From 7dc2f0fcff2d0e4e64f3c7d47b83b212ed3a3a09 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 15:20:44 +0200
Subject: [PATCH 01/17] Preserve signed one-dimensional torus integration

---
 .../hypertorus/hypertoroidal_uniform_distribution.py           | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/pyrecest/distributions/hypertorus/hypertoroidal_uniform_distribution.py b/src/pyrecest/distributions/hypertorus/hypertoroidal_uniform_distribution.py
index c9da343bcd..9e0fc48a93 100644
--- a/src/pyrecest/distributions/hypertorus/hypertoroidal_uniform_distribution.py
+++ b/src/pyrecest/distributions/hypertorus/hypertoroidal_uniform_distribution.py
@@ -206,7 +206,8 @@ def integrate(self, integration_boundaries=None) -> float:
             left, right = integration_boundaries
         left = _validate_boundary("left", left, self.dim)
         right = _validate_boundary("right", right, self.dim)
-        _validate_boundary_order(left, right)
+        if self.dim > 1:
+            _validate_boundary_order(left, right)
 
         volume = prod(right - left)
         return 1.0 / (2.0 * pi) ** self.dim * volume

From 33037f73ff7abe90dbb663bcbdd504228d27aef4 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 15:21:04 +0200
Subject: [PATCH 02/17] Restore signed scalar integration regression

---
 .../test_hypertoroidal_uniform_distribution.py            | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/tests/distributions/test_hypertoroidal_uniform_distribution.py b/tests/distributions/test_hypertoroidal_uniform_distribution.py
index 2ab1c1c3e5..f264ee262d 100644
--- a/tests/distributions/test_hypertoroidal_uniform_distribution.py
+++ b/tests/distributions/test_hypertoroidal_uniform_distribution.py
@@ -82,7 +82,6 @@ def test_integrate_validates_boundary_shapes():
 
     with pytest.raises(ShapeError, match="left"):
         dist.integrate((zeros((1,)), ones((2,))))
-
     with pytest.raises(ShapeError, match="right"):
         dist.integrate((zeros((2,)), ones((1,))))
 
@@ -94,11 +93,12 @@ def test_integrate_rejects_reversed_boundaries():
         dist.integrate((array([0.0, 1.0]), array([1.0, 0.5])))
 
 
-def test_integrate_rejects_reversed_scalar_boundaries():
+def test_integrate_preserves_signed_scalar_boundaries():
     dist = HypertoroidalUniformDistribution(1)
 
-    with pytest.raises(ValueError, match="increasing"):
-        dist.integrate((array(1.0), array(0.0)))
+    assert dist.integrate((array(1.0), array(0.0))) == pytest.approx(
+        -1.0 / (2.0 * pi)
+    )
 
 
 def test_integrate_accepts_scalar_boundaries_for_one_dimension():

From 3d54f480bf16dc9a267110d59f198a764126087c Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 15:21:33 +0200
Subject: [PATCH 03/17] Use backend conjugation in wrapped-Cauchy test

---
 tests/distributions/test_wrapped_cauchy_distribution.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/tests/distributions/test_wrapped_cauchy_distribution.py b/tests/distributions/test_wrapped_cauchy_distribution.py
index 4c309117f0..43f39a5fb2 100644
--- a/tests/distributions/test_wrapped_cauchy_distribution.py
+++ b/tests/distributions/test_wrapped_cauchy_distribution.py
@@ -6,7 +6,7 @@
 import pyrecest.backend
 
 # pylint: disable=no-name-in-module,no-member
-from pyrecest.backend import arange, array, pi
+from pyrecest.backend import arange, array, conj, pi
 from pyrecest.distributions.circle.custom_circular_distribution import (
     CustomCircularDistribution,
 )
@@ -91,7 +91,7 @@ def test_trigonometric_moment_accepts_negative_integer_orders(self):
         positive_moment = dist.trigonometric_moment(2)
         negative_moment = dist.trigonometric_moment(-2)
 
-        npt.assert_allclose(negative_moment, positive_moment.conjugate(), rtol=1e-12)
+        npt.assert_allclose(negative_moment, conj(positive_moment), rtol=1e-12)
 
     @unittest.skipIf(
         pyrecest.backend.__backend_name__ in ("pytorch", "jax"),

From d08f786186430b2c2ab51811d27726bf9bd102fd Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 15:23:27 +0200
Subject: [PATCH 04/17] Keep integration regression diff minimal

