diff --git a/.github/workflows/test_minimum_versions.yml b/.github/workflows/test_minimum_versions.yml
new file mode 100644
index 0000000..1e775fe
--- /dev/null
+++ b/.github/workflows/test_minimum_versions.yml
@@ -0,0 +1,49 @@
+name: Minimum version tests
+
+on:
+  push:
+    branches:
+      - main
+      - 'stable/**'
+  pull_request:
+    branches:
+      - main
+      - 'stable/**'
+
+jobs:
+  tests:
+    name: minimum version tests (${{ matrix.os }}, ${{ matrix.python-version }})
+    runs-on: ${{ matrix.os }}
+    timeout-minutes: 30
+    strategy:
+      fail-fast: false
+      max-parallel: 4
+      matrix:
+        os: [ubuntu-latest]
+        python-version: ["3.10"]
+    steps:
+      - uses: actions/checkout@v7
+        with:
+          # The reference energies the tests check against come from data in the
+          # vendored upstream checkout, and setup.py needs its headers to compile.
+          submodules: recursive
+      - name: Set up Python ${{ matrix.python-version }}
+        uses: actions/setup-python@v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Install MPI and BLAS (Linux)
+        if: runner.os == 'Linux'
+        run: |
+          sudo apt-get update
+          sudo apt-get install -y libopenmpi-dev openmpi-bin libopenblas-dev
+      - name: Install dependencies (minimum versions)
+        shell: bash
+        run: |
+          python -m pip install --upgrade pip
+          python -m pip install 'extremal-python-dependencies<2'
+          pip install "tox==$(extremal-python-dependencies get-tox-minversion)"
+          extremal-python-dependencies pin-dependencies-to-minimum --inplace
+      - name: Test using tox environments
+        shell: bash
+        run: |
+          tox -e py,notebook,mpi
diff --git a/pyproject.toml b/pyproject.toml
index a235da2..30b4ed8 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -1,5 +1,5 @@
 [build-system]
-requires = ["setuptools>=77", "wheel", "pybind11>=2.6.0", "mpi4py>=3.0.0"]
+requires = ["setuptools>=77.0.1", "wheel", "pybind11>=2.12.0", "mpi4py>=4.0.0"]
 build-backend = "setuptools.build_meta"
 
 [project]
@@ -27,8 +27,8 @@ classifiers = [
     "Programming Language :: C++",
 ]
 dependencies = [
-    "mpi4py>=3.0.0",
-    "numpy>=1.19.0",
+    "mpi4py>=4.0.0",
+    "numpy>=1.26.0",
 ]
 
 [project.optional-dependencies]
@@ -83,4 +83,4 @@ testpaths = ["test"]
 # skips by default: a bare count of skips would not distinguish "this case is slow" or
 # "this backend was not built" from "the reference data is missing". --durations shows
 # where the time goes, the cases differing by orders of magnitude in cost.
-addopts = "-rs --durations=10"
\ No newline at end of file
+addopts = "-rs --durations=10"
diff --git a/releasenotes/notes/minimum-supported-dependencies-b1e7c5649c3201d1.yaml b/releasenotes/notes/minimum-supported-dependencies-b1e7c5649c3201d1.yaml
new file mode 100644
index 0000000..e6c7250
--- /dev/null
+++ b/releasenotes/notes/minimum-supported-dependencies-b1e7c5649c3201d1.yaml
@@ -0,0 +1,8 @@
+---
+upgrade:
+  - |
+    The declared minimum versions of several dependencies have been raised to
+    the oldest ones this package actually builds and runs against:
+    ``numpy>=1.26.0`` (was 1.19.0), ``mpi4py>=4.0.0`` (was 3.0.0),
+    ``pybind11>=2.12.0`` (was 2.6.0), and ``setuptools>=77.0.1`` (was 77).
+    The previous floors were aspirational rather than tested.