+    and so lowers the threshold; for a large enough lambda the weak directions
+    stop being flagged at all).
+
+    Args:
+        H:   6×6 Hessian from calc_H_g_e2 (in PCR order), invertible — see Contract
+        g:   6-element gradient from calc_H_g_e2
+        deg: DegeneracyResult from analyse_hessian(H)
+
+    Returns:
+        6-element step dx in PCR order
+
+    Raises:
+        numpy.linalg.LinAlgError: if H (or the eigenvector matrix) is singular.
+    """
+    Vu = deg.eigenvectors.T.copy()          # (6, 6) rows = eigenvectors
+    for i in range(6):
+        if deg.degenerate_mask[i]:
+            Vu[i, :] = 0.0                  # zero degenerate rows
+
+    Vf_inv = np.linalg.inv(deg.eigenvectors.T)
+    H_inv_g = np.linalg.solve(H, g)
+    return -(Vf_inv @ Vu @ H_inv_g)
diff --git a/point_cloud_registration/icp.py b/point_cloud_registration/icp.py
index c26c57e..cde97ac 100644
--- a/point_cloud_registration/icp.py
+++ b/point_cloud_registration/icp.py
@@ -41,16 +41,19 @@ def calc_H_g_e2(self, cur_T, source):
         R = cur_T[:3, :3]
         S = skews(src_mask)
         S_sum = skew(np.sum(src_mask, axis=0))
+        # J = [R | -R@skew(p)]: the tangent-space increment dx is applied by
+        # plus() as T @ [expSO3(dx[3:]) | dx[:3]], so both blocks carry R.
+        # In H the R factors cancel (R.T@R = I), leaving these closed forms.
         H_ll = num * np.eye(3)
-        H_lr = - R @ S_sum
+        H_lr = - S_sum
         H_rr = skew2(src_mask)
         H = np.zeros((6, 6))
         H[:3, :3] = H_ll
         H[:3, 3:] = H_lr
         H[3:, :3] = H_lr.T
         H[3:, 3:] = H_rr
-        g0 = rs.sum(axis=0)
-        Rt_r = rs @ R.T
+        Rt_r = rs @ R  # row i is R.T @ rs[i]
+        g0 = Rt_r.sum(axis=0)
         g1 = np.einsum('nij,ni->j', S, -Rt_r)
         g = np.hstack([g0, g1])
         e2 = np.sum(rs * rs)
@@ -65,7 +68,9 @@ def calc_H_g_e2_no_parallel_ver(self, cur_T, source):
         src_trans = transform_points(cur_T, source)
         dist, idx = self.kdtree.query(src_trans.astype(np.float32))
         mask = dist < self.max_dist
+        idx = idx[mask]
         src_trans = src_trans[mask]
+        src_mask = source[mask]
         num = src_trans.shape[0]
         # Find corresponding target points
         qs = self.target[idx]
@@ -75,10 +80,10 @@ def calc_H_g_e2_no_parallel_ver(self, cur_T, source):
         e2 = 0
         for i in range(num):
             J = np.zeros((3, 6))
-            # Jacobian of the transformation
-            J[:, :3] = np.eye(3)
+            # Jacobian of the translation (body-frame increment: t += R @ dt)
+            J[:, :3] = R
             # Jacobian of the rotation
-            J[:, 3:] = -R @ skew(source[i])
+            J[:, 3:] = -R @ skew(src_mask[i])
             # residual
             r = src_trans[i] - qs[i]
             # Hessian
diff --git a/point_cloud_registration/math_tools.py b/point_cloud_registration/math_tools.py
index a1bba26..59e08e5 100644
--- a/point_cloud_registration/math_tools.py
+++ b/point_cloud_registration/math_tools.py
@@ -88,7 +88,13 @@ def expSO3(omega):
     nearZero = theta2 <= epsilon
     W = skew(omega)
     if (nearZero):
-        return np.eye(3) + W
+        # Second-order Taylor of sin(theta)/theta and (1-cos(theta))/theta^2:
+        # the first-order I + W is not in SO(3) (det = 1 + O(theta^2)), and
+        # repeated small steps applied through plus() accumulate that
+        # scale/shear error into the pose.
+        A = 1.0 - theta2 / 6.0
+        B = 0.5 - theta2 / 24.0
+        return np.eye(3) + A * W + B * W.dot(W)
     else:
         K = W/theta
         KK = K.dot(K)
diff --git a/point_cloud_registration/ndt.py b/point_cloud_registration/ndt.py
index 181aa86..fa59317 100644
--- a/point_cloud_registration/ndt.py
+++ b/point_cloud_registration/ndt.py
@@ -10,13 +10,23 @@
 
 
 class NDT(Registration):
-    def __init__(self, voxel_size=1.0, max_iter=30, max_dist=2, tol=1e-3):
+    def __init__(self, voxel_size=1.0, max_iter=30, max_dist=2, tol=1e-3,
+                 min_points=10, cov_reg=0.0):
+        """
+        :param min_points: Voxels holding fewer points than this are discarded.
+        :param cov_reg: Isotropic shift added to every per-voxel covariance.
+            Coplanar voxels otherwise yield singular covariances, which calc_icov
+            can only handle by clamping the determinant. 0.0 disables it.
+        """
         super().__init__(max_iter=max_iter, tol=tol)
         self.voxel_size = voxel_size
         self.max_dist = max_dist
+        self.min_points = min_points
+        self.cov_reg = cov_reg
 
     def set_target(self, target):
-        self.voxels = VoxelGrid(self.voxel_size)
+        self.voxels = VoxelGrid(self.voxel_size, min_points=self.min_points,
+                                cov_reg=self.cov_reg)
         self.voxels.set_points(target)
         self.voxels.calc_icov() # need for ndt
         self._is_target_set = True
@@ -37,10 +47,12 @@ def calc_H_g_e2(self, cur_T, source):
         src_trans = src_trans[mask]
 
         diff = src_trans - means  # shape: (N, 3)
+        # The tangent-space increment dx is applied by plus() as
+        # T @ [expSO3(dx[3:]) | dx[:3]], so J0 = R (not I) and J1 = -R@skew(p).
         J1 = -R @ skews(src_mask)
         icov_J1 = np.einsum('nij,njk->nik', icov, J1)
-        H_ll = np.sum(icov, axis=0) # sum (J0.T * icov * J0)
-        H_lr = np.sum(icov_J1, axis=0)
+        H_ll = R.T @ np.sum(icov, axis=0) @ R # sum (J0.T * icov * J0)
+        H_lr = R.T @ np.sum(icov_J1, axis=0)
         H_rr = np.einsum('nji,njk->ik', J1, icov_J1)
 
         H = np.zeros((6, 6))
@@ -50,7 +62,7 @@ def calc_H_g_e2(self, cur_T, source):
         H[3:, 3:] = H_rr
 
         icov_r = np.einsum('nij,nj->ni', icov, diff)
-        g0 = np.sum(icov_r, axis=0)  # J0.T * icov * diff
+        g0 = R.T @ np.sum(icov_r, axis=0)  # J0.T * icov * diff
         g1 = np.einsum('nji,nj->i', J1, icov_r) # J1.T * icov * diff
         g = np.hstack([g0, g1])  # shape: (6,)
         e2 = np.einsum('ni,ni->', diff, icov_r)  # r.T * icov * r
@@ -75,23 +87,21 @@ def calc_H_g_e2_no_parallel_ver(self, cur_T, source):
         mask = dist < self.max_dist
         means = means[mask]
         icov = icov[mask]
-        #src_mask = source[mask]
+        src_mask = source[mask]
         src_trans = src_trans[mask]
         H = np.zeros((6, 6))
         g = np.zeros(6)
         e2 = 0
 
-        for i in range(source.shape[0]):
+        for i in range(src_mask.shape[0]):
             J = np.zeros((3, 6))
-            # Jacobian of the transformation
-            J[:, :3] = np.eye(3)
+            # Jacobian of the translation (body-frame increment: t += R @ dt)
+            J[:, :3] = R
             # Jacobian of the rotation
-            J[:, 3:] = -R @ skew(source[i])
+            J[:, 3:] = -R @ skew(src_mask[i])
             # residual
             r = src_trans[i] - means[i]
 
-            if dist[i] > self.max_dist:
-                continue
             H += J.T @ icov[i] @  J
             g += J.T @ icov[i] @ r
             e2 += r @ icov[i] @ r
diff --git a/point_cloud_registration/plane_icp.py b/point_cloud_registration/plane_icp.py
index 52c1b63..62bbb46 100644
--- a/point_cloud_registration/plane_icp.py
+++ b/point_cloud_registration/plane_icp.py
@@ -47,10 +47,13 @@ def calc_H_g_e2(self, cur_T, source):
         diff = src_trans - means
         src_mask = source[mask]
         rs = np.einsum('ij,ij->i', norms, diff)
-        Jt = norms
         Rt_norms = R.T @ norms.T
+        # The tangent-space increment dx is applied by plus() as
+        # T @ [expSO3(dx[3:]) | dx[:3]], so the residual row is
+        # n.T @ [R | -R@skew(p)] = [(R.T@n).T | (skew(p)@R.T@n).T].
+        Jt = Rt_norms.T
         # # equal to skew_time_vector
-        # Jr = np.einsum('ijk,ki->ij', skews(src_mask), Rt_norms) 
+        # Jr = np.einsum('ijk,ki->ij', skews(src_mask), Rt_norms)
         Jr = skew_time_vector(src_mask, Rt_norms.T)
         H_ll = np.einsum('ij,ik->jk', Jt, Jt)
         H_lr = np.einsum('ij,ik->jk', Jt, Jr)
@@ -83,18 +86,17 @@ def calc_H_g_e2_no_parallel_ver(self, cur_T, source):
         means = self.target[idx]
         norms = self.normal[idx]
         src_trans = src_trans[mask]
+        src_mask = source[mask]
 