---
 tests/distributions/test_hypertoroidal_uniform_distribution.py | 1 +
 1 file changed, 1 insertion(+)

diff --git a/tests/distributions/test_hypertoroidal_uniform_distribution.py b/tests/distributions/test_hypertoroidal_uniform_distribution.py
index f264ee262d..6b95b0f967 100644
--- a/tests/distributions/test_hypertoroidal_uniform_distribution.py
+++ b/tests/distributions/test_hypertoroidal_uniform_distribution.py
@@ -82,6 +82,7 @@ def test_integrate_validates_boundary_shapes():
 
     with pytest.raises(ShapeError, match="left"):
         dist.integrate((zeros((1,)), ones((2,))))
+
     with pytest.raises(ShapeError, match="right"):
         dist.integrate((zeros((2,)), ones((1,))))
 

From 49ea581843fd2eb42a74824d8ec0046fd7e49658 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 15:44:13 +0200
Subject: [PATCH 05/17] Accept singleton mode vectors for von Mises updates

---
 .../circle/von_mises_distribution.py          | 33 ++++---------------
 1 file changed, 7 insertions(+), 26 deletions(-)

diff --git a/src/pyrecest/distributions/circle/von_mises_distribution.py b/src/pyrecest/distributions/circle/von_mises_distribution.py
index 20ed7d1489..da0e6f23d7 100644
--- a/src/pyrecest/distributions/circle/von_mises_distribution.py
+++ b/src/pyrecest/distributions/circle/von_mises_distribution.py
@@ -99,9 +99,13 @@ def set_mode(self, mode):
         """Return a copy with a replaced mode direction.
 
         For a von Mises distribution, the mode and mean direction are both
-        represented by ``mu``.  The zero-concentration case is uniform, where
-        setting ``mu`` still preserves the distribution family and API contract.
+        represented by ``mu``.  Generic manifold APIs represent a
+        one-dimensional mode as a singleton vector, so accept that form in
+        addition to the native scalar representation.
         """
+        mode = array(mode)
+        if mode.shape == (1,):
+            mode = mode[0]
         return self.set_mean(mode)
 
     @staticmethod
@@ -238,28 +242,5 @@ def trigonometric_moment_analytic(self, n: int):
                 * exp(1j * n * self.mu)
             )
         else:
-            raise NotImplementedError("Not implemented")
-
+            raise NotImplementedError()
         return m
-
-    @staticmethod
-    def from_moment(m):
-        """
-        Obtain a VM distribution from a given first trigonometric moment.
-
-        Parameters:
-            m (scalar): First trigonometric moment (complex number).
-
-        Returns:
-            vm (VMDistribution): Distribution obtained by moment matching.
-        """
-        kappa_ = VonMisesDistribution.besselratio_inverse(0, abs(m))
-        if VonMisesDistribution._as_float_scalar(kappa_, "kappa") == 0.0:
-            mu_ = array(0.0)
-        else:
-            mu_ = mod(arctan2(imag(m), real(m)), 2.0 * pi)
-        vm = VonMisesDistribution(mu_, kappa_)
-        return vm
-
-    def __str__(self) -> str:
-        return f"VonMisesDistribution: mu = {self.mu}, kappa = {self.kappa}"

From 05a8d5e2b22dbb1d2e300667fe183853643d4bde Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 15:46:41 +0200
Subject: [PATCH 06/17] Restore von Mises helpers after mode-shape fix

---
 .../circle/von_mises_distribution.py          | 31 ++++++++++++++++---
 1 file changed, 27 insertions(+), 4 deletions(-)

diff --git a/src/pyrecest/distributions/circle/von_mises_distribution.py b/src/pyrecest/distributions/circle/von_mises_distribution.py
index da0e6f23d7..d858b03e4f 100644
--- a/src/pyrecest/distributions/circle/von_mises_distribution.py
+++ b/src/pyrecest/distributions/circle/von_mises_distribution.py
@@ -99,9 +99,9 @@ def set_mode(self, mode):
         """Return a copy with a replaced mode direction.
 
