diff --git a/base/images/tests/cases/runtime/container-base/test_math/Dockerfile b/base/images/tests/cases/runtime/container-base/test_math/Dockerfile
new file mode 100644
index 00000000000..b5cdacfef9e
--- /dev/null
+++ b/base/images/tests/cases/runtime/container-base/test_math/Dockerfile
@@ -0,0 +1,4 @@
+ARG BASE_IMAGE
+FROM ${BASE_IMAGE}
+RUN dnf install -y python3 && dnf clean all
+COPY pi.py /opt/azl-tests/pi.py
diff --git a/base/images/tests/cases/runtime/container-base/test_math/pi.py b/base/images/tests/cases/runtime/container-base/test_math/pi.py
new file mode 100644
index 00000000000..977de27db41
--- /dev/null
+++ b/base/images/tests/cases/runtime/container-base/test_math/pi.py
@@ -0,0 +1,72 @@
+# SPDX-License-Identifier: MIT
+# Pi calculation adapted from https://github.com/MrBlaise/learnpython/blob/master/Numbers/pi.py
+"""Spigot-algorithm Pi calculation, run inside the container for compute validation."""
+
+from __future__ import annotations
+
+import time
+from collections.abc import Iterator
+
+PI_1000 = (
+    "3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412"
+    "737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051"
+    "320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473"
+    "035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989"
+)
+
+MAX_SECONDS_PER_COMPUTE = 20.0
+
+
+def calc_pi(limit: int) -> Iterator[object]:
+    """Calculate Pi digits one at a time via the spigot algorithm."""
+    q, r, t, k, n, step = 1, 0, 1, 1, 3, 3
+    counter = 0
+    while counter != limit + 1:
+        if 4 * q + r - t < n * t:
+            yield n
+            if counter == 0:
+                yield "."
+            if limit == counter:
+                break
+            counter += 1
+            nr = 10 * (r - n * t)
+            n = ((10 * (3 * q + r)) // t) - 10 * n
+            q *= 10
+            r = nr
+        else:
+            nr = (2 * q + r) * step
+            nn = (q * (7 * k) + 2 + (r * step)) // (t * step)
+            q *= k
+            t *= step
+            step += 2
+            k += 1
+            n = nn
+            r = nr
+
+
+def pi_to_places(places: int) -> str:
+    """Return Pi, accurate to the given number of decimal places."""
+    return "".join(str(d) for d in calc_pi(places))
+
+
+def verify_pi_1000() -> bool:
+    """Check that Pi to 1000 places matches the known reference value."""
+    return pi_to_places(1000) == PI_1000
+
+
+def verify_pi_n_times_1000(nrange: int = 10, mult: int = 1000) -> bool:
+    """Repeatedly compute Pi at growing precision and assert performance (max 20s per computation)."""
+    for count in range(nrange + 1):
+        places = max(count * mult, 3)
+        start = time.time()
+        answer = pi_to_places(places)
+        if len(answer) != places + 2 or time.time() - start > MAX_SECONDS_PER_COMPUTE:
+            return False
+    return True
+
+
+if __name__ == "__main__":
+    import sys
+
+    checks = {"1000": verify_pi_1000, "n1000": verify_pi_n_times_1000}
+    sys.exit(0 if checks[sys.argv[1]]() else 1)
diff --git a/base/images/tests/cases/runtime/container-base/test_math/test_math.py b/base/images/tests/cases/runtime/container-base/test_math/test_math.py
new file mode 100644
index 00000000000..44ff29c1574
--- /dev/null
+++ b/base/images/tests/cases/runtime/container-base/test_math/test_math.py
@@ -0,0 +1,22 @@
+# SPDX-License-Identifier: MIT
+"""Verify the container's python3 computes Pi correctly (Dockerfile adds python3)."""
+
+from __future__ import annotations
+
+import pytest
+
+_PI = "/opt/azl-tests/pi.py"
+
+
+@pytest.mark.dockerfile()
+def test_pi_1000_places(container_exec_shell) -> None:
+    """Pi computed to 1000 places must match the known value."""
+    result = container_exec_shell(f"python3 {_PI} 1000")
+    assert result.exit_code == 0, f"Pi(1000) verification failed: {result.output}"
+
+
+@pytest.mark.dockerfile()
+def test_pi_n_times_1000_places(container_exec_shell) -> None:
+    """Sustained Pi compute (10 x 1000 places) must stay correct and fast."""
+    result = container_exec_shell(f"python3 {_PI} n1000")
+    assert result.exit_code == 0, f"Pi series verification failed: {result.output}"
diff --git a/base/images/tests/cases/runtime/container-base/test_network/Dockerfile b/base/images/tests/cases/runtime/container-base/test_network/Dockerfile
new file mode 100644
index 00000000000..9b723bf2477
--- /dev/null
+++ b/base/images/tests/cases/runtime/container-base/test_network/Dockerfile
@@ -0,0 +1,4 @@
+ARG BASE_IMAGE
+FROM ${BASE_IMAGE}
+RUN dnf install -y python3 && dnf clean all
+COPY netcheck.py /opt/azl-tests/netcheck.py
diff --git a/base/images/tests/cases/runtime/container-base/test_network/netcheck.py b/base/images/tests/cases/runtime/container-base/test_network/netcheck.py
new file mode 100644
index 00000000000..851188a9041
--- /dev/null
+++ b/base/images/tests/cases/runtime/container-base/test_network/netcheck.py
@@ -0,0 +1,92 @@
+# SPDX-License-Identifier: MIT
+"""Outbound HTTPS fetch + parse helpers, run inside the container to validate networking."""