         H = np.zeros((6, 6))
         g = np.zeros(6)
         e2 = 0
-        for i in range(source.shape[0]):
+        for i in range(src_mask.shape[0]):
             n = norms[i]
             r = n @ (src_trans[i] - means[i])
             J = np.zeros((1, 6))
-            J[0, :3] = n
-            J[0, 3:] = skew(source[i]) @ (R.T @ n.T)
-            if np.abs(r) > self.max_dist:
-                continue
+            J[0, :3] = R.T @ n
+            J[0, 3:] = skew(src_mask[i]) @ (R.T @ n.T)
             H += J.T @ J
             g += J[0] * r
             e2 += r * r
diff --git a/point_cloud_registration/registration.py b/point_cloud_registration/registration.py
index f46dde0..10922aa 100644
--- a/point_cloud_registration/registration.py
+++ b/point_cloud_registration/registration.py
@@ -5,6 +5,7 @@
 
 import numpy as np
 from point_cloud_registration.math_tools import plus
+from point_cloud_registration.degeneracy import analyse_hessian, apply_sr_solve
 
 
 class Registration:
@@ -17,6 +18,7 @@ def __init__(self, max_iter=30, tol=1e-3):
         self.max_iter = max_iter
         self.tol = tol
         self._is_target_set = False
+        self._last_hessian = None
 
     def is_target_set(self):
         """
@@ -68,46 +70,105 @@ def calc_H_g_e2(self, cur_T, source, dx_norm=np.inf):
         return H, g, e2
 
 
-    def align(self, source, init_T=np.eye(4), verbose=False):
+    def align(self, source, init_T=np.eye(4), verbose=False,
+              use_solution_remapping=False, sr_lambda_threshold=None,
+              lm_damping=False):
         """
-        use Gauss-Newton method to find the transformation 
-        that aligns the source point cloud to the target point cloud.
+        Gauss-Newton alignment of the source cloud onto the target.
+
+        The per-iteration step dx is a body-frame right-tangent increment
+        applied as T @ [expSO3(dx[3:]) | dx[:3]] (see math_tools.plus).
+
         :param source: Source point cloud (Nx3 array).
         :param init_T: Initial transformation (4x4 array).
         :param verbose: Print error at each iteration.
+        :param use_solution_remapping: If True, zero the step in degenerate
+            Hessian eigenvector directions at each iteration (SR mode;
+            Hinduja, Ho & Kaess, IROS 2019, Algorithm 1 — solution remapping
+            per Zhang, Kaess & Singh, ICRA 2016; see degeneracy.py for full
+            references).
+        :param sr_lambda_threshold: Override the condition-number threshold
+            for SR. None = adaptive (sqrt(lambda_max / lambda_min)).
+        :param lm_damping: If True, solve the damped system (H + lambda*I) dx = -g
+            instead of H dx = -g, with lambda scaled to the trace of H
+            (Levenberg-Marquardt). This keeps the linear solve well posed on
+            geometry that leaves H singular or near-singular — a single flat
+            surface, a straight corridor, too few correspondences — where the
+            plain solve raises numpy.linalg.LinAlgError or returns a step
+            dominated by noise. On well-conditioned data the damping is small
+            enough to leave the solution unchanged. Pair it with
+            use_solution_remapping when the Hessian may be exactly rank
+            deficient: the degeneracy analysis runs on the raw H so the
+            classification stays honest, while the solve uses the damped one.
         :return: Final transformation (4x4 array).
         """
         if self.is_target_set() is False:
             raise ValueError("Target is not set.")
 
         source = source.astype(np.float32)
-        cur_T = init_T
-        # dx_norm = np.inf
-        # best_T = cur_T
-        # best_error = np.inf
-        # source = source.astype(np.float32)
+        # Copy: align() must not return the caller's array (or the shared
+        # mutable np.eye(4) default) when it converges before the first step.
+        cur_T = init_T.copy()
+        H_final = None
+        converged = False
+
         for i in range(self.max_iter):
             H, g, e2 = self.calc_H_g_e2(cur_T, source)
+            H_final = H
             if verbose:
                 print(f"iter {i}, error {e2}")
 
-            # # ensure the error is decreasing
-            # if e2 < best_error:
-            #     best_error = e2
-            #     best_T = cur_T.copy()
-            # else:
-            #     break
+            if lm_damping:
+                trace_H = np.trace(H)
+                lambda_lm = max(1e-4 * trace_H / 6.0 if trace_H > 0 else 1e-3, 1e-6)
+                H_solve = H + lambda_lm * np.eye(6)
+            else:
+                H_solve = H
+
+            if use_solution_remapping:
+                # Eigendecompose the raw H (correct degeneracy thresholding) but
+                # solve on H_solve (numerical stability): an isotropic +lambda*I
+                # shift leaves the eigenvectors identical.
+                deg = analyse_hessian(H.astype(float), lambda_threshold=sr_lambda_threshold)
+                dx = apply_sr_solve(H_solve.astype(float), g.astype(float), deg)
+            elif lm_damping:
+                try:
+                    dx = -np.linalg.solve(H_solve, g)
+                except np.linalg.LinAlgError:
+                    dx = -np.linalg.lstsq(H_solve, g, rcond=None)[0]
+            else:
+                dx = -np.linalg.solve(H, g)
 
-            
-            # solve the linear system
-            dx = -np.linalg.solve(H, g)
-    
             # check convergence
             dx_norm = np.linalg.norm(dx)
             if dx_norm < self.tol:
+                converged = True
                 break
 
             # Update transformation
             cur_T = plus(cur_T, dx)
 
-        return cur_T
\ No newline at end of file
+        # If max_iter was exhausted, cur_T advanced past the last
+        # linearization, so recompute the Hessian at the returned pose.
+        if not converged and H_final is not None:
+            H_final, _, _ = self.calc_H_g_e2(cur_T, source)
+        self._last_hessian = H_final
+        return cur_T
+
+    @property
+    def last_hessian(self):
+        """
+        The 6x6 Gauss-Newton Hessian (J^T W J) evaluated at the pose returned
+        by the most recent align() call, or None before any align().
+
+        It is expressed in the right-tangent (body) frame of that pose, DOF
+        order [tx, ty, tz, wx, wy, wz] — the increment coordinates consumed by
+        math_tools.plus().  Consumers that reason about world-frame axes must
+        map directions through the returned pose: a body increment
+        [dt, w] moves the estimate by R @ dt (translation) and R @ w
+        (rotation axis) in the world; a full left-tangent (world-frame)
+        Hessian additionally carries the translation-rotation coupling and
+        is obtained with the SE(3) adjoint of the pose,
+        H_world = Ad^{-T} @ H @ Ad^{-1}.
+        """
+        return self._last_hessian
\ No newline at end of file
diff --git a/point_cloud_registration/voxel.py b/point_cloud_registration/voxel.py
index 504b61e..a7c8f7d 100644
--- a/point_cloud_registration/voxel.py
+++ b/point_cloud_registration/voxel.py
@@ -53,10 +53,19 @@ class VoxelGrid:
     An efficient VoxelGrid structure using hash table
     """
 
-    def __init__(self, voxel_size, min_points=10):
+    def __init__(self, voxel_size, min_points=10, cov_reg=0.0):
+        """
+        :param voxel_size: Edge length of a voxel.
+        :param min_points: Voxels holding fewer points than this are discarded.
+        :param cov_reg: Isotropic shift added to every per-voxel covariance.
+            0.0 (the default) leaves the covariances untouched.
+        """
+        if cov_reg < 0.0:
+            raise ValueError(f"cov_reg must be >= 0.0, got {cov_reg}")
         self.voxel_size = voxel_size
         self.kdtree = None
         self.min_points = min_points
+        self.cov_reg = cov_reg
 
     def calc_sqrt_icov(self):
         """
@@ -146,13 +155,21 @@ def set_points(self, points):
             [[c00, c01, c02],
              [c01, c11, c12],
              [c02, c12, c22]]).transpose(2, 0, 1)
-        
+
+        if self.cov_reg > 0.0:
+            # Isotropic shift keeps eigenvectors (normals) intact; prevents singular
+            # covariances when a voxel's points are exactly coplanar/collinear.
+            covs = covs + np.eye(3) * self.cov_reg
+
         # Filter out voxels with too few points
         mask = counts >= self.min_points
         means = means[mask]
         covs = covs[mask]
         # t6 = time.time()
 
+        if len(means) == 0:
+            raise ValueError("data_pts should be non-empty")
+
         # get the normal of each voxel
         _, eigenvectors = np.linalg.eigh(covs)
         norms = eigenvectors[:, :, 0]
diff --git a/point_cloud_registration/voxelized_plane_icp.py b/point_cloud_registration/voxelized_plane_icp.py
index 36c380a..bc63230 100644
--- a/point_cloud_registration/voxelized_plane_icp.py
+++ b/point_cloud_registration/voxelized_plane_icp.py
@@ -10,13 +10,22 @@
 
 
 class VPlaneICP(Registration):
-    def __init__(self, voxel_size=1.0, max_iter=30, max_dist=2, tol=1e-3):
+    def __init__(self, voxel_size=1.0, max_iter=30, max_dist=2, tol=1e-3,
+                 min_points=10, cov_reg=0.0):
+        """
+        :param min_points: Voxels holding fewer points than this are discarded.
+        :param cov_reg: Isotropic shift added to every per-voxel covariance,
+            which keeps coplanar voxels non-singular. 0.0 disables it.
+        """
         super().__init__(max_iter=max_iter, tol=tol)
         self.voxel_size = voxel_size
         self.max_dist = max_dist
+        self.min_points = min_points
+        self.cov_reg = cov_reg
 
     def set_target(self, target):
-        self.voxels = VoxelGrid(self.voxel_size)
+        self.voxels = VoxelGrid(self.voxel_size, min_points=self.min_points,
+                                cov_reg=self.cov_reg)
         self.voxels.set_points(target)
         self._is_target_set = True
 