         For a von Mises distribution, the mode and mean direction are both
-        represented by ``mu``.  Generic manifold APIs represent a
-        one-dimensional mode as a singleton vector, so accept that form in
-        addition to the native scalar representation.
+        represented by ``mu``. Generic manifold APIs represent a one-dimensional
+        mode as a singleton vector, so accept that form in addition to the native
+        scalar representation.
         """
         mode = array(mode)
         if mode.shape == (1,):
@@ -242,5 +242,28 @@ def trigonometric_moment_analytic(self, n: int):
                 * exp(1j * n * self.mu)
             )
         else:
-            raise NotImplementedError()
+            raise NotImplementedError("Not implemented")
+
         return m
+
+    @staticmethod
+    def from_moment(m):
+        """
+        Obtain a VM distribution from a given first trigonometric moment.
+
+        Parameters:
+            m (scalar): First trigonometric moment (complex number).
+
+        Returns:
+            vm (VMDistribution): Distribution obtained by moment matching.
+        """
+        kappa_ = VonMisesDistribution.besselratio_inverse(0, abs(m))
+        if VonMisesDistribution._as_float_scalar(kappa_, "kappa") == 0.0:
+            mu_ = array(0.0)
+        else:
+            mu_ = mod(arctan2(imag(m), real(m)), 2.0 * pi)
+        vm = VonMisesDistribution(mu_, kappa_)
+        return vm
+
+    def __str__(self) -> str:
+        return f"VonMisesDistribution: mu = {self.mu}, kappa = {self.kappa}"

From d61968efb0ebfb4c5552c1c720d3053bf2c7650f Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 16:13:05 +0200
Subject: [PATCH 07/17] Use explicit coarse-grid alignment tolerance

---
 .../filters/test_hyperhemispherical_grid_filter_vmf_update.py  | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/tests/filters/test_hyperhemispherical_grid_filter_vmf_update.py b/tests/filters/test_hyperhemispherical_grid_filter_vmf_update.py
index a3c0e2ac4c..ad0796b507 100644
--- a/tests/filters/test_hyperhemispherical_grid_filter_vmf_update.py
+++ b/tests/filters/test_hyperhemispherical_grid_filter_vmf_update.py
@@ -27,7 +27,8 @@ def test_accepts_numerically_equatorial_vmf_measurement(self):
 
         estimate = filter_.get_point_estimate()
         self.assertAlmostEqual(float(linalg.norm(estimate)), 1.0, places=5)
-        self.assertGreater(abs(float(estimate[0])), 0.9)
+        alignment = abs(float(estimate @ measurement))
+        self.assertGreater(alignment, math.cos(math.radians(30.0)))
 
     def test_rejects_vmf_measurement_outside_equator_tolerance(self):
         filter_ = HyperhemisphericalGridFilter(50, 2)

From 41b0de166932f72b79b3c85f5dde63dc6b50770d Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:31:35 +0200
Subject: [PATCH 08/17] Match SciPy boolean axis handling in PyTorch
 fftconvolve

---
 src/pyrecest/_backend/pytorch/signal.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/pyrecest/_backend/pytorch/signal.py b/src/pyrecest/_backend/pytorch/signal.py
index 43dbfa4a8e..85bd8f0cff 100644
--- a/src/pyrecest/_backend/pytorch/signal.py
+++ b/src/pyrecest/_backend/pytorch/signal.py
@@ -5,7 +5,7 @@
 
 
 def _is_boolean_scalar(axis):
-    return isinstance(axis, (bool, _np.bool_)) or (
+    return isinstance(axis, _np.bool_) or (
         isinstance(axis, _np.ndarray) and axis.shape == () and axis.dtype == _np.bool_
     )
 

From 7d9375c45aee412a6e6ac17ebd04b8ccf4fd315f Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:31:46 +0200
Subject: [PATCH 09/17] Remove contradictory boolean-axis expectations

---
 .../test_pytorch_fftconvolve_axes_validation.py               | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/tests/backend_support/test_pytorch_fftconvolve_axes_validation.py b/tests/backend_support/test_pytorch_fftconvolve_axes_validation.py
index bd9393ed84..a585886cab 100644
--- a/tests/backend_support/test_pytorch_fftconvolve_axes_validation.py
+++ b/tests/backend_support/test_pytorch_fftconvolve_axes_validation.py
@@ -41,9 +41,9 @@ def test_pytorch_fftconvolve_rejects_non_integer_axes(axes):
 
 @pytest.mark.parametrize(
     "axes",
-    [True, False, np.bool_(True), np.array(True)],
+    [np.bool_(True), np.bool_(False), np.array(True), np.array(False)],
 )
-def test_pytorch_fftconvolve_rejects_boolean_axes(axes):
+def test_pytorch_fftconvolve_rejects_numpy_boolean_axes(axes):
     _skip_unless_pytorch()
 
     first = backend.asarray([1.0, 2.0])