+
+from __future__ import annotations
+
+import re
+import time
+import urllib.request
+from urllib.error import HTTPError, URLError
+
+WEATHER_URLS = (
+    "https://iotadbselfhostreportstor.blob.core.windows.net/marinerextendedtests/mockweathergov.html",
+    "https://forecast.weather.gov/MapClick.php?lat=47.6786&lon=-122.1316",
+)
+REPO_CONFIG_URLS = (
+    "https://packages.microsoft.com/azurelinux/3.0/prod/base/x86_64/config.repo",
+    "https://packages.microsoft.com/cbl-mariner/2.0/prod/base/x86_64/config.repo",
+)
+
+
+def fetch(url: str, retries: int = 4) -> str:
+    """Fetch a URL, retrying transient HTTP errors with exponential backoff, and a 30s timeout to avoid hangs."""
+    for attempt in range(1, retries + 1):
+        try:
+            with urllib.request.urlopen(urllib.request.Request(url), timeout=30) as resp:  # noqa: S310 — https literals only
+                return resp.read().decode()
+        except HTTPError:
+            time.sleep(attempt * 2)
+        except URLError:
+            break
+    return ""
+
+
+def fetch_first(urls: tuple[str, ...]) -> str:
+    """Return the first non-empty page across the given URLs."""
+    for url in urls:
+        page = fetch(url)
+        if page:
+            return page
+    return ""
+
+
+def substring_between(source: str, before: str, after: str) -> str:
+    """Return the markup-stripped text between two markers, or empty string."""
+    beg = source.find(before)
+    if beg < 0:
+        return ""
+    sub = source[beg + len(before) :]
+    end = sub.find(after)
+    if end >= 0:
+        sub = sub[:end]
+    return re.sub(r"<.*?>", "", sub).strip()
+
+
+def verify_weather(*, strict: bool = True) -> bool:
+    """Fetch Redmond weather; strict requires all forecast fields present."""
+    page = fetch_first(WEATHER_URLS)
+    visibility = substring_between(page, "<b>Visibility</b></td>", "</td>")
+    dewpoint = substring_between(page, "<b>Dewpoint</b></td>", "</td>")
+    humidity = substring_between(page, "<b>Humidity</b></td>", "</td>")
+    wind = substring_between(page, "<b>Wind Speed</b></td>", "</td>")
+    forecast = substring_between(page, 'alt="Today:', '"')
+    print(
+        f"weather: bytes={len(page)} humidity={humidity!r} wind={wind!r} "
+        f"visibility={visibility!r} dewpoint={dewpoint!r} forecast={forecast!r}"
+    )
+    if not (wind and humidity and visibility and dewpoint) and strict:
+        return False
+    return len(page) > 0
+
+
+def verify_sustained_https(iterations: int = 50, *, strict: bool = True) -> bool:
+    """Repeat repo-config fetch; strict requires name+enabled fields each time."""
+    for i in range(iterations):
+        page = fetch_first(REPO_CONFIG_URLS)
+        name = substring_between(page, "name", "\n")
+        enabled = substring_between(page, "enabled", "\n")
+        print(f"fetch {i + 1}/{iterations}: bytes={len(page)} name={name!r} enabled={enabled!r}")
+        if not (name and enabled) and strict:
+            return False
+        if len(page) == 0:
+            return False
+        time.sleep(1)
+    return True
+
+
+if __name__ == "__main__":
+    import sys
+
+    checks = {"weather": verify_weather, "sustained": verify_sustained_https}
+    strict = "--strict" in sys.argv
+    sys.exit(0 if checks[sys.argv[1]](strict=strict) else 1)
diff --git a/base/images/tests/cases/runtime/container-base/test_network/test_network.py b/base/images/tests/cases/runtime/container-base/test_network/test_network.py
new file mode 100644
index 00000000000..0fe1aa7fdf2
--- /dev/null
+++ b/base/images/tests/cases/runtime/container-base/test_network/test_network.py
@@ -0,0 +1,22 @@
+# SPDX-License-Identifier: MIT
+"""Verify outbound networking from the container (Dockerfile adds python3)."""
+
+from __future__ import annotations
+
+import pytest
+
+_NET = "/opt/azl-tests/netcheck.py"
+
+
+@pytest.mark.dockerfile()
+def test_online_service_weather(container_exec_shell) -> None:
+    """Test Online Services: a single outbound HTTPS fetch must return a non-empty page."""
+    result = container_exec_shell(f"python3 {_NET} weather")
+    assert result.exit_code == 0, f"Outbound HTTPS fetch failed: {result.output}"
+
+
+@pytest.mark.dockerfile()
+def test_sustained_https_fetch(container_exec_shell) -> None:
+    """Test Core Networking: 50 sequential outbound HTTPS fetches must all return valid repo config."""
+    result = container_exec_shell(f"python3 {_NET} sustained --strict")
+    assert result.exit_code == 0, f"Sustained HTTPS fetch failed: {result.output}"