@@ -42,10 +51,13 @@ def calc_H_g_e2(self, cur_T, source):
         diff = src_trans - means
         src_mask = source[mask]
         rs = np.einsum('ij,ij->i', norms, diff)
-        Jt = norms
         Rt_norms = R.T @ norms.T
+        # The tangent-space increment dx is applied by plus() as
+        # T @ [expSO3(dx[3:]) | dx[:3]], so the residual row is
+        # n.T @ [R | -R@skew(p)] = [(R.T@n).T | (skew(p)@R.T@n).T].
+        Jt = Rt_norms.T
         # # equal to skew_time_vector
-        # Jr = np.einsum('ijk,ki->ij', skews(src_mask), Rt_norms) 
+        # Jr = np.einsum('ijk,ki->ij', skews(src_mask), Rt_norms)
         Jr = skew_time_vector(src_mask, Rt_norms.T)
         H_ll = np.einsum('ij,ik->jk', Jt, Jt)
         H_lr = np.einsum('ij,ik->jk', Jt, Jr)
@@ -83,18 +95,17 @@ def calc_H_g_e2_no_parallel_ver(self, cur_T, source):
         means = query_data['mean'][mask]
         norms = query_data['norm'][mask]
         src_trans = src_trans[mask]
+        src_mask = source[mask]
 
         H = np.zeros((6, 6))
         g = np.zeros(6)
         e2 = 0
-        for i in range(source.shape[0]):
+        for i in range(src_mask.shape[0]):
             n = norms[i]
             r = n @ (src_trans[i] - means[i])
             J = np.zeros((1, 6))
-            J[0, :3] = n
-            J[0, 3:] = skew(source[i]) @ (R.T @ n.T)
-            if np.abs(r) > self.max_dist:
-                continue
+            J[0, :3] = R.T @ n
+            J[0, 3:] = skew(src_mask[i]) @ (R.T @ n.T)
             H += J.T @ J
             g += J[0] * r
             e2 += r * r
diff --git a/tests/test_degeneracy.py b/tests/test_degeneracy.py
new file mode 100644
index 0000000..6e6fd2a
--- /dev/null
+++ b/tests/test_degeneracy.py
@@ -0,0 +1,162 @@
+"""
+Unit tests for point_cloud_registration.degeneracy.
+
+Hessian fixtures:
+  - 1-DOF: diag([ε,ε,λ_big,ε,ε,ε]) — only tz constrained
+  - 2-DOF: diag([λ_med,ε,λ_big,ε,ε,ε]) — tx + tz constrained
+  - well-conditioned: A.T@A + 100*I — all 6 DOFs constrained
+"""
+import numpy as np
+import pytest
+
+from point_cloud_registration.degeneracy import (
+    DegeneracyResult,
+    analyse_hessian,
+    apply_sr_solve,
+)
+
+# PCR DOF order: [tx, ty, tz, ωx, ωy, ωz]
+#
+# For a direction to be non-degenerate under the condition-number criterion:
+#   lambda_i >= lambda_cn = sqrt(lambda_max / lambda_min)
+# With lambda_min = EPS = 1e-6 and lambda_max = LAMBDA_BIG = 1e8:
+#   lambda_cn = sqrt(1e8/1e-6) = 1e7
+# So LAMBDA_BIG = 1e8 >= 1e7 → non-degenerate; EPS = 1e-6 << 1e7 → degenerate.
+# LAMBDA_MED = 5e7 >= 1e7 → also non-degenerate (used for 2-DOF case).
+EPS = 1e-6
+LAMBDA_BIG = 1e8
+LAMBDA_MED = 5e7
+
+
+def _h_1dof() -> np.ndarray:
+    """Only tz constrained — index 2 in PCR order."""
+    return np.diag([EPS, EPS, LAMBDA_BIG, EPS, EPS, EPS])
+
+
+def _h_2dof() -> np.ndarray:
+    """tx (index 0) and tz (index 2) constrained."""
+    return np.diag([LAMBDA_MED, EPS, LAMBDA_BIG, EPS, EPS, EPS])
+
+
+def _h_well_conditioned() -> np.ndarray:
+    rng = np.random.default_rng(42)
+    A = rng.standard_normal((6, 6))
+    return A.T @ A + 100.0 * np.eye(6)
+
+
+class TestAnalyseHessian:
+    def test_1dof_constrained_count(self):
+        deg = analyse_hessian(_h_1dof())
+        assert deg.num_constrained_dof == 1
+
+    def test_2dof_constrained_count(self):
+        deg = analyse_hessian(_h_2dof())
+        assert deg.num_constrained_dof == 2
+
+    def test_well_conditioned_no_degeneracy(self):
+        deg = analyse_hessian(_h_well_conditioned())
+        assert not deg.is_degenerate
+        assert deg.num_constrained_dof == 6
+
+    def test_1dof_is_degenerate(self):
+        assert analyse_hessian(_h_1dof()).is_degenerate
+
+    def test_condition_number_formula(self):
+        # lambda_max = 100, lambda_min = 1 → cn = sqrt(100/1) = 10
+        H = np.diag([1.0, 1.0, 100.0, 1.0, 1.0, 1.0])
+        deg = analyse_hessian(H)
+        expected = np.sqrt(100.0 / 1.0)
+        assert abs(deg.condition_number - expected) < 0.01
+
+    def test_eigenvalues_ascending(self):
+        deg = analyse_hessian(_h_1dof())
+        assert np.all(np.diff(deg.eigenvalues) >= 0)
+
+    def test_V_constrained_shape(self):
+        deg = analyse_hessian(_h_1dof())
+        assert deg.V_constrained.shape == (1, 6)
+
+        deg2 = analyse_hessian(_h_2dof())
+        assert deg2.V_constrained.shape == (2, 6)
+
+        deg6 = analyse_hessian(_h_well_conditioned())
+        assert deg6.V_constrained.shape == (6, 6)
+
+    def test_V_constrained_rows_are_unit_vectors(self):
+        deg = analyse_hessian(_h_2dof())
+        norms = np.linalg.norm(deg.V_constrained, axis=1)
+        np.testing.assert_allclose(norms, 1.0, atol=1e-12)
+
+    def test_fixed_lambda_threshold(self):
+        H = _h_1dof()
+        deg_low = analyse_hessian(H, lambda_threshold=EPS / 2)
+        # Threshold below all eigenvalues → nothing degenerate
+        assert deg_low.num_constrained_dof == 6
+
+    def test_completely_degenerate(self):
+        H = np.zeros((6, 6))
+        deg = analyse_hessian(H)
+        assert deg.is_degenerate
+        assert deg.num_constrained_dof == 0
+        assert deg.V_constrained.shape == (0, 6)
+
+    def test_structural_zero_alone_is_degenerate(self):
+        """
+        A structural zero stays degenerate even when the adaptive threshold
+        lands below it: five equal 1e12 eigenvalues give a threshold of
+        sqrt(1e12/1e12) = 1, yet the 10.0 direction is eleven orders of
+        magnitude weaker and must not be reported constrained.
+        """
+        H = np.diag([1e12, 1e12, 1e12, 1e12, 1e12, 10.0])
+        deg = analyse_hessian(H)
+        assert deg.num_constrained_dof == 5
+        assert deg.degenerate_mask[0]  # ascending order puts 10.0 first
+        assert deg.condition_number == pytest.approx(1.0)
+
+    def test_structural_zero_combines_with_cn_test(self):
+        """Phase 1 (structural zeros) and phase 2 (cn test) OR together."""
+        H = np.diag([1e12, 1e12, 1e12, 1e12, 1e3, 1e-6])
+        deg = analyse_hessian(H)
+        # 1e-6 is a structural zero; 1e3 fails the cn test (~3.2e4).
+        assert deg.num_constrained_dof == 4
+
+
+class TestApplySRSolve:
+    def test_degenerate_direction_zeroed(self):
+        H = _h_1dof()
+        deg = analyse_hessian(H)
+        g = np.ones(6)
+        dx = apply_sr_solve(H, g, deg)
+
+        # Component of dx along each degenerate eigenvector should be ~0
+        for i in range(6):
+            if deg.degenerate_mask[i]:
+                v = deg.eigenvectors[:, i]
+                component = float(v @ dx)
+                assert abs(component) < 1e-8, f"Degenerate dir {i} not zeroed: {component}"
+
+    def test_no_degeneracy_matches_standard_solve(self):
+        H = _h_well_conditioned()
+        deg = analyse_hessian(H)
+        g = np.ones(6)
+        dx_sr = apply_sr_solve(H, g, deg)
+        dx_std = np.linalg.solve(H, -g)
+        np.testing.assert_allclose(dx_sr, dx_std, rtol=1e-6)
+
+    def test_exactly_singular_hessian_classified_but_solve_raises(self):
+        """
+        Pins the contract boundary: analyse_hessian CLASSIFIES an exactly
+        rank-deficient Hessian, but apply_sr_solve does not solve one.
+
+        H is singular yet partially constrained (only tz observed).  The
+        analysis correctly reports 1 constrained DOF, but the remapping still
+        goes through np.linalg.solve(H, g), which raises before any projection
+        happens.  Callers whose raw Hessian can be exactly singular must pass a
+        regularized H to apply_sr_solve while analysing the raw one.
+        """
+        H = np.diag([0.0, 0.0, LAMBDA_BIG, 0.0, 0.0, 0.0])
+        deg = analyse_hessian(H)
+        assert deg.num_constrained_dof == 1
+
+        with pytest.raises(np.linalg.LinAlgError):
+            apply_sr_solve(H, np.ones(6), deg)
diff --git a/tests/test_icp.py b/tests/test_icp.py
index 2b12983..dd74eb5 100644
--- a/tests/test_icp.py
+++ b/tests/test_icp.py
@@ -1,7 +1,7 @@
 import numpy as np
 import pytest
 from point_cloud_registration.icp import ICP
-from point_cloud_registration.math_tools import expSO3
+from point_cloud_registration.math_tools import expSO3, makeT
 