From 3a81811e4430156461bdb4e30a4fa5f29bdd5e83 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:32:22 +0200
Subject: [PATCH 10/17] Accept scalar array axes in PyTorch flip

---
 .../backend_support/_pytorch_allclose_device_contract.py     | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/src/pyrecest/backend_support/_pytorch_allclose_device_contract.py b/src/pyrecest/backend_support/_pytorch_allclose_device_contract.py
index bfd0a13198..4544d2f961 100644
--- a/src/pyrecest/backend_support/_pytorch_allclose_device_contract.py
+++ b/src/pyrecest/backend_support/_pytorch_allclose_device_contract.py
@@ -96,7 +96,10 @@ def _flip_axes(axis, ndim):
             return list(range(ndim))
         if isinstance(axis, (int, np.integer)):
             return [int(axis)]
-        return [int(_operator_index(one_axis)) for one_axis in axis]
+        try:
+            return [int(_operator_index(axis))]
+        except TypeError:
+            return [int(_operator_index(one_axis)) for one_axis in axis]
 
     def flip(x, axis):
         x = pytorch_backend.array(x)

From 18a54afe582fed5f163d0315c72f825623d2ded7 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:33:06 +0200
Subject: [PATCH 11/17] Accept backend integer scalars for wrapped-normal
 moments

---
 .../circle/wrapped_normal_distribution.py             | 11 +++++++++--
 1 file changed, 9 insertions(+), 2 deletions(-)

diff --git a/src/pyrecest/distributions/circle/wrapped_normal_distribution.py b/src/pyrecest/distributions/circle/wrapped_normal_distribution.py
index 7cf9d5ac24..0e4d1f5319 100644
--- a/src/pyrecest/distributions/circle/wrapped_normal_distribution.py
+++ b/src/pyrecest/distributions/circle/wrapped_normal_distribution.py
@@ -1,5 +1,6 @@
 from math import isfinite
 from numbers import Integral
+from operator import index as _operator_index
 from typing import Union
 
 import pyrecest.backend
@@ -187,9 +188,15 @@ def ncdf(from_, to):
         return squeeze(val)
 
     def trigonometric_moment(self, n: Union[int, int32, int64]):
-        if isinstance(n, bool) or not isinstance(n, Integral):
+        dtype = getattr(n, "dtype", None)
+        if isinstance(n, bool) or (
+            dtype is not None and str(dtype).lower().endswith("bool")
+        ):
             raise ValueError("n must be an integer")
-        n = int(n)
+        try:
+            n = int(_operator_index(n))
+        except (TypeError, ValueError) as exc:
+            raise ValueError("n must be an integer") from exc
         return exp(1j * n * self.scalar_mu - n**2 * self.sigma**2 / 2)
 
     def multiply(

From 2e5f5a181269a2373df52863c0f49fb71860ec21 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:34:44 +0200
Subject: [PATCH 12/17] Keep SE3 plotting backend independent

---
 .../abstract_se3_distribution.py              | 32 +++++++++++++++----
 1 file changed, 25 insertions(+), 7 deletions(-)

diff --git a/src/pyrecest/distributions/abstract_se3_distribution.py b/src/pyrecest/distributions/abstract_se3_distribution.py
index f7daa6e36c..d158520fcf 100644
--- a/src/pyrecest/distributions/abstract_se3_distribution.py
+++ b/src/pyrecest/distributions/abstract_se3_distribution.py
@@ -13,7 +13,6 @@
     int64,
     max,
     min,
-    spatial,
 )
 
 from .cart_prod.abstract_lin_bounded_cart_prod_distribution import (
@@ -63,13 +62,32 @@ def plot_state(
     @staticmethod
     def plot_point(se3point):  # pylint: disable=too-many-locals
         """Visualize just a point in the SE(3) domain (no uncertainties are considered)"""
-        # se3point[:4] is (w, x, y, z)
+        # se3point[:4] is (w, x, y, z). Compute the rotation matrix directly so
+        # plotting remains available on backends that do not expose SciPy Rotation.
         w, x, y, z = se3point[:4]
-
-        # Rotation.from_quat expects (x, y, z, w)
-        q_xyzw = array([x, y, z, w])
-        rot = spatial.Rotation.from_quat(q_xyzw)
-        rotMat = rot.as_matrix()  # 3x3 rotation matrix
+        norm_squared = w * w + x * x + y * y + z * z
+        if not bool(norm_squared > 0):
+            raise ValueError("Quaternion must have nonzero norm.")
+        scale = 2.0 / norm_squared
+        rotMat = array(
+            [
+                [
+                    1.0 - scale * (y * y + z * z),
+                    scale * (x * y - z * w),
+                    scale * (x * z + y * w),
+                ],
+                [
+                    scale * (x * y + z * w),
+                    1.0 - scale * (x * x + z * z),
+                    scale * (y * z - x * w),
+                ],
+                [
+                    scale * (x * z - y * w),
+                    scale * (y * z + x * w),
+                    1.0 - scale * (x * x + y * y),
+                ],
+            ]
+        )
 
         pos = se3point[4:]
 

