diff --git a/scripts/jax_assertions/matern_regularization.py b/scripts/jax_assertions/matern_regularization.py
new file mode 100644
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+++ b/scripts/jax_assertions/matern_regularization.py
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+"""
+Jax Assertions: Matern-Kernel Regularization (tfp bessel_kve path)
+==================================================================
+
+Verifies that ``MaternKernel`` regularization builds end to end on the JAX
+backend and matches the eager NumPy reference. This is the regression guard for
+the 2026-07-13 release-validation failure (PyAutoArray #385): the JAX Matern
+path lazily imports ``tensorflow_probability.substrates.jax`` for the
+modified-Bessel ``bessel_kve`` term, and the last *stable* tfp release (0.25.0)
+crashes at import under the resolved ``jax>=0.7`` stack
+(``jax.interpreters.xla.pytype_aval_mappings`` was removed from modern JAX).
+The fix pins the tfp nightly; this script fails loudly if that combination ever
+regresses.
+
+It exercises:
+
+* ``matern_cov_matrix_from`` + ``inv_via_cholesky`` (the regularization-matrix
+  math) with ``xp=np`` vs ``xp=jnp``, agreeing to ``atol=1e-8``.
+* the same under ``jax.jit`` (confirms ``kv_xp`` traces, not just runs eagerly).
+* ``MaternKernel.regularization_matrix_from`` — the real entry point used by
+  pixelizations — via a minimal mesh-grid stub.
+* two ``nu`` values: a half-integer (0.5) and a non-half-integer (1.7). ``nu``
+  is a continuous free prior (``Uniform(0.5, 5.5)``), so the Bessel term must be
+  correct for arbitrary real order, not just closed-form half-integers.
+
+Library unit tests stay numpy-only by project convention; this cross-xp
+coverage lives here in workspace_test.
+"""
+
+import jax
+import jax.numpy as jnp
+import numpy as np
+import numpy.testing as npt
+
+import autoarray as aa
+from autoarray.inversion.regularization.matern_kernel import (
+    matern_cov_matrix_from,
+    inv_via_cholesky,
+)
+
+
+"""
+__Setup: a small irregular set of source-plane pixel coordinates [N, 2]__
+"""
+rng = np.random.default_rng(0)
+pixel_points = rng.standard_normal((12, 2)).astype(np.float64)
+
+scale = 0.7
+
+
+"""
+__matern_cov_matrix_from + inv_via_cholesky: fp64 JAX matches NumPy to atol=1e-8__
+
+Runs both nu = 0.5 (half-integer) and nu = 1.7 (arbitrary real order) through
+the tfp bessel_kve path.
+"""
+for nu in (0.5, 1.7):
+    cov_np = matern_cov_matrix_from(
+        scale=scale, nu=nu, pixel_points=pixel_points, xp=np
+    )
+    reg_np = inv_via_cholesky(cov_np, xp=np)
+
+    cov_jnp = matern_cov_matrix_from(
+        scale=scale, nu=nu, pixel_points=jnp.asarray(pixel_points), xp=jnp
+    )
+    reg_jnp = inv_via_cholesky(cov_jnp, xp=jnp)
+
+    npt.assert_allclose(np.asarray(cov_np), np.asarray(cov_jnp), atol=1.0e-8)
+    npt.assert_allclose(np.asarray(reg_np), np.asarray(reg_jnp), atol=1.0e-8)
+
+
+"""
+__Under jax.jit: kv_xp traces (not just runs eagerly)__
+
+A tracer flowing through ``kv_xp`` is what would surface a broken tfp import at
+compile time; assert the jitted result still matches NumPy.
+"""
+
+
+def _reg_matrix_jax(points, scale, nu):
+    cov = matern_cov_matrix_from(scale=scale, nu=nu, pixel_points=points, xp=jnp)
+    return inv_via_cholesky(cov, xp=jnp)
+
+
+reg_jit = jax.jit(_reg_matrix_jax, static_argnums=(1, 2))(
+    jnp.asarray(pixel_points), scale, 1.7
+)
+
+reg_np_ref = inv_via_cholesky(
+    matern_cov_matrix_from(scale=scale, nu=1.7, pixel_points=pixel_points, xp=np),
+    xp=np,
+)
+npt.assert_allclose(np.asarray(reg_jit), np.asarray(reg_np_ref), atol=1.0e-8)
+
+
+"""
+__MaternKernel.regularization_matrix_from: the real pixelization entry point__
+
+The class reads ``linear_obj.source_plane_mesh_grid.array`` (an [N, 2] coord
+array) and ``linear_obj.params``; a minimal stub supplies just those.
+"""
+
+
+class _MeshGridStub:
+    def __init__(self, array):
+        self.array = array
+
+
+class _LinearObjStub:
+    def __init__(self, points):
+        self.source_plane_mesh_grid = _MeshGridStub(points)
+        self.params = points.shape[0]
+
+
+regularization = aa.reg.MaternKernel(coefficient=1.0, scale=scale, nu=1.7)
+
+reg_matrix_np = regularization.regularization_matrix_from(
+    linear_obj=_LinearObjStub(pixel_points), xp=np
+)
+reg_matrix_jnp = regularization.regularization_matrix_from(
+    linear_obj=_LinearObjStub(jnp.asarray(pixel_points)), xp=jnp
+)
+
+npt.assert_allclose(
+    np.asarray(reg_matrix_np), np.asarray(reg_matrix_jnp), atol=1.0e-8
+)
+
+
+print("matern_regularization: all assertions passed")