 
 @pytest.fixture
@@ -17,16 +17,24 @@ def generate_test_data():
     return target, source
 
 
-def test_calc_H_g_e2(generate_test_data):
+@pytest.mark.parametrize(
+    "cur_T",
+    [np.eye(4), makeT(expSO3(np.array([0.3, -0.2, 0.4])), np.array([0.1, 0.2, -0.1]))],
+    ids=["identity_pose", "rotated_pose"],
+)
+@pytest.mark.parametrize("with_outliers", [False, True], ids=["all_inliers", "with_outliers"])
+def test_calc_H_g_e2(generate_test_data, cur_T, with_outliers):
     """
     Test that calc_H_g_e2 and calc_H_g_e2_no_parallel_ver produce the same results.
     """
     target, source = generate_test_data
+    if with_outliers:
+        # Points far outside the target's reach: rejected by the max_dist gate.
+        source = np.vstack([source[:10] + 50.0, source])
     source = source.astype(np.float32)
     icp = ICP(max_iter=10, max_dist=2.0, tol=1e-3)
     icp.set_target(target)
 
-    cur_T = np.eye(4)  # Initial transformation (identity matrix)
 
     # Compute results using both methods
     H1, g1, e2_1 = icp.calc_H_g_e2(cur_T, source)
diff --git a/tests/test_math_tools.py b/tests/test_math_tools.py
new file mode 100644
index 0000000..d3f7d38
--- /dev/null
+++ b/tests/test_math_tools.py
@@ -0,0 +1,43 @@
+"""
+expSO3 must return a member of SO(3) in both of its branches.
+
+The small-angle branch used to return the first-order I + W, which is
+not orthonormal (det = 1 + O(theta^2)): at |omega| = 3e-3 the
+determinant error is ~9e-6, and every Gauss-Newton step taken through
+plus() bakes that scale/shear into the pose estimate.
+"""
+import numpy as np
+
+from point_cloud_registration.math_tools import expSO3, skew
+
+
+def _rodrigues(omega):
+    """Exact Rodrigues formula, valid for any nonzero angle (float64)."""
+    theta = np.linalg.norm(omega)
+    K = skew(omega) / theta
+    return np.eye(3) + np.sin(theta) * K + (1.0 - np.cos(theta)) * (K @ K)
+
+
+def _assert_in_SO3(R, atol):
+    np.testing.assert_allclose(R.T @ R, np.eye(3), atol=atol)
+    assert abs(np.linalg.det(R) - 1.0) < atol
+
+
+def test_small_angle_branch_is_orthonormal():
+    # theta^2 = 9e-6 <= epsilon = 1e-5: exercises the near-zero branch.
+    R = expSO3(np.array([3e-3, 0.0, 0.0]))
+    _assert_in_SO3(R, atol=1e-12)
+
+
+def test_small_angle_branch_matches_rodrigues():
+    omega = np.array([1.5e-3, -2e-3, 1e-3])
+    np.testing.assert_allclose(expSO3(omega), _rodrigues(omega), atol=1e-12)
+
+
+def test_large_angle_branch_is_orthonormal():
+    R = expSO3(np.array([0.5, -0.3, 0.2]))
+    _assert_in_SO3(R, atol=1e-12)
+
+
+def test_zero_rotation_is_identity():
+    np.testing.assert_allclose(expSO3(np.zeros(3)), np.eye(3), atol=1e-15)
diff --git a/tests/test_ndt.py b/tests/test_ndt.py
index f21c757..a834b16 100644
--- a/tests/test_ndt.py
+++ b/tests/test_ndt.py
@@ -2,6 +2,7 @@
 import pytest
 from point_cloud_registration import NDT
 from point_cloud_registration import expSO3
+from point_cloud_registration.math_tools import makeT
 
 
 @pytest.fixture
@@ -19,17 +20,24 @@ def generate_test_data():
     return target, normals, source
 
 
-def test_calc_H_g_e2(generate_test_data):
+@pytest.mark.parametrize(
+    "cur_T",
+    [np.eye(4), makeT(expSO3(np.array([0.3, -0.2, 0.4])), np.array([0.1, 0.2, -0.1]))],
+    ids=["identity_pose", "rotated_pose"],
+)
+@pytest.mark.parametrize("with_outliers", [False, True], ids=["all_inliers", "with_outliers"])
+def test_calc_H_g_e2(generate_test_data, cur_T, with_outliers):
     """
-    Test that calc_H_g_e2 and calc_H_g_e2x produce the same results.
+    Test that calc_H_g_e2 and calc_H_g_e2_no_parallel_ver produce the same results.
     """
-    target, normals, source = generate_test_data
+    target, _, source = generate_test_data
+    if with_outliers:
+        # Points far outside the target's reach: rejected by the max_dist gate.
+        source = np.vstack([source[:10] + 50.0, source])
     source = source.astype(np.float32)
     plane_icp = NDT(voxel_size=1.0, max_iter=10, max_dist=2.0, tol=1e-3)
     plane_icp.set_target(target)
-    plane_icp.normal = normals
 
-    cur_T = np.eye(4)  # Initial transformation (identity matrix)
 
     # Compute results using both methods
     H1, g1, e2_1 = plane_icp.calc_H_g_e2(cur_T, source)
diff --git a/tests/test_picp.py b/tests/test_picp.py
index 7d90cf9..1bc3987 100644
--- a/tests/test_picp.py
+++ b/tests/test_picp.py
@@ -1,7 +1,7 @@
 import numpy as np
 import pytest
 from point_cloud_registration.plane_icp import PlaneICP
-from point_cloud_registration.math_tools import expSO3
+from point_cloud_registration.math_tools import expSO3, makeT
 
 
 @pytest.fixture
@@ -17,16 +17,24 @@ def generate_test_data():
     return target, source
 
 
-def test_calc_H_g_e2(generate_test_data):
+@pytest.mark.parametrize(
+    "cur_T",
+    [np.eye(4), makeT(expSO3(np.array([0.3, -0.2, 0.4])), np.array([0.1, 0.2, -0.1]))],
+    ids=["identity_pose", "rotated_pose"],
+)
+@pytest.mark.parametrize("with_outliers", [False, True], ids=["all_inliers", "with_outliers"])
+def test_calc_H_g_e2(generate_test_data, cur_T, with_outliers):
     """
     Test that calc_H_g_e2 and calc_H_g_e2_no_parallel_ver produce the same results.
     """
     target, source = generate_test_data
+    if with_outliers:
+        # Points far outside the target's reach: rejected by the max_dist gate.
+        source = np.vstack([source[:10] + 50.0, source])
     source = source.astype(np.float32)
     vpicp = PlaneICP(max_iter=10, max_dist=2.0, tol=1e-3)
     vpicp.set_target(target)
 
-    cur_T = np.eye(4)  # Initial transformation (identity matrix)
 