From 9ce8c8d76002bcb62a3bf8f0b13f0bd1bc7bae16 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:35:15 +0200
Subject: [PATCH 13/17] Compare wrapped-Cauchy moments through the active
 backend

---
 tests/distributions/test_wrapped_cauchy_distribution.py | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/tests/distributions/test_wrapped_cauchy_distribution.py b/tests/distributions/test_wrapped_cauchy_distribution.py
index 43f39a5fb2..47ab9ba8bc 100644
--- a/tests/distributions/test_wrapped_cauchy_distribution.py
+++ b/tests/distributions/test_wrapped_cauchy_distribution.py
@@ -6,7 +6,7 @@
 import pyrecest.backend
 
 # pylint: disable=no-name-in-module,no-member
-from pyrecest.backend import arange, array, conj, pi
+from pyrecest.backend import allclose, arange, array, conj, pi
 from pyrecest.distributions.circle.custom_circular_distribution import (
     CustomCircularDistribution,
 )
@@ -91,7 +91,9 @@ def test_trigonometric_moment_accepts_negative_integer_orders(self):
         positive_moment = dist.trigonometric_moment(2)
         negative_moment = dist.trigonometric_moment(-2)
 
-        npt.assert_allclose(negative_moment, conj(positive_moment), rtol=1e-12)
+        self.assertTrue(
+            allclose(negative_moment, conj(positive_moment), rtol=1e-12)
+        )
 
     @unittest.skipIf(
         pyrecest.backend.__backend_name__ in ("pytorch", "jax"),

From 197974da0faf8f38cb512374bbe438da2d94d7da Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:35:47 +0200
Subject: [PATCH 14/17] Compare wrapped-Laplace moments through the active
 backend

---
 tests/distributions/test_wrapped_laplace_distribution.py | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/tests/distributions/test_wrapped_laplace_distribution.py b/tests/distributions/test_wrapped_laplace_distribution.py
index da51713a6c..349846eac6 100644
--- a/tests/distributions/test_wrapped_laplace_distribution.py
+++ b/tests/distributions/test_wrapped_laplace_distribution.py
@@ -7,7 +7,7 @@
 import pyrecest.backend
 
 # pylint: disable=no-name-in-module,no-member
-from pyrecest.backend import arange, array, exp, linspace, pi
+from pyrecest.backend import allclose, arange, array, conj, exp, linspace, pi
 from pyrecest.distributions.circle.wrapped_laplace_distribution import (
     WrappedLaplaceDistribution,
 )
@@ -96,7 +96,9 @@ def test_trigonometric_moment_accepts_negative_integer_orders(self):
         positive_moment = self.wl.trigonometric_moment(2)
         negative_moment = self.wl.trigonometric_moment(-2)
 
-        npt.assert_allclose(negative_moment, positive_moment.conjugate(), rtol=1e-12)
+        self.assertTrue(
+            allclose(negative_moment, conj(positive_moment), rtol=1e-12)
+        )
 
     @unittest.skipIf(
         pyrecest.backend.__backend_name__ in ("pytorch", "jax"),

From 2cb60803cd690cdfa15eeab1179638d61787ddfa Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 17:36:21 +0200
Subject: [PATCH 15/17] Handle optional RTS control entries on PyTorch

---
 src/pyrecest/smoothers/abstract_smoother.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/pyrecest/smoothers/abstract_smoother.py b/src/pyrecest/smoothers/abstract_smoother.py
index 0b4c498bf3..6c9f1e9e90 100644
--- a/src/pyrecest/smoothers/abstract_smoother.py
+++ b/src/pyrecest/smoothers/abstract_smoother.py
@@ -109,7 +109,7 @@ def _normalize_vector_sequence(  # pylint: disable=too-many-return-statements
 
         try:
             values_arr = asarray(values)
-        except (TypeError, ValueError):
+        except (TypeError, ValueError, RuntimeError):
             values_arr = None
         if values_arr is not None:
             if ndim(values_arr) == 0:

From 466b7c8df4c974d90fb1b2ba6d6f3027238247c2 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 13:16:06 +0200
Subject: [PATCH 16/17] Avoid Merwe scale cancellation

---
 src/pyrecest/sampling/sigma_points.py | 20 ++++++++++++++++----
 1 file changed, 16 insertions(+), 4 deletions(-)

diff --git a/src/pyrecest/sampling/sigma_points.py b/src/pyrecest/sampling/sigma_points.py
index 673f04cff9..8ddfb800dc 100644
--- a/src/pyrecest/sampling/sigma_points.py
+++ b/src/pyrecest/sampling/sigma_points.py
@@ -90,6 +90,18 @@ def _validate_finite_scalar(value, name: str) -> float:
     return result
 
 
+def _merwe_scale(n: int, alpha: float, kappa: float) -> float:
+    """Return ``alpha**2 * (n + kappa)`` without subtractive cancellation."""
+
+    try:
+        scale = alpha * alpha * (n + kappa)
+    except OverflowError as exc:
+        raise ValueError("alpha**2 * (n + kappa) must be finite and positive") from exc
+    if not math.isfinite(scale) or scale <= 0.0:
+        raise ValueError("alpha**2 * (n + kappa) must be finite and positive")
+    return scale
+
+
 def _validate_sigma_inputs(x, P, n: int):
     if _has_complex_dtype(x):
         raise ValueError("x must contain real values")
@@ -138,8 +150,8 @@ def __init__(self, n: int, alpha: float, beta: float, kappa: float):
 
     def _compute_weights(self):
         n = self.n
-        lam = self.alpha**2 * (n + self.kappa) - n
-        scale = n + lam
+        scale = _merwe_scale(n, self.alpha, self.kappa)
+        lam = scale - n
 
         self.Wm = concatenate(
             [
@@ -168,11 +180,11 @@ def sigma_points(self, x, P):
             State covariance, shape ``(n, n)``.
         """
         n = self.n
-        lam = self.alpha**2 * (n + self.kappa) - n
+        scale = _merwe_scale(n, self.alpha, self.kappa)
 
         x, P = _validate_sigma_inputs(x, P, n)
 
-        U = linalg.cholesky((n + lam) * P)  # lower-triangular
+        U = linalg.cholesky(scale * P)  # lower-triangular
 
         positive = [x + U[:, i] for i in range(n)]
         negative = [x - U[:, i] for i in range(n)]

From 2e1b34ba4db4147bca3f9679322294adad9f2daf Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 13:16:21 +0200
Subject: [PATCH 17/17] Test small Merwe alpha scaling

---
 tests/test_sigma_points_small_alpha.py | 30 ++++++++++++++++++++++++++
 1 file changed, 30 insertions(+)
 create mode 100644 tests/test_sigma_points_small_alpha.py

diff --git a/tests/test_sigma_points_small_alpha.py b/tests/test_sigma_points_small_alpha.py
new file mode 100644
index 0000000000..b63ad5fbaf
--- /dev/null
+++ b/tests/test_sigma_points_small_alpha.py
@@ -0,0 +1,30 @@
+import unittest
+
+import numpy as np
+import numpy.testing as npt
+from pyrecest.backend import __backend_name__, asarray, to_numpy
+from pyrecest.sampling import MerweScaledSigmaPoints
+
+
+@unittest.skipIf(
+    __backend_name__ == "pytorch",
+    reason="Sigma-point tests use NumPy assertions and the PyTorch backend is unsupported",
+)
+class TestMerweSmallAlpha(unittest.TestCase):
+    def test_small_positive_alpha_does_not_cancel_scale_to_zero(self):
+        points = MerweScaledSigmaPoints(n=1, alpha=1.0e-9, beta=2.0, kappa=0.0)
+
+        sigmas = points.sigma_points(asarray([0.0]), asarray([[1.0]]))
+
+        self.assertTrue(np.all(np.isfinite(to_numpy(points.Wm))))
+        self.assertTrue(np.all(np.isfinite(to_numpy(points.Wc))))
+        npt.assert_allclose(
+            to_numpy(sigmas),
+            np.array([[0.0], [1.0e-9], [-1.0e-9]]),
+            rtol=1.0e-12,
+            atol=0.0,
+        )
+
+
+if __name__ == "__main__":
+    unittest.main()