     # Compute results using both methods
     H1, g1, e2_1 = vpicp.calc_H_g_e2(cur_T, source)
diff --git a/tests/test_registration_align.py b/tests/test_registration_align.py
new file mode 100644
index 0000000..a2fb9bc
--- /dev/null
+++ b/tests/test_registration_align.py
@@ -0,0 +1,234 @@
+"""
+Behavioral tests for Registration.align().
+"""
+import numpy as np
+import pytest
+
+from point_cloud_registration.icp import ICP
+from point_cloud_registration.plane_icp import PlaneICP
+from point_cloud_registration.math_tools import expSO3, makeT
+
+
+@pytest.fixture
+def well_conditioned_pair():
+    """
+    Three mutually orthogonal planes ("corner") — normals span R^3, so all six
+    DOF are constrained and the Hessian is well conditioned.
+
+    Returns (target, source, T_true) where T_true is the transform align()
+    should recover, i.e. the one that maps source onto target.
+    """
+    rng = np.random.default_rng(0)
+    n = 400
+    a = rng.uniform(0.0, 4.0, size=(n, 2))
+    b = rng.uniform(0.0, 4.0, size=(n, 2))
+    c = rng.uniform(0.0, 4.0, size=(n, 2))
+    face_z = np.column_stack([a[:, 0], a[:, 1], np.zeros(n)])
+    face_x = np.column_stack([np.zeros(n), b[:, 0], b[:, 1]])
+    face_y = np.column_stack([c[:, 0], np.zeros(n), c[:, 1]])
+    target = np.vstack([face_z, face_x, face_y])
+
+    R = expSO3(np.array([0.02, -0.03, 0.05]))
+    t = np.array([0.15, -0.1, 0.08])
+    source = (R @ target.T).T + t
+    T_true = np.linalg.inv(makeT(R, t))
+    return target, source, T_true
+
+
+@pytest.fixture
+def degenerate_pair():
+    """
+    A perfectly flat z=0 grid as target, source the same grid shifted in +z.
+
+    Every surface normal is (0, 0, 1), so the plane-ICP Jacobian rows are
+    [0, 0, 1, y, -x, 0]: tx, ty and wz never appear and the Hessian is exactly
+    rank 3.  z, wx and wy are the only observable directions.
+    """
+    g = np.arange(-5.0, 5.0 + 1e-9, 1.0)
+    xx, yy = np.meshgrid(g, g)
+    target = np.column_stack([xx.ravel(), yy.ravel(), np.zeros(xx.size)])
+    source = target + np.array([0.0, 0.0, 0.5])
+    return target, source
+
+
+def _yaw(T):
+    return float(np.arctan2(T[1, 0], T[0, 0]))
+
+
+def test_last_hessian_lifecycle(well_conditioned_pair):
+    """last_hessian is None before any align and a 6x6 array afterwards."""
+    target, source, _ = well_conditioned_pair
+    engine = PlaneICP(max_iter=30, max_dist=2.0, tol=1e-6)
+    engine.set_target(target)
+
+    assert engine.last_hessian is None
+
+    engine.align(source)
+
+    H = engine.last_hessian
+    assert isinstance(H, np.ndarray)
+    assert H.shape == (6, 6)
+    assert np.all(np.isfinite(H))
+
+
+def test_last_hessian_recomputed_when_max_iter_exhausted(well_conditioned_pair):
+    """
+    When align() runs out of iterations, cur_T has advanced past the last
+    linearization, so the stored Hessian must be recomputed at the returned
+    pose rather than left at the pre-step one.
+    """
+    target, source, _ = well_conditioned_pair
+    engine = PlaneICP(max_iter=1, max_dist=2.0, tol=1e-6)
+    engine.set_target(target)
+
+    T = engine.align(source)
+
+    H_at_returned = engine.calc_H_g_e2(T, source)[0]
+    H_at_init = engine.calc_H_g_e2(np.eye(4), source)[0]
+
+    assert np.allclose(engine.last_hessian, H_at_returned), (
+        "last_hessian is not the Hessian at the returned pose; max diff "
+        f"{np.max(np.abs(engine.last_hessian - H_at_returned))}"
+    )
+    assert not np.allclose(engine.last_hessian, H_at_init), (
+        "last_hessian is stale (equals the pre-step Hessian)"
+    )
+
+
+def test_align_does_not_alias_init_T():
+    """
+    align() must return a transform the caller owns.
+
+    With source == target the very first step is ~zero, so align()
+    converges before ever calling plus(): without the defensive copy it
+    returns the init_T object itself — and with the mutable np.eye(4)
+    default, a caller mutating the result silently corrupts the default
+    for every subsequent align() call in the process.
+    """
+    np.random.seed(1)
+    target = np.random.rand(100, 3)
+    icp = ICP(max_iter=10, max_dist=2.0, tol=1e-3)
+    icp.set_target(target)
+    source = target.astype(np.float32)
+
+    init_T = np.eye(4)
+    T = icp.align(source, init_T=init_T)
+    assert T is not init_T
+
+    # Mutating the result must not corrupt the shared default argument.
+    T_default = icp.align(source)
+    T_default[0, 3] = 123.0
+    T_again = icp.align(source)
+    np.testing.assert_allclose(T_again, np.eye(4), atol=1e-6)
+
+
+def test_lm_damping_rescues_singular_hessian(degenerate_pair):
+    """
+    On exactly rank-deficient geometry the plain Gauss-Newton solve fails; the
+    damped solve returns a finite transform.
+    """
+    target, source = degenerate_pair
+
+    engine = PlaneICP(max_iter=10, max_dist=2.0, tol=1e-6)
+    engine.set_target(target)
+
+    # The fixture really is singular: three eigenvalues are exactly zero.
+    H = engine.calc_H_g_e2(np.eye(4), source)[0]
+    eigenvalues = np.linalg.eigvalsh(H)
+    assert np.count_nonzero(eigenvalues <= 0.0) == 3, f"eigenvalues: {eigenvalues}"
+
+    # Undamped: np.linalg.solve rejects the exactly singular Hessian.
+    with pytest.raises(np.linalg.LinAlgError):
+        engine.align(source, lm_damping=False)
+
+    # Damped: H + lambda*I is invertible, so align completes.
+    T = engine.align(source, lm_damping=True)
+    assert T.shape == (4, 4)
+    assert np.all(np.isfinite(T))
+
+
+def test_lm_damping_parity_on_well_conditioned_data(well_conditioned_pair):
+    """Damping must not move the solution when the problem is well conditioned."""
+    target, source, _ = well_conditioned_pair
+
+    engine = PlaneICP(max_iter=30, max_dist=2.0, tol=1e-6)
+    engine.set_target(target)
+    T_plain = engine.align(source, lm_damping=False)
+
+    engine_damped = PlaneICP(max_iter=30, max_dist=2.0, tol=1e-6)
+    engine_damped.set_target(target)
+    T_damped = engine_damped.align(source, lm_damping=True)
+
+    assert np.allclose(T_plain, T_damped, atol=1e-3), (
+        f"damping changed the solution; max diff {np.max(np.abs(T_plain - T_damped))}"
+    )
+
+
+def test_solution_remapping_zeroes_degenerate_directions(degenerate_pair):
+    """
+    On a flat plane the observable directions are z, wx and wy.  Solution
+    remapping must leave the unobservable ones (x, y, yaw) untouched while z
+    still converges onto the target plane.
+
+    lm_damping is required here: apply_sr_solve inverts the Hessian it is given,
+    and the raw one is singular.
+    """
+    target, source = degenerate_pair
+
+    engine = PlaneICP(max_iter=30, max_dist=2.0, tol=1e-6)
+    engine.set_target(target)
+
+    T = engine.align(source, use_solution_remapping=True, lm_damping=True)
+
+    assert np.all(np.isfinite(T))
+    assert abs(T[0, 3]) < 1e-6, f"x moved along a degenerate direction: {T[0, 3]}"
+    assert abs(T[1, 3]) < 1e-6, f"y moved along a degenerate direction: {T[1, 3]}"
+    assert abs(_yaw(T)) < 1e-6, f"yaw moved along a degenerate direction: {_yaw(T)}"
+    # The source sits 0.5 above the target plane, so align must pull it back down.
+    assert T[2, 3] == pytest.approx(-0.5, abs=1e-3), f"z did not converge: {T[2, 3]}"
+
+
+def test_solution_remapping_no_world_leak_under_rotated_init(degenerate_pair):
+    """
+    The same flat plane, but starting from a rotated initial guess.
+
+    The unobservable directions of the scene are world x, y and yaw; SR zeroes
+    the step in the body frame, and because H and the step share the body
+    frame of the retraction, the accumulated world-frame motion must still
+    have no component along them.  Before the body-frame Jacobian fix this
+    leaked ~0.13 m into world y at a 0.25 rad initial roll.
+    """
+    from point_cloud_registration.math_tools import transform_points
+
+    target, source = degenerate_pair
+
+    engine = PlaneICP(max_iter=50, max_dist=2.0, tol=1e-8)
+    engine.set_target(target)
+
+    init_T = makeT(expSO3(np.array([0.25, 0.0, 0.0])), np.zeros(3))
+    T = engine.align(source, init_T=init_T,
+                     use_solution_remapping=True, lm_damping=True)
+
+    assert np.all(np.isfinite(T))
+    assert abs(T[0, 3]) < 1e-6, f"x leaked: {T[0, 3]}"
+    assert abs(T[1, 3]) < 1e-6, f"y leaked: {T[1, 3]}"
+    assert abs(_yaw(T)) < 1e-6, f"yaw leaked: {_yaw(T)}"
+    # The observable directions must still do their job: the aligned source
+    # has to land on the z=0 target plane.
+    aligned = transform_points(T, source)
+    assert np.max(np.abs(aligned[:, 2])) < 1e-3
+
+
+def test_default_behaviour_converges_to_ground_truth(well_conditioned_pair):
+    """With both options off, align() is the plain Gauss-Newton solver."""
+    target, source, T_true = well_conditioned_pair
+
+    engine = PlaneICP(max_iter=30, max_dist=2.0, tol=1e-6)
+    engine.set_target(target)
+
+    T = engine.align(source, use_solution_remapping=False, lm_damping=False)
+
+    assert T.shape == (4, 4)
+    assert np.allclose(T, T_true, atol=1e-4), (
+        f"max diff from ground truth {np.max(np.abs(T - T_true))}"
+    )
diff --git a/tests/test_retraction_consistency.py b/tests/test_retraction_consistency.py
new file mode 100644
index 0000000..7dd6643
--- /dev/null
+++ b/tests/test_retraction_consistency.py
@@ -0,0 +1,221 @@
+"""
+Retraction-consistency oracle for every solver's calc_H_g_e2.
+
+align() applies its Gauss-Newton step through the right-multiplicative
+retraction plus(T, dx) = T @ [expSO3(dx[3:]) | dx[:3]], so the (H, g)
+returned by calc_H_g_e2 must be the Gauss-Newton pair of the solver's
+own cost with respect to THAT increment (a body-frame right-tangent
+6-vector, DOF order [tx, ty, tz, wx, wy, wz]).
+
+The oracle freezes the correspondences the solver selected at cur_T
+(Gauss-Newton linearizes with correspondences held fixed), rebuilds the
+residual model in float64, differentiates each residual through plus()
+by central differences, and checks
+
+    g == sum_i J_i.T @ W_i @ r_i        H == sum_i J_i.T @ W_i @ J_i
+
+independently.  This catches frame errors that fast-vs-reference parity
+tests cannot see, because those compare two implementations that could
+share the same wrong convention.
+"""
+import numpy as np
+import pytest
+
+from point_cloud_registration.icp import ICP
+from point_cloud_registration.plane_icp import PlaneICP
+from point_cloud_registration.voxelized_plane_icp import VPlaneICP
+from point_cloud_registration.ndt import NDT
+from point_cloud_registration.math_tools import expSO3, makeT, plus, transform_points
+
+H_STEP = 3e-3
+CUR_T = makeT(expSO3(np.array([0.3, -0.2, 0.4])), np.array([0.1, 0.2, -0.1]))
+
+
+def _numeric_J(residual_fn, dim):
+    """Central-difference Jacobian of residual_fn: R^6 -> R^(N,dim)."""
+    r0 = residual_fn(np.zeros(6))
+    J = np.zeros((r0.shape[0], dim, 6))
+    for k in range(6):
+        dx = np.zeros(6)
+        dx[k] = H_STEP
+        rp = residual_fn(dx)
+        rm = residual_fn(-dx)
+        J[:, :, k] = (rp - rm) / (2.0 * H_STEP)
+    return r0, J
+
+
+def _assert_H_g_match(H, g, r0, J, W=None):
+    """Compare solver (H, g) with the numeric Gauss-Newton pair."""
+    if W is None:
+        Wr = r0
+        WJ = J
+    else:
+        Wr = np.einsum('nij,nj->ni', W, r0)
+        WJ = np.einsum('nij,njk->nik', W, J)
+    g_num = np.einsum('nik,ni->k', J, Wr)
+    H_num = np.einsum('nik,nil->kl', J, WJ)
+    g_scale = max(np.max(np.abs(g_num)), 1.0)
+    H_scale = max(np.max(np.abs(H_num)), 1.0)
+    assert np.allclose(g, g_num, atol=2e-3 * g_scale), (
+        f"g mismatch:\nsolver {g}\nnumeric {g_num}")
+    assert np.allclose(H, H_num, atol=2e-3 * H_scale), (
+        f"H mismatch: max |dH| = {np.max(np.abs(H - H_num))}")
+
+
+@pytest.fixture
+def cloud_pair():
+    np.random.seed(42)
+    target = np.random.rand(500, 3) * 4.0
+    R = expSO3(np.array([0.1, 0.2, 0.3]))
+    t = np.array([0.5, -0.3, 0.2])
+    source = ((R @ target.T).T + t).astype(np.float32)
+    return target, source
+
+
+def test_icp_matches_numeric_gauss_newton(cloud_pair):
+    target, source = cloud_pair
+    icp = ICP(max_iter=10, max_dist=2.0)
+    icp.set_target(target)
+    H, g, _ = icp.calc_H_g_e2(CUR_T, source)
+
+    # Freeze the correspondences exactly as the solver selected them.
+    src_trans = transform_points(CUR_T.astype(np.float32), source)
+    dist, idx = icp.kdtree.query(src_trans)
+    mask = dist < icp.max_dist
+    p = source[mask].astype(np.float64)
+    q = target[idx[mask]].astype(np.float64)
+
+    def residual(dx):
+        T = plus(CUR_T, dx)
+        return transform_points(T, p) - q
+
+    r0, J = _numeric_J(residual, 3)
+    _assert_H_g_match(H, g, r0, J)
+
+
+def test_plane_icp_matches_numeric_gauss_newton(cloud_pair):
+    target, source = cloud_pair
+    picp = PlaneICP(max_iter=10, max_dist=2.0, k=10)
+    picp.set_target(target)
+    H, g, _ = picp.calc_H_g_e2(CUR_T, source)
+
+    src_trans = transform_points(CUR_T.astype(np.float32), source)
+    dist, idx = picp.kdtree.query(src_trans)
+    mask = dist < picp.max_dist
+    p = source[mask].astype(np.float64)
+    m = picp.target[idx[mask]].astype(np.float64)
+    n = picp.normal[idx[mask]].astype(np.float64)
+
+    def residual(dx):
+        T = plus(CUR_T, dx)
+        return np.einsum('ij,ij->i', n, transform_points(T, p) - m)[:, None]
+
+    r0, J = _numeric_J(residual, 1)
+    _assert_H_g_match(H, g, r0, J)
+
+
+def test_vplane_icp_matches_numeric_gauss_newton(cloud_pair):
+    target, source = cloud_pair
+    vpicp = VPlaneICP(voxel_size=1.0, max_iter=10, max_dist=2.0)
+    vpicp.set_target(target)
+    H, g, _ = vpicp.calc_H_g_e2(CUR_T, source)
+
+    src_trans = transform_points(CUR_T.astype(np.float32), source)
+    query = vpicp.voxels.query(src_trans, ['mean', 'norm'])
+    mask = query['dist'] < vpicp.max_dist
+    p = source[mask].astype(np.float64)
+    m = query['mean'][mask].astype(np.float64)
+    n = query['norm'][mask].astype(np.float64)
+
+    def residual(dx):
+        T = plus(CUR_T, dx)
+        return np.einsum('ij,ij->i', n, transform_points(T, p) - m)[:, None]
+
+    r0, J = _numeric_J(residual, 1)
+    _assert_H_g_match(H, g, r0, J)
+
+
+def test_ndt_matches_numeric_gauss_newton(cloud_pair):
+    target, source = cloud_pair
+    ndt = NDT(voxel_size=1.0, max_iter=10, max_dist=2.0)
+    ndt.set_target(target)
+    H, g, _ = ndt.calc_H_g_e2(CUR_T, source)
+
+    src_trans = transform_points(CUR_T.astype(np.float32), source)
+    query = ndt.voxels.query(src_trans, ['icov', 'mean'])
+    mask = query['dist'] < ndt.max_dist
+    p = source[mask].astype(np.float64)
+    m = query['mean'][mask].astype(np.float64)
+    W = query['icov'][mask].astype(np.float64)
+
+    def residual(dx):
+        T = plus(CUR_T, dx)
+        return transform_points(T, p) - m
+
+    r0, J = _numeric_J(residual, 3)
+    _assert_H_g_match(H, g, r0, J, W=W)
+
+
+class TestAlignRecovery:
+    """End-to-end: align() must reach the true pose, not just a stationary one."""
+
+    def test_icp_recovers_large_rotation(self):
+        np.random.seed(7)
+        target = np.random.rand(200, 3) * 2.0
+        T_true = makeT(expSO3(np.array([0.5, 0.0, 0.0])),
+                       np.array([0.3, -0.2, 0.1]))
+        source = transform_points(np.linalg.inv(T_true), target)
+        icp = ICP(max_iter=100, max_dist=5.0, tol=1e-9)
+        icp.set_target(target)
+        T = icp.align(source.astype(np.float32))
+        assert np.allclose(T, T_true, atol=1e-3), (
+            f"pose error {np.max(np.abs(T - T_true))}")
+
+    def test_plane_icp_recovers_from_rotated_init(self):
+        rng = np.random.default_rng(0)
+        pts = []
+        for _ in range(3):
+            u = rng.uniform(0.0, 4.0, (400, 2))
+            pts.append(np.column_stack([u[:, 0], u[:, 1], np.zeros(400)]))
+        corner = np.vstack([
+            pts[0],
+            pts[1][:, [0, 2, 1]],
+            pts[2][:, [2, 0, 1]],
+        ])
+        T_true = makeT(expSO3(np.array([0.02, -0.03, 0.05])),
+                       np.array([0.15, -0.1, 0.08]))
+        source = transform_points(np.linalg.inv(T_true), corner)
+        picp = PlaneICP(max_iter=60, max_dist=1.0, tol=1e-8, k=10)
+        picp.set_target(corner)
+        init_T = makeT(expSO3(np.array([0.1, -0.05, 0.08])), np.zeros(3))
+        T = picp.align(source.astype(np.float32), init_T=init_T)
+        assert np.allclose(T, T_true, atol=1e-3), (
+            f"pose error {np.max(np.abs(T - T_true))}")
+
+    @pytest.mark.parametrize("engine_cls", [VPlaneICP, NDT])
+    def test_voxel_solvers_reduce_pose_error(self, engine_cls):
+        # Three gently textured faces of a corner, offset by +0.5 so the
+        # surfaces sit mid-voxel: faces lying exactly on voxel boundaries
+        # give voxels that straddle two faces and blend their normals,
+        # which biases the voxelized cost minimum away from the true pose.
+        rng = np.random.default_rng(3)
+        faces = []
+        for axes in ((0, 1, 2), (0, 2, 1), (2, 0, 1)):
+            u = rng.uniform(0.0, 4.0, (3000, 2))
+            face = np.zeros((3000, 3))
+            face[:, axes[0]] = u[:, 0]
+            face[:, axes[1]] = u[:, 1]
+            face[:, axes[2]] = 0.05 * np.sin(u[:, 0]) * np.cos(u[:, 1])
+            faces.append(face)
+        target = np.vstack(faces) + 0.5
+        T_true = makeT(expSO3(np.array([0.03, -0.02, 0.04])),
+                       np.array([0.2, -0.15, 0.1]))
+        source = transform_points(np.linalg.inv(T_true), target)
+        engine = engine_cls(voxel_size=1.0, max_iter=60, max_dist=2.0, tol=1e-8)
+        engine.set_target(target)
+        T = engine.align(source.astype(np.float32))
+        err_before = np.linalg.norm(np.eye(4) - T_true)
+        err_after = np.linalg.norm(T - T_true)
+        assert err_after < err_before / 10.0, (
+            f"before {err_before}, after {err_after}")
+        assert np.allclose(T[:3, 3], T_true[:3, 3], atol=3e-2)
diff --git a/tests/test_synthetic_data.py b/tests/test_synthetic_data.py
new file mode 100644
index 0000000..f2fe1b3
--- /dev/null
+++ b/tests/test_synthetic_data.py
@@ -0,0 +1,108 @@
+"""
+Pins the degeneracy story of the synthetic clouds in data/.
+
+demo_degeneracy.py and the README claim that the staircase constrains
+exactly five DOFs (everything but cross-step translation ty) and the
+flat plane exactly three (tz, roll, pitch). These tests hold the
+generators — and the committed .pcd files — to that claim, with margin
+assertions so a normals/threshold drift shows up here rather than as a
+silently broken demo.
+"""
+import importlib.util
+import os
+
+import numpy as np
+import pytest
+
+from point_cloud_registration import PlaneICP, analyse_hessian
+
+DATA_DIR = os.path.join(os.path.dirname(os.path.dirname(
+    os.path.abspath(__file__))), "data")
+
+_spec = importlib.util.spec_from_file_location(
+    "generate_synthetic", os.path.join(DATA_DIR, "generate_synthetic.py"))
+generate_synthetic = importlib.util.module_from_spec(_spec)
+_spec.loader.exec_module(generate_synthetic)
+
+
+def _analyse(points):
+    """Degeneracy analysis of a cloud registered against itself at identity."""
+    engine = PlaneICP(max_iter=1, max_dist=1.0, k=10)
+    engine.set_target(points)
+    H = engine.calc_H_g_e2(np.eye(4), points.astype(np.float32))[0]
+    return analyse_hessian(H.astype(float))
+
+
+def _dominant_dof(eigenvector):
+    return int(np.argmax(np.abs(eigenvector)))
+
+
+def test_staircase_flags_exactly_ty():
+    deg = _analyse(generate_synthetic.make_staircase())
+
+    assert deg.num_constrained_dof == 5
+    flagged = np.where(deg.degenerate_mask)[0]
+    assert flagged.shape == (1,)
+    # The single degenerate direction is dominated by ty (DOF index 1).
+    assert _dominant_dof(deg.eigenvectors[:, flagged[0]]) == 1
+    # Margins: the flagged eigenvalue sits clearly below the threshold and
+    # the next one clearly above, so small sampling/normals drift cannot
+    # silently flip the classification.
+    assert deg.eigenvalues[flagged[0]] < 0.8 * deg.lambda_threshold
+    assert deg.eigenvalues[1] > 5.0 * deg.lambda_threshold
+
+
+def test_plane_flags_three_dof():
+    deg = _analyse(generate_synthetic.make_plane())
+
+    assert deg.num_constrained_dof == 3
+    flagged = np.where(deg.degenerate_mask)[0]
+    doms = sorted(_dominant_dof(deg.eigenvectors[:, i]) for i in flagged)
+    assert doms == [0, 1, 5]  # tx, ty, yaw
+    assert np.all(deg.eigenvalues[flagged] < 0.8 * deg.lambda_threshold)
+
+
+def test_generators_are_deterministic():
+    a = generate_synthetic.make_staircase()
+    b = generate_synthetic.make_staircase()
+    np.testing.assert_array_equal(a, b)
+
+    c = generate_synthetic.make_plane()
+    d = generate_synthetic.make_plane()
+    np.testing.assert_array_equal(c, d)
+
+
+def test_pcd_roundtrip(tmp_path):
+    points = generate_synthetic.make_plane(num_points=100)
+    path = tmp_path / "roundtrip.pcd"
+    generate_synthetic.save_pcd(path, points)
+    loaded = generate_synthetic.load_pcd(path)
+    # save_pcd documents float32 / six-decimal storage: measured worst-case
+    # roundtrip error across all shipped clouds is ~7.3e-7.
+    np.testing.assert_allclose(loaded, points, atol=2e-6)
+
+
+@pytest.mark.parametrize("name,expected_degenerate_doms", [
+    ("synthetic_staircase_target.pcd", [1]),
+    ("synthetic_staircase_source.pcd", [1]),
+    ("synthetic_plane_target.pcd", [0, 1, 5]),
+    ("synthetic_plane_source.pcd", [0, 1, 5]),
+])
+def test_committed_pcds_keep_their_degeneracy(name, expected_degenerate_doms):
+    """
+    The files on disk — not just fresh arrays — carry the story: the same
+    DOF identities and the same classification margin, for every committed
+    cloud (each has its own seed and therefore its own margin).
+    """
+    path = os.path.join(DATA_DIR, name)
+    points = generate_synthetic.load_pcd(path)
+    deg = _analyse(points)
+
+    assert deg.num_constrained_dof == 6 - len(expected_degenerate_doms)
+    flagged = np.where(deg.degenerate_mask)[0]
+    doms = sorted(_dominant_dof(deg.eigenvectors[:, i]) for i in flagged)
+    assert doms == expected_degenerate_doms
+    assert np.all(deg.eigenvalues[flagged] < 0.8 * deg.lambda_threshold), (
+        f"weak margin: {deg.eigenvalues[flagged]} vs threshold "
+        f"{deg.lambda_threshold}"
+    )
diff --git a/tests/test_voxel_grid.py b/tests/test_voxel_grid.py
new file mode 100644
index 0000000..a38decc
--- /dev/null
+++ b/tests/test_voxel_grid.py
@@ -0,0 +1,305 @@
+"""
+Tests for VoxelGrid covariance regularization (cov_reg), the explicit
+raise when the min_points mask empties the voxel set, and the threading of
+both knobs through VPlaneICP and NDT.
+"""
+
+import numpy as np
+import pytest
+
+from point_cloud_registration.voxel import VoxelGrid
+from point_cloud_registration.voxelized_plane_icp import VPlaneICP
+from point_cloud_registration.ndt import NDT
+
+VOXEL_SIZE = 1.0
+COV_REG = 1e-3
+
+
+@pytest.fixture
+def coplanar_cloud():
+    """
+    A dense flat z=0 grid, spacing 0.2 over [0, 4).  With voxel_size 1.0 every
+    voxel holds 25 exactly coplanar points, so each per-voxel covariance is
+    exactly singular (smallest eigenvalue 0) and NDT's fast inverse hits its
+    det_A == 0 clamp.
+    """
+    g = np.arange(0.0, 4.0, 0.2)
+    xx, yy = np.meshgrid(g, g)
+    return np.column_stack([xx.ravel(), yy.ravel(), np.zeros(xx.size)])
+
+
+@pytest.fixture
+def dense_cloud():
+    """A well-conditioned random cloud: 4000 points filling 64 voxels."""
+    rng = np.random.default_rng(7)
+    return rng.uniform(0.0, 4.0, size=(4000, 3))
+
+
+@pytest.fixture
+def sparse_cloud():
+    """
+    Six voxels holding four points each.  min_points=10 masks every voxel away
+    (empty voxel set); min_points=3 keeps all six.
+    """
+    return np.array(
+        [[i * 2.0 + 0.1 * j, i * 2.0, 0.3 * j] for i in range(6) for j in range(4)],
+        dtype=float,
+    )
+
+
+@pytest.fixture
+def tiny_two_voxel_cloud():
+    """Eight points in two voxels — small enough to check covariances by hand."""
+    return np.array([
+        [0.1, 0.1, 0.1], [0.2, 0.4, 0.3], [0.5, 0.2, 0.7], [0.3, 0.9, 0.2],
+        [1.1, 0.1, 0.1], [1.4, 0.6, 0.3], [1.9, 0.2, 0.8], [1.3, 0.7, 0.4],
+    ], dtype=float)
+
+
+@pytest.fixture
+def bumpy_pair():
+    """
+    Three gently bumpy, mutually orthogonal faces ("corner"), plus a purely
+    translated copy.  Normals span R^3, so all six DOF are constrained.
+
+    Returns (target, source, T_true) where T_true is the transform align()
+    should recover, i.e. the one mapping source back onto target.
+    """
+    rng = np.random.default_rng(3)
+    n = 3000
+    a = rng.uniform(0.0, 4.0, size=(n, 2))
+    b = rng.uniform(0.0, 4.0, size=(n, 2))
+    c = rng.uniform(0.0, 4.0, size=(n, 2))
+
+    def bump(u, v):
+        return 0.05 * np.sin(u) * np.cos(v)
+
+    face_z = np.column_stack([a[:, 0], a[:, 1], bump(a[:, 0], a[:, 1])])
+    face_x = np.column_stack([bump(b[:, 0], b[:, 1]), b[:, 0], b[:, 1]])
+    face_y = np.column_stack([c[:, 0], bump(c[:, 0], c[:, 1]), c[:, 1]])
+    target = np.vstack([face_z, face_x, face_y])
+
+    t = np.array([0.12, -0.09, 0.07])
+    source = target + t
+    T_true = np.eye(4)
+    T_true[:3, 3] = -t
+    return target, source, T_true
+
+
+# --------------------------------------------------------------------------
+# 1. cov_reg regularizes coplanar voxels
+# --------------------------------------------------------------------------
+
+def test_cov_reg_regularizes_coplanar_voxels(coplanar_cloud):
+    """
+    An isotropic +cov_reg*I shift lifts the null direction of an exactly
+    coplanar voxel, so the fast inverse becomes a true inverse instead of
+    hitting the det_A == 0 clamp.
+    """
+    # Fixture guard: without regularization the covariances really are singular.
+    plain = VoxelGrid(VOXEL_SIZE, min_points=3)
+    plain.set_points(coplanar_cloud)
+    assert np.allclose(np.linalg.eigvalsh(plain.cov)[:, 0], 0.0, atol=1e-12), (
+        "fixture is not coplanar; smallest eigenvalues "
+        f"{np.linalg.eigvalsh(plain.cov)[:, 0]}"
+    )
+
+    vg = VoxelGrid(VOXEL_SIZE, min_points=3, cov_reg=COV_REG)
+    vg.set_points(coplanar_cloud)
+    vg.calc_icov()
+
+    assert np.all(np.isfinite(vg.icov)), "icov has non-finite entries"
+
+    eigenvalues = np.linalg.eigvalsh(vg.cov)
+    assert np.all(eigenvalues >= COV_REG - 1e-12), (
+        f"covariance eigenvalues fell below the shift: min {eigenvalues.min()}"
+    )
+
+    # The clamp was not taken: icov really inverts cov.
+    identity = np.broadcast_to(np.eye(3), vg.cov.shape)
+    assert np.allclose(vg.icov @ vg.cov, identity, atol=1e-6), (
+        "icov is not the inverse of cov; the singular clamp was still hit"
+    )
+
+
+def test_cov_reg_preserves_normals(coplanar_cloud):
+    """An isotropic shift leaves the eigenvectors — and so the normals — intact."""
+    plain = VoxelGrid(VOXEL_SIZE, min_points=3)
+    plain.set_points(coplanar_cloud)
+
+    regularized = VoxelGrid(VOXEL_SIZE, min_points=3, cov_reg=COV_REG)
+    regularized.set_points(coplanar_cloud)
+
+    assert np.allclose(np.abs(plain.norm), np.abs(regularized.norm), atol=1e-12), (
+        "cov_reg rotated the voxel normals"
+    )
+
+
+# --------------------------------------------------------------------------
+# 2. cov_reg=0 preserves upstream values exactly
+# --------------------------------------------------------------------------
+
+def test_default_cov_reg_equals_explicit_zero(dense_cloud):
+    """The default must be bit-for-bit the unregularized path."""
+    default = VoxelGrid(VOXEL_SIZE, min_points=3)
+    default.set_points(dense_cloud)
+
+    explicit = VoxelGrid(VOXEL_SIZE, min_points=3, cov_reg=0.0)
+    explicit.set_points(dense_cloud)
+
+    assert default.cov_reg == 0.0
+    assert np.array_equal(default.cov, explicit.cov)
+    assert np.array_equal(default.mean, explicit.mean)
+    assert np.array_equal(default.norm, explicit.norm)
+
+
+def test_negative_cov_reg_is_rejected():
+    """A negative shift would make covariances indefinite: reject, not ignore."""
+    with pytest.raises(ValueError, match="cov_reg"):
+        VoxelGrid(VOXEL_SIZE, cov_reg=-1e-3)
+
+
+def test_default_cov_reg_matches_hand_computed_statistics(tiny_two_voxel_cloud):
+    """
+    With cov_reg at its default the per-voxel mean/covariance/normal must equal
+    the textbook values (ddof=1 sample covariance), i.e. upstream behavior.
+    """
+    vg = VoxelGrid(VOXEL_SIZE, min_points=3)
+    vg.set_points(tiny_two_voxel_cloud)
+
+    assert len(vg.mean) == 2
+
+    for i, sl in enumerate([slice(0, 4), slice(4, 8)]):
+        points = tiny_two_voxel_cloud[sl]
+        expected_mean = points.mean(axis=0)
+        expected_cov = np.cov(points.T, ddof=1)
+        expected_norm = np.linalg.eigh(expected_cov)[1][:, 0]
+
+        assert np.allclose(vg.mean[i], expected_mean, atol=1e-12)
+        assert np.allclose(vg.cov[i], expected_cov, atol=1e-12)
+        assert np.allclose(np.abs(vg.norm[i]), np.abs(expected_norm), atol=1e-9)
+
+
+# --------------------------------------------------------------------------
+# 3. raise-on-empty
+# --------------------------------------------------------------------------
+
+def test_set_points_raises_when_min_points_empties_the_grid(sparse_cloud):
+    """
+    Every voxel holds four points, so min_points=10 leaves nothing behind.  The
+    empty grid must be reported explicitly rather than relying on whichever
+    KDTree backend happens to be compiled in.
+    """
+    with pytest.raises(ValueError, match="data_pts should be non-empty"):
+        VoxelGrid(VOXEL_SIZE, min_points=10).set_points(sparse_cloud)
+
+
+def test_set_points_succeeds_when_min_points_admits_voxels(sparse_cloud):
+    """The same cloud with a reachable threshold builds a usable grid."""
+    vg = VoxelGrid(VOXEL_SIZE, min_points=3)
+    vg.set_points(sparse_cloud)
+
+    assert len(vg.mean) > 0
+    assert len(vg.mean) == len(vg.cov) == len(vg.norm)
+
+
+# --------------------------------------------------------------------------
+# 4. VPlaneICP threading
+# --------------------------------------------------------------------------
+
+def test_vplane_icp_threads_min_points_and_cov_reg(sparse_cloud):
+    """Both knobs must reach the VoxelGrid that set_target builds."""
+    engine = VPlaneICP(voxel_size=VOXEL_SIZE, min_points=3, cov_reg=COV_REG)
+    engine.set_target(sparse_cloud)
+
+    assert engine.is_target_set()
+    assert len(engine.voxels.mean) > 0
+    assert engine.voxels.min_points == 3
+    assert engine.voxels.cov_reg == COV_REG
+
+
+def test_vplane_icp_propagates_empty_voxel_error(sparse_cloud):
+    """An unreachable min_points surfaces as a ValueError through set_target."""
+    engine = VPlaneICP(voxel_size=VOXEL_SIZE, min_points=10)
+    with pytest.raises(ValueError, match="data_pts should be non-empty"):
+        engine.set_target(sparse_cloud)
+
+
+def test_vplane_icp_defaults_preserve_upstream(dense_cloud):
+    """Untouched constructor arguments keep the upstream (10, 0.0) behavior."""
+    engine = VPlaneICP(voxel_size=VOXEL_SIZE)
+    engine.set_target(dense_cloud)
+
+    assert engine.voxels.min_points == 10
+    assert engine.voxels.cov_reg == 0.0
+
+
+# --------------------------------------------------------------------------
+# 5. NDT threading
+# --------------------------------------------------------------------------
+
+def test_ndt_threads_cov_reg_for_coplanar_target(coplanar_cloud):
+    """
+    NDT.set_target runs calc_icov itself, so cov_reg has to be in place by then;
+    with it the inverse covariances are finite, moderate and genuine inverses.
+    """
+    engine = NDT(voxel_size=VOXEL_SIZE, min_points=3, cov_reg=COV_REG)
+    engine.set_target(coplanar_cloud)
+
+    assert engine.is_target_set()
+    assert engine.voxels.min_points == 3
+    assert engine.voxels.cov_reg == COV_REG
+    assert np.all(np.isfinite(engine.voxels.icov))
+
+    identity = np.broadcast_to(np.eye(3), engine.voxels.cov.shape)
+    assert np.allclose(engine.voxels.icov @ engine.voxels.cov, identity, atol=1e-6)
+
+    # The null direction is now weighted at ~1/cov_reg rather than clamped away.
+    max_icov = np.max(np.abs(engine.voxels.icov))
+    assert 1e2 < max_icov < 1e4, f"icov magnitude out of range: {max_icov}"
+
+
+def test_ndt_defaults_hit_singular_clamp_on_coplanar_target(coplanar_cloud):
+    """
+    Without cov_reg the det_A == 0 clamp replaces the determinant with 1e6, which
+    collapses icov to ~0: the voxel contributes no information at all.  This is
+    the behavior cov_reg exists to avoid, and it must remain the default.
+    """
+    engine = NDT(voxel_size=VOXEL_SIZE)
+    engine.set_target(coplanar_cloud)
+
+    assert engine.voxels.cov_reg == 0.0
+
+    icov = engine.voxels.icov
+    identity = np.broadcast_to(np.eye(3), engine.voxels.cov.shape)
+    assert not np.allclose(icov @ engine.voxels.cov, identity, atol=1e-6), (
+        "clamped icov unexpectedly inverts cov"
+    )
+    assert np.max(np.abs(icov)) < 1e-6, (
+        f"expected the clamp to collapse icov, got max {np.max(np.abs(icov))}"
+    )
+
+    # Contrast: the regularized grid carries ~9 orders of magnitude more weight.
+    regularized = NDT(voxel_size=VOXEL_SIZE, min_points=3, cov_reg=COV_REG)
+    regularized.set_target(coplanar_cloud)
+    assert np.max(np.abs(regularized.voxels.icov)) > 1e6 * np.max(np.abs(icov))
+
+
+# --------------------------------------------------------------------------
+# 6. end-to-end sanity
+# --------------------------------------------------------------------------
+
+def test_vplane_icp_end_to_end_alignment_with_cov_reg(bumpy_pair):
+    """Threading the new knobs must not disturb registration itself."""
+    target, source, T_true = bumpy_pair
+
+    engine = VPlaneICP(voxel_size=0.5, max_iter=50, max_dist=2.0, tol=1e-6,
+                       min_points=3, cov_reg=COV_REG)
+    engine.set_target(target)
+
+    T = engine.align(source)
+
+    assert np.all(np.isfinite(T))
+    assert np.allclose(T, T_true, atol=2e-2), (
+        f"max diff from ground truth {np.max(np.abs(T - T_true))}"
+    )
diff --git a/tests/test_vpicp.py b/tests/test_vpicp.py
index 8ee86ef..6d9d644 100644
--- a/tests/test_vpicp.py
+++ b/tests/test_vpicp.py
@@ -2,6 +2,7 @@
 import pytest
 from point_cloud_registration import VPlaneICP
 from point_cloud_registration import expSO3
+from point_cloud_registration.math_tools import makeT
 
 
 @pytest.fixture
@@ -19,17 +20,24 @@ def generate_test_data():
     return target, normals, source
 
 
-def test_calc_H_g_e2(generate_test_data):
+@pytest.mark.parametrize(
+    "cur_T",
+    [np.eye(4), makeT(expSO3(np.array([0.3, -0.2, 0.4])), np.array([0.1, 0.2, -0.1]))],
+    ids=["identity_pose", "rotated_pose"],
+)
+@pytest.mark.parametrize("with_outliers", [False, True], ids=["all_inliers", "with_outliers"])
+def test_calc_H_g_e2(generate_test_data, cur_T, with_outliers):
     """
-    Test that calc_H_g_e2 and calc_H_g_e2x produce the same results.
+    Test that calc_H_g_e2 and calc_H_g_e2_no_parallel_ver produce the same results.
     """
-    target, normals, source = generate_test_data
+    target, _, source = generate_test_data
+    if with_outliers:
+        # Points far outside the target's reach: rejected by the max_dist gate.
+        source = np.vstack([source[:10] + 50.0, source])
     source = source.astype(np.float32)
     plane_icp = VPlaneICP(voxel_size=1.0, max_iter=10, max_dist=2.0, tol=1e-3)
     plane_icp.set_target(target)
-    plane_icp.normal = normals
 
-    cur_T = np.eye(4)  # Initial transformation (identity matrix)
 
     # Compute results using both methods
     H1, g1, e2_1 = plane_icp.calc_H_g_e2(cur_T, source)